Superfluidity and phase transitions in a resonant Bose gas
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Superfluidity and phase transitions in a resonant Bose gas
Leo Radzihovsky , Peter B. Weichman , and Jae I. Park , Department of Physics, University of Colorado, Boulder, CO 80309 BAE Systems, Advanced Information Technologies,6 New England Executive Park, Burlington, MA 01803 and National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305-3328 (Dated: October 22, 2018)The atomic Bose gas is studied across a Feshbach resonance, mapping out its phase diagram,and computing its thermodynamics and excitation spectra. It is shown that such a degenerate gasadmits two distinct atomic and molecular superfluid phases, with the latter distinguished by theabsence of atomic off-diagonal long-range order, gapped atomic excitations, and deconfined atomic π -vortices. The properties of the molecular superfluid are explored, and it is shown that across aFeshbach resonance it undergoes a quantum Ising transition to the atomic superfluid, where bothatoms and molecules are condensed. In addition to its distinct thermodynamic signatures anddeconfined half-vortices, in a trap a molecular superfluid should be identifiable by the absence of anatomic condensate peak and the presence of a molecular one. PACS numbers: Valid PACS appear here
I. INTRODUCTIONA. Background
Remarkable experimental advances in manipulatingdegenerate atomic gases have opened a new era in stud-ies of highly coherent, interacting quantum many-bodysystems. One of the most striking advances is theability to finely control atomic two-body interactionsby tuning with a magnetic field the energy (detuning)of the molecular Feshbach resonance (FR) through theatomic continuum.
This technique has led to a realiza-tion of a long-sought-after s-wave paired superfluidity inbosonic and fermionic atomic gases.
For fermionicatoms, it also allowed the system to be tuned betweenthe BCS regime of weakly-paired, strongly overlappingCooper pairs (familiar from solid-state superconductors),and the BEC regime of tightly bound, weakly-interactingBose-condensed diatomic molecules.Although this crossover has received considerableattention, because of the absence of qual-itative differences between the BCS and BEC s-wavepaired fermionic superfluids, their equilibrium proper-ties are already qualitatively well described by early sem-inal works. In fact for a narrow FR (unfortunatelynot realized by most current experimental systems), thecrossover can even be computed quantitatively, as a per-turbation series in the ratio of the FR width to the Fermienergy.
In such narrow FR systems the crossoverto BEC takes place when the FR detuning ν (quasi-molecule’s rest energy) ranges from twice the Fermi en-ergy 2 ǫ F (when it first becomes favorable to converta finite fraction of the Fermi-sea into molecules stabi-lized by Pauli-blocking) down to zero energy, where allthe fermions have become bound into Bose-condenseddiatomic molecules. The complementary broad reso-nance regime of most experiments, particularly neara universal unitary point has been successfully stud- ied using quantum Monte Carlo and field theo-retic ǫ -expansion and 1 /N -expansion methodsborrowed from critical phenomena.As was recently pointed out and is the subjectof this paper, the phenomenology of resonantly inter-acting degenerate bosonic atoms contrasts strongly andqualitatively with this picture. For a large positive de-tuning, molecules are strongly energetically suppressedand unpaired atoms (as in any bosonic system at zerotemperature) form an atomic superfluid (ASF), exhibit-ing atomic off-diagonal long-range order (ODLRO). Inthe opposite extreme of a large negative detuning, freeatoms are strongly disfavored (gapped), pairing up intostable bosonic molecules, that then, at T = 0, form adiatomic molecular superfluid characterized by a molec-ular ODLRO. The MSF does not exhibit atomic ODLRO,nor the associated atomic superfluidity. Together with agapped atomic excitation spectrum and correlation func-tions (characteristics that extend to finite temperature),these features qualitatively distinguish it from the ASF.In a trapped, dilute atomic gas the existence of thesetwo qualitatively distinct superfluid phases should bemost directly detectable through independent images ofatomic and molecular density profiles. As illustrated inFig. 1(a), the atomic component should exhibit a BECpeak in the ASF phase, that is absent in the MSF phase,shown in Fig. 1(b). Both superfluid phases are distin-guished from the normal state by the BEC peak in the molecular density profile, as illustrated in the insets tothese figures.Because of its paired nature, a complementary dis-tinguishing characteristic of a MSF are deconfined π − (half-) vortices, topological defects that, in contrast, arelinearly confined in the ASF state. Consequently, asillustrated in Fig. 2, a thermodynamically sharp quan-tum phase transition, at an intermediate critical Fesh-bach resonance detuning ν c , must separate the MSF andASF phases. Each in turn is also separated by a finite- n (r)r r T0 r n (r) (a) ASF n (r) r T rn (r) (a) MSF FIG. 1: Atomic density profiles, n ( r ) in (a) the ASF and(b) the MSF phases. These are distinguished by the presenceand absence of atomic BEC peak, respectively. Each of thesesuperfluid phases is distinguished from the “normal” (ther-mal) state by the BEC peak in the molecular density profile, n ( r ), illustrated in insets. In the dilute limit, the width r σ ( σ = 1 ,
2) of the BEC peak (set by the single-particle Gaus-sian ground state wavefunction), and the extent r Tσ of thethermal part of the atomic cloud, are given by Eqs. (5.33)and (5.34), respectively. temperature transition from the “normal” (N) state lack-ing any order (i.e., breaking no symmetries).Experimental observations of these and associated pre-dictions have so far been precluded by a short lifetimeof the vibrationally hot molecular state. The latteris believed to be limited by 3-body recombination andstrongly enhanced atom-molecule scattering near the res-onance. In contrast to Fermi systems, where Pauliexclusion greatly extends the molecular lifetime for apositive scattering length and stabilizes the Fermi-seafor negative scattering lengths by suppressing multi-bodycollisions, the resonantly interacting bosonic atomic gasis observed to be highly unstable in the negative two-body scattering length regime. Viable proposals forsurmounting these problems are currently being investi-gated. These include use of an adiabatic ramp of thedetuning through resonance, or a two-photon Ramantransition to transfer the Feshbach molecular states to a ν ASF T ν (n,0) c ν (n,T) c NormalMSF T c0 T c1 T c2 FIG. 2: The phase diagram (at fixed total density, n ) for a uni-form condensate as a function of Feshbach resonance detuning ν and temperature T . A curve of critical detuning ν c ( n, T )separates the atomic (ASF) and molecular (MSF) superfluidphases by a phase transition, which is continuous between the( T = 0) quantum critical point ν c ( n,
0) and a tricritical point
T C . The section of the critical curve (gray) between the twotricritical points T C and T C denotes a first order transi-tion boundary, that terminates the continuous MSF–N phaseboundary at a critical end point T c , where three phases meet.The critical temperatures T ∞ cσ , σ = 1 ,
2, correspond to the fardetuned limits, ν/k B T → ±∞ . The dashed curve inside theASF phase corresponds to a crossover line, ν × ( T ), at whichthe molecules would condense on their own if there were noFeshbach resonance coupling them to the atoms. lower lying vibrational state. Although no direct evidence for an equilibrium Bosemolecular condensate exists, observed resonant atomicloss in a stimulated Raman transition in Rb (Ref. 37)and time domain density oscillations in Rb (Ref. 4) areconsistent with a coherent transfer of population fromfree bosonic atoms to diatomic molecules.
It is notat the moment clear (at least to the present authors)whether current experimental difficulties of stabilizingbosonic atom-molecule mixtures near a FR are funda-mental or technical and system specific. One possiblefundamental source of instability in bosonic atom sys-tems is the existence of Efimov bound states of bosonicatom triplets.
At least at a theoretical model levelthese can be suppressed by a sufficiently strong three-body repulsion. Even in the unfavorable scenario, wheresuch phases of bosons near a FR are indeed metastable(as most states of degenerate atoms ultimately are) oneexpects that ideas discussed here should be important onsufficiently short time scales and for understanding of theassociated nonequilibrium dynamics.The rest of the paper is organized as follows. TheIntroduction is concluded with a summary of the mainresults and their experimental implications. In SectionII a microscopic two-channel model, that is believe toaccurately describe resonantly-interacting atomic bosegas is introduced. The model is first used to computethe two-body s-wave scattering, showing that it cor-rectly captures the Feshbach resonance phenomenology.Matching the computed scattering amplitude to its mea-sured counterpart allows one to relate parameters ap-pearing in the Hamiltonian to experimental observables.In Section III a general symmetry-based discussion ofthe expected phases and associated phase transitions inthis system is presented. In Section IV, by minimizingthe corresponding imaginary-time coherent state action,the generic mean field phase diagram for the system ismapped out. In Sections V and VI, this Landau analy-sis is supplemented by detailed microscopic calculationsof phase boundaries, spectra, condensate depletion andsuperfluid density for a dilute, weakly-interacting gas.The asymptotic nature of the ASF–MSF phase transi-tion is discussed in Section VII. In Section VIII themean field and perturbative analyses, performed within atwo-channel model, are supplemented with a variationaltheory of a one-channel model. The latter is a betterdescription of bosons in which the paired state is absent(i.e., there is no long lived metastable paired state withdistinct internal quantum numbers) once the two-bodyattraction becomes too weak to bind atom pairs (whichincludes, of course, the more familiar regime of two-bodyrepulsion). In Section IX topological defects, vortices anddomain walls, in the ASF are studied, and the ASF–MSFand SF-to-normal fluid transitions are characterized interms of a proliferation of these topological defects. Thepaper is concluded in Section X.
B. Summary of results
In this paper a considerable elaboration and extensionof predictions reported in a recent Letter are presented.The primary results are summarized by the density pro-files in Fig. 1 and the phase diagrams in Figs. 2 and10, characterizing the phases and phase transitions ofa resonant Bose gas. As illustrated there, it is foundthat Feshbach-resonantly interacting atomic Bose gas, inaddition to the normal state exhibits two distinct low-temperature superfluid states. The first, appearing atpositive detuning, is the more conventional atomic su-perfluid, characterized by coexisting atomic and molec-ular BEC and their associated ODLROs, with finite or-der parameters Ψ and Ψ , respectively. The other,more exotic, MSF state, appearing at low temperatureand negative detuning, is characterized by superfluidityof diatomic molecules, with a finite molecular condensateorder parameter Ψ . It is distinguished from the ASFby the absence of atomic ODLRO, i.e., inside the MSFphase Ψ = 0.As illustrated in detail in Sec. IV B, a finite Ψ alwaysimplies a finite Ψ . In the presence of an atomic con-densate, Ψ , the Feshbach resonance coupling allows ascattering of two Bose-condensed atoms out of the atomic BEC into the molecular BEC (i.e., ASF is really a su-perposition of Bose-condensed open-channel atoms andBose-condensed closed-channel molecules) and thereforeacts like an ordering “field” on the molecular order pa-rameter. This implies that a state in which atoms arecondensed but molecules are not is forbidden by generalsymmetry principles. As noted above, a vivid signature of two distinct su-perfluid orders should be detectable via time-of-flightshadow images. In the dilute regime (described by aBEC approximation), the resulting images are schemat-ically illustrated in Fig. 1. At higher densities, where alocal density approximation is more appropriate, it is ex-pected that for a range of atom number and detuning,phase boundaries as a function of chemical potential inthe bulk system (see e.g., Fig. 10) will translate into shellstructure which should also be observable experi-mentally in time-of-flight shadow images.As for any neutral superfluid, ASF and MSF are eachcharacterized by an acoustic (Bogoliubov) “sound” mode,illustrated in Fig. 3, corresponding to long wavelengthcondensate phase fluctuations, with long wavelength dis-persions E + σ ( k ) ≈ c σ ~ k, (1.1)where c σ (with σ = ASF or MSF, or equivalently 1 or2) are the associated sound speeds with c MSF given by(6.29) and c ASF given by (6.46) in terms of the interactionparameters of the model (see Sec. VI B).In the ASF state the gapless mode corresponds to in-phase fluctuations of the atomic and molecular conden-sates. However, as illustrated in Fig. 3, in contrast toordinary superfluids, ASF and MSF also each exhibit agapped branch of excitations, E − σ ( k ) ≈ E gap σ + b σ k , (1.2)with gaps E gap σ given explicitly by (6.31) and (6.40),while the quadratic corrections b σ may be inferred fromthe general forms (6.8) and (6.21) of the spectra. In theASF the gap is controlled by out-of-phase fluctuations ofthe atomic and molecular condensates, and is set by theFeshbach coupling α .In the MSF, gapped excitations are single atom-like quasiparticles akin to Bogoliubov excitations in thepaired BCS state, that however do not carry a defi-nite atom number. These single-particle excitations are“squeezed” by the presence of the molecular condensate,offering a mechanism to realize atomic squeezed states. We expect that these gapped, atomic quantum fluctua-tions associated with the presence of the molecular con-densate can be measured by interference experiments,similar to those reported in Ref. 47. As detailed below,the low-energy nature of these excitations is guaranteedby the vanishing of the gap at the MSF–ASF transition, ν c , with E gapMSF ( ν c ) = 0.In the dilute weakly interacting limit appropriate toatomic gases, the ASF–N and MSF–N transition tem- E k E (−) (−) EEE (gap) E (gap) MSFASF ASF ASFMSF MSF (+)(+) a c o u s t i c gapped FIG. 3: Schematic low-energy excitation spectra character-izing ASF and MSF phases. In the ASF the acoustic andgapped branches of excitations correspond to in-phase andout-of-phase fluctuations of atomic and molecular conden-sates, respectively. In the MSF state the acoustic branch isthe standard gapless Bogoliubov mode. The gapped branchcorresponds to atom-like quasiparticle excitations that aresqueezed by the presence of the molecular condensate. Atthe critical detuning ν c , the gap closes, signaling a quantumMSF–ASF phase transition. peratures, T c ( ν ) and T c ( ν ), respectively, for a three-dimensional (3d) bulk uniform system are well approxi-mated by T cσ ( ν ) ≈ T c (cid:20) a σ (cid:16) | ν | k B T c (cid:17) (cid:21) , | ν | ≪ k B T c T ∞ cσ = b σ T c , | ν | ≫ k B T c , (1.3)with a = 2 / π / / cζ (3 / a = a / b = c / , b =2 − / c / , and c = 1+2 / . One sees that T c > T c , withthe asymptotic ratio b /b = 2 / set by the mass andboson number that both differ by a factor of two betweenthe two phases. When interactions are included, for ν = 0the asymptotic nature of these thermal transitions is inthe well-studied classical 3d XY universality class.As illustrated in Fig. 2 (see also Figs. 10 and 11), in thevicinity of the critical endpoint ν = 0, T = T c , where the three phases, N, MSF, ASF meet, a coupling of themolecular and atomic superfluid order parameters con-verts a section (between the tricritical points TC andTC ) of the (otherwise) continuous N–ASF and MSF–ASF transitions to first order. The resulting crossingpoint at T c , that terminates a continuous N–MSF tran-sition is a critical endpoint (CEP). In the dilute limit,the CEP temperature is given by T c ≈ h πm k B (cid:20) ncζ (3 / (cid:21) / , (1.4)As illustrated in Secs. IV and VII, the appearance of afirst order transition (even in mean field theory) near N–MSF continuous boundary, is a generic feature resultingfrom the coupling of the ASF order parameter to thecritical MSF order parameter. The corresponding transition temperatures in a trap (here distinguished from bulk quantities by a tilde) arealso easily computed and in 3d are given by˜ T cσ ( ν ) ≈ ˜ T c h a σ | ν | k B ˜ T c i , | ν | ≪ k B ˜ T c ˜ T ∞ cσ h − b σ e −| ν | /σ ˜ T ∞ cσ i , | ν | ≫ k B ˜ T c , (1.5)with a σ = 2 ζ (2) / σ ζ (3), and b σ = 2 / σ ζ (3). Thetransition temperatures in the limit of asymptoticallylarge positive ( σ = 1) and negative ( σ = 2) detuning( | ν | /k B T cσ ≫ ν = 0), aregiven, respectively, by˜ T ∞ cσ = ~ ω (cid:20) Nσζ (3) (cid:21) / , (1.6)˜ T c = ~ ω (cid:20) N ζ (3) (cid:21) / , (1.7)where ω is the trap frequency. Comparing the first linesof (1.3) and (1.5), note that the latter is now approached linearly with, rather than as the square-root of the re-duced detuning from either side.In the dilute limit the thermodynamics is also easilyworked out. In the 3d bulk system, the condensate den-sities for the atomic and molecular BEC are given, re-spectively, by n ( T, ν ) = n " − (cid:18) TT c (cid:19) / ζ (3 /
2) + 2 / g / ( e − ν/k B T ) ζ (3 /
2) + 2 / g / ( e − ν/k B T c ) , ν > , T < T c ( ν ) , (1.8) n ( T, ν ) = 12 n " − (cid:18) TT c (cid:19) / / ζ (3 /
2) + g / ( e ν/ k B T )2 / ζ (3 /
2) + g / ( e ν/ k B T c ) , ν < , T < T c ( ν ) , (1.9)where ζ (3 / ≃ .
612 and g α ( x ) = P ∞ n =1 x n /n α is the extended zeta function. As illustrated in Fig. 2 the MSF–ASF transition takesplace at a critical value of detuning ν c ( T, n ) determinedby the strength of atomic and molecular interactions,shifting it away from its noninteracting value of 0. Atzero temperature this is a continuous quantum phasetransition that for a d -dimensional system is in the ( d +1)-dimensional classical Ising universality class with ν c (0 , n ) ≈ − ( g / − g ) n − α √ n, (1.10)where g , g , and g are, respectively, the atom-atom, atom-molecule and molecule-molecule interactionstrengths, related in the standard way to the correspond-ing scattering lengths, and α is the Feshbach resonancecoupling. The transition at ν c is characterized, upon ap-proach from the MSF side, by the vanishing of the single-atom excitation gap E gapMSF ( ν ), and, upon approach fromthe ASF side, by the disappearance of the atomic conden-sate n ( ν ). At zero temperature, in the critical regionthese are predicted to vanish according to n (0 , ν ) ∼ | ν − ν c | β I , E gapMSF (0 , ν ) ∼ | ν − ν c | ν I , (1.11)where β I and ν I are, respectively, the order parame-ter and correlation length exponents for the ( d + 1)-dimensional Ising model. One may hope that when long-lived molecular condensates are produced, nontrivial be-havior of E gapMSF ( ν ) and the full excitation spectra, E ± σ ( k )may be observed in Ramsey fringes and in Bragg andRF spectroscopy experiments .At finite temperature, away from the critical endpoint T c the transition is in the classical d -dimensional Isinguniversality class. Scaling, together with the relevance(in the renormalization group sense) of T at the quantumcritical point, also implies a universal shape of the low T part of the MSF–ASF phase boundary ν c ( n, T ) ∼ ν c ( n,
0) + a T /ν I , (1.12)illustrated in Fig. 2.The dashed curve, ν × ( T ) inside the ASF phase of thephase diagram (Figs. 2 and 10) denotes a crossover (thatbecomes sharp with a vanishing Feshbach resonance cou-pling α ) between ASF regimes with low and high valuesof the molecular condensate n . In the absence of thecoupling, the molecules would condense on their own for ν < ν c ( T ). For small α the weak symmetry breaking fieldgenerated by the atomic condensate smears this transi-tion into a ASF-AMSF crossover, and leads to small, butfinite, n even for ν > ν c ( T ).As for any superfluid, the ASF and MSF phases alsoexhibit interaction-driven condensate depletion δn σ ≡ n − n − n , quantifying the fact that, even at T = 0,not all atoms are in the condensate. At T = 0 theseare computed in Sec. VI D. An interesting feature, illus-trated in Fig. 4, is that δn σ ( ν ) exhibit a cusp maximumat ν c , δn σ ( ν ) = δn ( ν c ) − c σ | ν − ν c | p , (1.13) ν MSF ASF δ n δ n < δ n > ν c FIG. 4: Schematic of the zero-temperature depletion δn (0 , ν )as a function of detuning. The far detuned limits are given by δn >,< = σ (8 / √ a σ n σ ) / , where >, < correspond to σ =1 , | ν − ν c | p , with p = 1 in d = 3. associated with enhanced role of quantum fluctuationsat the MSF–ASF transition. The maximum depletion isgiven by δn ( ν c ) ≈ √ π ( n a ) / + 13 π (cid:18) m α √ n ~ (cid:19) / (1.14)where a is the molecule-molecule s-wave scatteringlength. Outside the critical region one expects p = 1,crossing over to p = 1 − α I inside it, where α I is the( d + 1)-dimensional Ising specific heat exponent.Another important qualitative distinction between theASF and MSF phases is the nature of their topologi-cal excitations, namely vortices. The paired nature ofthe MSF allows for π − (half-) vortices (as in a BCS su-perconductor), while the ASF, being (from a symmetrypoint of view) a standard “charge-one” superfluid, admitsonly standard 2 π -vortices. However, pairing correlationspresent in the ASF lead to an interesting π -vortex exper-imental signature even in the atomic superfluid. Thus,in the ASF phase a seemingly standard 2 π -vortex in anatomic condensate, Ψ >
0, will generically split intotwo π -vortices (see Fig. 5) confined by a domain wall oflength R ≈ π ~ q mαn / r n n , (1.15)that diverges as the ASF–MSF phase boundary is ap-proached from the ASF side. Sufficiently close to thetransition, it is expected to track the associated correla-tion length.This confinement arises because in the large Feshbachcoupling limit a 2 π -vortex in the atomic condensate in-duces a 4 π -vortex in the molecular condensate. Such adouble molecular vortex is unstable to two fundamental π π R FIG. 5: 2 π atomic condensate vortex in the ASF splits into a π + π vortex pair connected by a “normal” domain wall, whoselength R increases as the FR coupling α becomes weaker. π molecular vortices that, in 2d, repel logarithmically,but are confined linearly inside the ASF phase. Thisprovides a complementary formulation of the ASF–MSFtransition as a confinement-deconfinement transition of π (half) atomic vortices. II. TWO-CHANNEL FESHBACH RESONANTMODEL
The goal here is to model a resonantly interact-ing atomic Bose gas of the type realized in recent experiments . A resonant interaction is the key fea-ture special to a select class of atomic (fermionic andbosonic ) systems. A fully microscopic description ofsuch resonant interactions is quite complex, involving afull set of internal nuclear and electronic spin degrees offreedom characterized by hyperfine states, mixed uponscattering by the interatomic exchange interaction. How-ever, in the vicinity of a resonance, two-atom scatter-ing in the “open” channel is dominated by hybridizationwith a two-atom molecular bound state in the “closed”channel, thereby allowing one to neglect all other off-resonant channels. As illustrated in Fig. 6, the two chan-nels are distinguished by the two-atom electron spins,with the open-channel an approximate spin-triplet andclosed-channel an approximate singlet. Consequentlythey have different Zeeman energies, allowing the cen-ter of mass rest energy ν of the closed-channel molecule(bound state) to be tuned, relative to the open-channeltwo-atom continuum, via an external magnetic field.This yields an unprecedented tunability of the effectiveatomic interaction strength by varying a magnetic field.The two-channel model describing the resonant atom-molecule system is characterized by the following grand-canonical Hamiltonian:ˆ H = Z d r ( X σ =1 (cid:20) ˆ ψ † σ ( r )ˆ h σ ˆ ψ σ ( r ) + 12 g σ ˆ ψ † σ ( r ) ˆ ψ σ ( r ) (cid:21) + g ˆ ψ † ( r ) ˆ ψ † ( r ) ˆ ψ ( r ) ˆ ψ ( r ) − α h ˆ ψ † ( r ) ˆ ψ † ( r ) ˆ ψ ( r ) + h . c . i) (2.1)where ˆ ψ † σ ( r ) , ˆ ψ σ ( r ) are bosonic creation and annihilationfield operators for atoms ( σ = 1) and molecules ( σ = 2).They are described by respective single-particle Hamil-tonians ˆ h σ = − ~ m σ ∇ + µ σ + V σ ( r ) , (2.2)with atomic and molecular masses m = m and m = 2 m and effective chemical potentials µ = µ and µ = 2 µ − ν .The (bare) detuning parameter ν is related to the energyof a (closed-channel) molecule at rest, that can be exper-imentally controlled with an external magnetic field. Inthe ensemble of fixed total number of atoms N (free andbound into molecules), relevant to trapped atomic gasexperiments, the chemical potential µ is determined bythe total atom number equation N = Z d r h h ˆ ψ † ( r ) ˆ ψ ( r ) i + 2 h ˆ ψ † ( r ) ˆ ψ ( r ) i i . (2.3)The positive local pseudo-potential parameters g , g , g measure background (nonresonant) repulsive atom-atom, atom-molecule and molecule-molecule interactions, re-spectively, and in the dilute limit are proportional to cor-responding background 2-body s-wave scattering lengths.The Feshbach resonance coupling α characterizes the co-herent atom-molecule interconversion rate (hyperfine in-teraction driven hybridization between open and closedchannels), encoding the fact that a molecule can de-cay into two open-channel atoms . The external po-tentials V σ ( r ) describe the atomic and molecular traps,which for most of the paper will be taken to be a “box”,modeled (for convenience) using periodic boundary con-ditions.In principle it is possible to obtain the above Hamilto-nian ˆ H from a microscopic analysis of atoms interactingvia a Feshbach resonance. However, its validity is ul-timately justified by the fact that for two atoms in avacuum (for which one takes µ = 0) it reproduces the ex-perimentally observed Feshbach-resonance phenomenol-ogy. Namely, it predicts an atomic scattering resonancefor positive detuning, a true molecular bound state for rU(r) Zeemansplittingclosed channelopen channel bg 0 B B a a FIG. 6:
Above:
Schematic illustration of a Feshbach reso-nance, modeled by two coupled channel interaction potentials(distinguished by two-atom electronic spin states) as a func-tion of inter-particle separation. The (so-called) “open” chan-nel is too shallow to support a bound state, while the other,“closed” channel supports a bound state or a resonance (in-dicated by a dashed line) that is tuned with a magnetic fieldvia the Zeeman splitting between the two channels.
Below:
At resonance ν = 0, when a bound state first appears, thes-wave atomic scattering length diverges according to (2.4) or(2.10). negative detuning (illustrated in Fig. 7), and an s-wavescattering length of the experimentally observed form a s = a bg (cid:18) − B w B − B (cid:19) . (2.4)Here, a bg is the background (nonresonant) scattering length, B w is the experimental width (not to be confusedwith the width of the Feshbach resonance ), and B isthe value of the magnetic field at which the Feshbachresonance is tuned to zero energy.These properties follow directly from the s-wave atomicscattering amplitude f ( E ) that for two atoms in a vac-uum can be computed exactly. Focusing for sim-plicity on the resonant part of the interaction, (i.e., tak-ing g σ = g = 0) the scattering amplitude is given by f ( E ) = − ~ √ m √ Γ E − ν + i √ Γ E , (2.5)with ν the renormalized (physical) detuning and Γ aparameter measuring the width of the resonance. Theseare given by ν = ν − α m ~ Z d p (2 π ) p , (2.6)Γ ≡ α m π ~ . (2.7)The latter is related to an effective range parameter r = − ~ / p m Γ . (2.8)The integral in (2.6) is implicitly cut off by the ultravi-olet (uv) scale Λ ≈ π/d , set by the inverse of the size d of the closed-channel (molecular) bound state [belowwhich the point interaction approximation inherent inthe Hamiltonian (2.1) breaks down], so that ν = ν − α m Λ2 π ~ (2.9)relates the bare and physical detuning.The s-wave scattering length a s = − f (0) is then givenby a s = − r Γ m ~ ν . (2.10)Thus, to reproduce the experimentally observed scatter-ing length variation with magnetic field (2.4), one fixesthe detuning to be ν ≈ µ B ( B − B ), with the approx-imate Bohr magneton proportionality constant set bythe Zeeman energy difference between approximate elec-tronic spin-triplet (open) and spin-singlet (closed) chan-nels. Matching (2.10) to (2.4) also allows one to relate theFeshbach resonance coupling to the “width” B w , givingΓ ≈ mµ B a B w / ~ . (2.11)Interpretation of the scattering physics in terms of anintermediate molecular bound or quasi-bound state fol-lows from the poles of f ( E ), together with appropri-ate constraints arising from boundary conditions on themolecular wavefunction. From (2.5) the physical pole isgiven by E p = E r − i Γ / , (2.12) resonance E
12r 1r virtualbound statebound state Re FIG. 7: The real part of the pole of the scattering ampli-tude f ( E ), (2.12), as a function of detuning, parameterizedhere by − /a s = ( ν/ ~ ) p m/ Γ , with − /r = √ m Γ / (2 ~ ).As discussed in the text, bound states and resonances mustcorrespond to physical solutions of the Schr¨odinger equationwith proper boundary conditions. The thin dotted line indi-cates asymptotic linear behavior of the bound state for smallpositive a s . where E r = ν − Γ / (4 ν/ Γ − / . Fornegative detuning, ν <
0, the pole is purely real andnegative, corresponding to a bound state with energy E − p ≈ ( − ν / Γ , | ν | ≪ Γ ,ν, | ν | ≫ Γ . (2.13)and infinite lifetime. The corresponding wavevector k − p = i p m | E p | lies on the positive imaginary axis, and cor-responds, as required, to a wavefunction ∼ e −| k − p | r thatdecays exponentially at infinity. As ν → + one has E p → − , and the bound state coincides with the bot-tom of the continuum.For ν >
0, one would like to be able to interpret thestate created by ˆ ψ as a metastable molecule with a finitedecay time into two atoms. Such an interpretation makessense only if the real and imaginary parts of E p are posi-tive and negative, respectively, and Re E p > | Im E p | — E p is called a resonance in this case, with the inequality be-ing the condition that a well defined resonance have awidth that is narrower than its energy. However, as seenin Fig. 7, for a range of small positive ν , rather thanmoving to positive values, E p remains real and movesback to negative values. But this does not indicate a re-stored bound state because k + p = − i p m | E p | now lies onthe negative imaginary axis and corresponds to a wave-function ∼ e + | k − p | r that grows at infinity. Thus, in thisrange of detuning the pole no longer corresponds to a truebound state and is often referred to as a virtual boundstate. Although for Γ / < ν < Γ /
2, one has Im E p < E p remainsnegative. Only for ν > Γ / E p >
0, and only for ν significantly larger than Γ / ψ field.This behavior is summarized in Fig. 7, where Re E p is plotted as a function of detuning. It should be em-phasized that the issue here is strictly one of physicalinterpretation of the microscopic scattering states. Themodel remains well defined, and is a valid descriptionof experiments in the FR regime, over the full parameterrange. . The thermodynamic phases that will be derivedin later sections are also well defined for all parameters.In addition to a gas parameters n / a ( i )bg , correspond-ing to background scattering lengths associated with cou-plings g , g , and g (that are constant in the neighbor-hood of the Feshbach resonance), the two-channel model(2.1) is characterized by a dimensionless parameter γ ≡ r Γ k B T BEC = r π n / | r | , (2.14)that measures the effective strength of the Feshbach-resonant interaction relative to the kinetic energy k B T BEC = (2 π ~ /m ) n / set by the total atomic density n . As long as these are all small, i.e., the gas is dilutewith respect to background scattering lengths and theFeshbach resonance is narrow, the description of phases (i.e., properties away from any phase transitions) can beaccurately given by a perturbative expansion in these di-mensionless interaction parameters. This will beverified through explicit calculations in Sec. VI D. Phys-ically, this narrow resonance limit, γ ≪ γ ≫
1) limit themolecular wave function is strongly hybridized with theopen-channel, and is characterized by a high densityof continuum states above threshold. For such a sys-tem, although for negative detuning a bound molecu-lar state still exists, no resonance remains for positivedetuning—the physics of this regime can no longer beinterpreted in terms of populations of coexisting atomsand (metastable) molecules. In this limit the disper-sion of the bare molecular field can be neglected andˆ ψ can be adiabatically eliminated (integrated out, ig-noring the subdominant atom-molecule and molecule-molecule density interactions) in favor of two open-channel bosons. The resulting single-channel broadresonance model Hamiltonian is then given byˆ H − ch = Z d r (cid:26) ˆ ψ † ( r )ˆ h ˆ ψ ( r ) + 12 g ˆ ψ † ( r ) ˆ ψ ( r ) + 16 w ˆ ψ † ( r ) ˆ ψ ( r ) (cid:27) , (2.15)where g is the effective atom-atom interaction coupling,approximately given by g = g − α /ν , and a stabiliz-ing (against collapse) three-body interaction has beenadded, with a coupling w >
0. In contrast to the γ ≪ g , the one-channel model is strongly interacting when the resonanceis tuned to low energy, ν ≈
0, and the s-wave scatteringlength exceeds the inter-particle spacing, i.e., na s ≫ | g | and w in-crease, quantitative predictive power would require oneto include higher than three-body interactions, so (2.15)really only makes sense when | g | is not too large.Considered as a function of µ and g , the single channelmodel (2.15) also exhibits the same three N, MSF andASF phases (with MSF requiring g < | g | is not too large (i.e., so long as the associated scatteringlength obeys n | a s | ≪ H , Eq. (2.1), the ther-modynamics as a function of a chemical potential µ (orequivalently atom density, n ), detuning ν and tempera-ture T can be worked out in a standard way by computingthe partition function Z = Tr[ e − β ˆ H ] ( β ≡ /k B T ) andthe corresponding free energy F = − k B T ln Z . The traceover quantum mechanical states can be conveniently re-formulated in terms of an imaginary-time ( τ ) functionalintegral over coherent-state atomic ( σ = 1) and molecu-lar ( σ = 2) fields ψ σ ( r , τ ), Z = Z D ¯ ψ σ Dψ σ e − S/ ~ , (2.16)where the imaginary-time action is given by S = Z β ~ dτ Z d r " X σ =1 ψ ∗ σ ~ ∂ τ ψ σ + H ( ψ ∗ σ , ψ σ ) . (2.17)The total atom number constraint (2.3) that allows oneto eliminate µ in favor of N is then simply given by N = − ∂F∂µ . (2.18) III. SYMMETRIES, PHASES AND PHASETRANSITIONS
Before delving into detailed calculations it is instruc-tive to first consider the general symmetry-based charac-terization of phases and transitions between them. Sinceboth atoms ( ˆ ψ ) and molecules ( ˆ ψ ) are bosonic andcan therefore Bose condense, the system’s thermodynam-ics is determined by two Bose-condensate order param-eters, Ψ and Ψ , respectively. As usual, microscopi-cally, these label thermodynamic averages of the corre-sponding field operators, or, for weakly interacting sys-tem, equivalently, are single-particle wavefunctions into which all bosons (atoms and molecules, respectively)Bose-condense.The condensate fields Ψ , Ψ are legitimate orderparameters that uniquely characterize the nature of thepossible phases. Naively one would expect four phases:(i) Normal (N) (Ψ = Ψ = 0), (ii) (Ψ = 0, Ψ = 0),(iii) MSF (Ψ = 0, Ψ = 0), and (iv) AMSF (Ψ = 0,Ψ = 0), corresponding to four different combinations ofvanishing and finite order parameters. However, a finiteFeshbach resonance interaction explicitly breaks U (1) × U (1) symmetry of the α = 0 Hamiltonian down to U (1) × Z . Physically, this corresponds to the fact that only the total number of atoms N = N + 2 N , (3.1)= ( h ˆ ψ † ˆ ψ i + 2 h ˆ ψ † ˆ ψ i ) V ( V is the system volume) is conserved in the presenceof Feshbach resonant scattering (break up) of a moleculeinto two atoms, rather than a separately conserved num-ber of atoms, N and a number of molecules, N . Thisis why only a single chemical potential µ is introduced inˆ H , Eq. (2.1). Consequently, a Feshbach resonant inter-action requires a condensation of molecules (ordering ofˆ ψ ) whenever atoms are Bose-condensed ( ˆ ψ is ordered)and thereby forbids the existence of the state (Ψ = 0,Ψ = 0). As a result, the system of resonantly inter-acting bosonic atoms exhibits only three distinct phases:N, MSF, and AMSF; since the atom-only condensate isimpossible, for brevity of notation the AMSF state willoften be referred to as simply ASF, using the two namesinterchangeably.In addition to the fully “disordered” normal statethat does not break any symmetries, the above dis-tinct thermodynamic phases are associated with differ-ent ways that the U (1) × Z symmetry is broken. Bose-condensation of molecules breaks the U (1) subgroup andcorresponds to a N–MSF transition to the MSF phase.Since it is characterized by ordering of a complex scalarfield, ˆ ψ , this transition is in an extensively-exploredand well-understood XY-model universality class . Thelow-energy excitations in the MSF phase are gaplessGoldstone-mode phase fluctuations of the condensateΨ , associated with the broken U (1) symmetry. In ad-dition there are gapped excitations associated with themagnitude fluctuations of Ψ .The breaking of the remaining Z symmetry is associ-ated with Bose-condensation of atoms, ˆ ψ , in the pres-ence of a molecular condensate. As will be shown explic-itly in Sec. VII, the corresponding MSF–AMSF transi-tion is associated with the ordering of a real scalar field,and one would therefore expect the MSF–AMSF transi-tion to be in the well-explored Ising universality class.However, as will be seen, a coupling of the scalar orderparameter to the strongly-fluctuating Goldstone mode ofthe MSF phase has a nontrivial effect on the Ising tran-sition, quite likely driving it first order sufficiently closeto the transition. ψ ) directlyfrom the normal state breaks the full U (1) × Z symm-metry and corresponds to a direct, continuous N–AMSFphase transition. Since it is associated with the orderingof a complex scalar field, one expects (and finds) it alsoto be in the well-studied XY-model universality class,and to exhibit a single Goldstone mode correspondingto common (locked) phase fluctuations of the condensatefields Ψ and Ψ .There is an instructive isomorphism of this descrip-tion, in terms of two complex scalar order parametersΨ and Ψ , to that in terms of a two-dimensionalrank-1 vector ( M ), together with a rank-2, traceless,symmetric tensor ( Q ) order parameter. The latter de-scription is well known in the studies of ferroelectric ne-matic liquid crystals. There, ordering of Q describesthe isotropic-nematic transition, where the principal axesof mesogenic molecules align macroscopically, breakingthe 2d rotational symmetry modulo π rotation. Thelatter remains unbroken in the nematic phase. This isisomorphic to the N–MSF transition discussed above.For polar molecules, at lower temperature this isotropic-nematic transition can be followed by vector ordering (of,e.g., the molecular electric dipole moments) of M . Inthe non-rotationally invariant, quadrupolar environmentof the nematic phase, this corresponds to spontaneousbreaking of the remaining Z symmetry. Therefore, thesubsequent nematic-polar (ferroelectric) transition corre-sponds to a spontaneous selection between two equivalent0 and π orientations of M relative to the molecular princi-pal axes characterized by the nematic Q order. Clearly,this latter transition can be identified with the MSF–AMSF transition in the atomic system. The mathemat-ical mapping between the two descriptions is elaboratedon in more detail in Appendix A. IV. MEAN FIELD THEORY
The first goal is to determine the nature of the phasesand corresponding phase transitions exhibited by the res-onant bosonic atom-molecule model introduced above.To this end one must evaluate the free energy, which inthe presence of interactions and fluctuations can only becarried out perturbatively. However, away from contin-uous phase transition boundaries, i.e., well within theordered phases, fluctuations are small. The functionalintegral in Z is then dominated by field configurations ψ σ ( r , τ ) ≈ Ψ σ ( r ) that minimize the action S , and there-fore can be evaluated via a saddle-point approximation.For time-independent solutions that characterize ther-modynamic phases, the order parameters Ψ σ ( r ) equiva-lently minimize the variational energy functional H [Ψ σ ],in which one substitutes the classical order parametersfor the field operators in (2.1). For the case of a uniformbulk system with V σ ( r ) = 0 (that is the focus of this sec-tion) one expects that H [Ψ σ ] is minimized by a spatiallyuniform solution (see however Ref. 65) Ψ σ = | Ψ σ | e iθ σ . A mean field analysis then reduces to a minimization ofthe energy density H mf = H [Ψ , Ψ ] /V (4.1)= − µ | Ψ | + g | Ψ | − µ | Ψ | + g | Ψ | + g | Ψ | | Ψ | − α Re[Ψ ∗ Ψ ] , Total atom number conservation ensures a global U (1)symmetry with respect to uniform, σ -independent phaserotation. The Feshbach resonance interaction α Re[Ψ ∗ Ψ ] = α | Ψ || Ψ | cos(2 θ − θ ) (4.2)clearly locks atomic and molecular phases together, anal-ogously to two Josephson-coupled superconductors, withthe energy minimized by θ = 2 θ (mod 2 π ) . (4.3)where without loss of generality α is taken to be positive;for α <
0, the molecular phase θ is simply shifted by π .The corresponding saddle point equations are given by0 = ∂ H mf ∂ Ψ ∗ (4.4a)= − α Ψ ∗ Ψ + Ψ (cid:0) − µ + g | Ψ | + g | Ψ | (cid:1) ∂ H mf ∂ Ψ ∗ (4.4b)= − α Ψ + Ψ (cid:0) − µ + g | Ψ | + g | Ψ | (cid:1) . For a trapped system with a fixed total number of atomsappropriate to atomic gas experiments, these equationsmust be supplemented by the total atom number (2.3)—given by
N/V = | Ψ | + 2 | Ψ | within the mean fieldapproximation—so as to map out the phase diagram asa function of atom number N and detuning ν . However,it is simpler to instead treat the atomic ( µ ) and molecu-lar ( µ ) chemical potentials as independent variables andfirst map out the phase behavior as a function of µ and µ . A. Vanishing Feshbach resonance coupling, α = 0 To complete a minimization of H mf it is instructive tofirst consider a special case of vanishing Feshbach reso-nance coupling, α = 0, for which saddle-point equationsreduce to0 = Ψ (cid:0) − µ + g | Ψ | + g | Ψ | (cid:1) (4.5a)0 = Ψ (cid:0) − µ + g | Ψ | + g | Ψ | (cid:1) . (4.5b)In this special case the model corresponds to an easy-plane (XY-model) limit of two energetically-coupled fer-romagnets, a model that has appeared in a broad varietyof physical contexts. α = 0 limit, the model exhibits four differentphases, corresponding to the four different combinationsof zero or nonzero order parameters Ψ , Ψ .For µ < µ < H mf is convex with a uniqueminimum at | Ψ | = | Ψ | = 0 , for µ , µ < , (4.6)corresponding to the normal state.For µ > µ <
0, the minimum continuouslyshifts to | Ψ | = p µ /g , | Ψ | = 0 , for µ > , µ < µ g g , (4.7)corresponding to a continuous transition at µ = 0 , µ < = 0 is indeed aminimum so long as µ < µ g /g .In a complementary fashion, for µ >
0, Ψ becomesnonzero, while Ψ continues to vanish so long as µ <µ g /g . Hence the normal state undergoes a transitionto the molecular superfluid (MSF) along the line µ < , µ = 0 (N–MSF transition line) . (4.9)The MSF phase is characterized by order parameters | Ψ | = p µ /g , | Ψ | = 0 , for µ > , µ < µ g g , (4.10)i.e., Bose condensed molecules but uncondensed atoms.Along the transition line µ = µ g g , µ > µ = µ g g , µ > g < g g the re-gion defined by above two boundaries is finite and corre-sponds to atomic-molecular superfluid (AMSF) in whichboth atoms and molecules are condensed, with | Ψ | = s g µ − g µ g g − g , | Ψ | = s g µ − g µ g g − g , for µ , µ > , g g < µ /µ < g g . (4.13)Within this mean field analysis, for this range of param-eters all transitions above are second order. For g > g g this fourth AMSF phase is absentbecause energies − µ σ / g σ of the ASF and MSF min-ima cross before either becomes locally unstable. Conse-quently, instead of continuous transition to AMSF, thesystem undergoes a first order transition between theatomic and molecular superfluids at µ µ = r g g , for g > g g , (first order ASF–MSF line) , (4.14)where the ASF and MSF minima become degenerate. Inthis case, the lines µ = µ g /g and µ = µ g /g arespinodals, beyond which the phase on the opposite sideof the first order line becomes locally, not just globally,unstable. As usual with a first order transition, for afixed total atom number (relevant to trapped atomic gasexperiments), along this first order line corresponds to acoexistence region where the system phase separates intocoexisting atomic and molecular superfluids. The twopossible phase diagrams for α = 0 are illustrated in Fig.8. B. Finite Feshbach resonance coupling, α = 0 As is clear from the Hamiltonian (2.1), and its meanfield form (4.1), the full system is characterized by a finiteFeshbach resonance coupling α >
0. If µ , µ < = Ψ =0 remains a local minimum of H mf , and a nonzero α doesnot affect the boundaries of the normal phase, with (4.8)and (4.9) remaining valid.A key physical consequence of a finite α is that, ina phase where atoms are condensed, a finite Feshbachcoupling [that mathematically acts like an ordering fieldon the molecular condensates; see (4.4b)] scatters pairsof condensed atoms into a molecular BEC. Equivalently,it hybridizes states of a pair of atoms and a molecule,and as a result a finite molecular condensate is always induced in a state where atoms are condensed. Conse-quently (much like an external magnetic field eliminatesthe distinction between the paramagnetic and ferromag-netic states), as anticipated in Sec. III, a finite α elim-inates the ASF phase, replacing it by the AMSF. A fi-nite Feshbach coupling thereby converts the ASF–AMSFtransition into a crossover , µ = µ g g , µ > , (4.15)between two regimes of AMSF with low and high densityof a molecular condensate, with the quantitative distinc-tion and crossover between these becoming sharp in thesmall α limit. This is illustrated in Fig. 9. For simplicityof notation, from here on, both of these regimes will bereferred to as simply ASF.For small α the value of the atomic and molecu-lar condensates throughout the ASF phase can be esti-mated from the saddle point equations (4.4). As for the2 (a) g g > g µ − µ − Normal ψ = 0 ψ = 0 ASF ψ = 0 ψ = 0 AMSF MSF ψ = 0 ψ = 0 ψ = 0 ψ = 0 µ − µ − Normal ψ = 0 ψ = 0 MSF ψ = 0 ψ = 0 ASF ψ = 0 ψ = 0 (b) g g < g FIG. 8: The two possible µ - µ mean field phase diagramsfor a special case of a vanishing Feshbach resonance coupling, α = 0. (a) For g g > g there are four distinct phasesseparated by four second order transition lines, meeting at atetracritical point. (b) For g g < g there are only threedistinct phases, meeting at the bicritical point. The uniformAMSF phase, exhibiting both atomic and molecular super-fluidity, is unstable and is replaced by a direct first orderASF–MSF transition denoted by the hatched double line. Onthis line the dominant atom-molecule repulsion drives the sys-tem to phase separate into ASF and MSF regions. The othertwo transitions remain continuous. In (b) the dashed lines µ = µ g /g and µ = µ g /g denote spinodals. case of α = 0, for µ > ≈ p | µ | /g ,which (through the atom-molecule repulsion g ) actsto shift the effective molecular chemical potential to µ eff2 = µ − µ g /g . Vanishing of µ eff2 defines theASF crossover line (4.15). To the left of and above thiscrossover boundary, the effective molecular chemical po-tential is negative and the molecular condensate is small. ν ASFAMSF crossover α ( n ) n µ / eff g FIG. 9: Schematic illustration of the crossover from smallmolecular condensate fraction (in a region in which n = | Ψ | would vanish exactly for α = 0) to a large molecularBEC. The horizontal axis represents a path, parameterizedby µ and µ , which intersects the crossover line in Fig. 10.The estimates for the exhibited sizes of n deep in the AMSFphase, and in the vicinity of the crossover line, follow from thediscussion surrounding equation (4.20). It is induced to be finite, via (4.4b), only by virtue of afinite Feshbach resonance coupling to the atomic conden-sate. This givesΨ = α Ψ − µ + g | Ψ | ) + O ( α ) (4.16) ≈ αµ g µ − g µ ) , for µ > , µ < µ g g . On the other hand below the crossover line, µ eff2 >
0, awould-be spontaneous (for α = 0) molecular condensateis only weakly modified from its α = 0 value | Ψ | = µ eff2 g (4.17) ≈ g µ − g µ g g , for µ > , µ > µ g g . In the intermediate regime, in the vicinity of the crossoverline itself, one expects Ψ still to vanish with α , butmore slowly than linearly. Thus, if | Ψ | ≫ α/g , then(4.4a) becomes µ − g | Ψ | ≈ g | Ψ | . Substitutingthis into the second term on the right hand side of (4.4b)one obtains to leading order the relation∆ = τ | Ψ | + | Ψ | . (4.18)in which∆ = αµ g g − g ) , τ = g µ − g µ g g − g (4.19)are the scaled Feshbach coupling and deviation from thecrossover line, respectively. The solution may be ob-tained in the scaling form | Ψ | = ∆ / X ( τ / ∆ / ) (4.20)3in which the scaling function X ( x ) is the solution to thecubic equation 1 = xX ( x ) + X ( x ) , (4.21)with X (0) = 1. Close to the crossover line, where | τ | / ∆ / = O (1), one sees that Ψ is of order( α | Ψ | ) / . For large positive τ / ∆ / one enters thelinear scaling regime where | Ψ | ≈ ∆ /τ , which is con-sistent with (4.16) so long as the constraint | Ψ | ≫ α/g is obeyed. This leads to the condition ( g µ − g µ ) /g µ ≪ = 0 is still a solution, sothat the N–MSF transition line (4.9) is not modified bythe Feshbach resonance coupling. Then equation (4.4b)still gives | Ψ | = p µ /g in the MSF phase. However,the subsequent MSF–ASF transition boundary is modi-fied, as a finite α shifts the effective chemical potentialof the atomic condensate (in addition to the shift due tothe atom-molecule repulsion, g ) to µ eff1 = µ − g | Ψ | + α | Ψ | . (4.22)with the MSF–ASF transition located by µ eff1 = 0. Using | Ψ | = p µ /g of the MSF one obtains an estimate ofthe MSF–ASF transition boundary µ = g g µ − α r µ g (MSF–ASF transition line) . (4.23)For small µ (specifically, 0 ≤ µ ≪ α g /g ), the sec-ond term on the right hand side dominates, and theboundary displays a sharp square-root singularity intonegative values of µ (near the origin preempted by afirst order transition: see below) illustrated in the phasediagrams for g g > g and g g < g in Figs. 10 and11, respectively. In the opposite limit ( µ ≫ α g /g ),the boundary asymptotes to the α = 0 phase boundary,(4.12).Another important consequence of a finite Feshbachresonance coupling is that for small chemical potentials itdrives the N–ASF and MSF–ASF transitions first order.The somewhat technical calculation of the correspondingfirst order phase boundaries, illustrated in Figs. 10 and11, are relegated to Appendix B. Here a more approx-imate, but more transparent, analysis is presented. Tothis end, one can use an approximation to (4.4b)Ψ ≈ α | µ | Ψ , (4.24)valid for sufficiently negative µ and µ , to eliminate Ψ from H mf [Ψ , Ψ ] in favor of Ψ . The resulting energydensity in the normal state is well approximated by H mf ≈ − µ | Ψ | + 12 (cid:18) g − α | µ | (cid:19) | Ψ | + α g µ | Ψ | . (4.25) µ − µ − Normal ψ = 0 ψ = 0 MSF ψ = 0 ψ = 0 ASF ψ = 0 ψ = 0 FIG. 10: Mean field phase diagram in the µ - µ plane fora finite Feshbach resonance coupling α = 0 and g g > g .In contrast with the α = 0 limit shown in Fig. 8(a), a fi-nite α eliminates the distinction between the ASF and AMSFphases, converting the ASF–AMSF transition, indicated bythe dashed line, into a crossover. Feshbach resonance scat-tering also strongly modifies the MSF–ASF phase bound-ary, and for small chemical potentials drives the N–ASFand MSF–ASF transitions first order (indicated by hatchedcurves), with the first order section terminated by two tricrit-ical points. The point where the three phases meet, and thecontinuous N–MSF phase boundary terminates at the firstorder boundary, is a critical endpoint. Clearly, for µ < negative effec-tive quartic coupling, u = g − α / | µ | a secondaryminimum at Ψ = 0 develops, that can compete withthe normal state Ψ = Ψ = 0 minimum. It is easyto show that the corresponding ASF state minimum be-comes degenerate with the normal state at a criticalvalue µ c = − u / u ( u = α g / µ is the effective | Ψ | coupling), that translates into a first order N–ASFboundary µ c ≈ − α g (cid:18) | µ | g α − (cid:19) , (4.26)as illustrated in Figs. 10 and 11. On the other hand,for sufficiently small µ >
0, the relation between Ψ and Ψ following from (4.4b) becomes (keeping only the g | Ψ | term inside the parentheses on the right handside), Ψ ≈ (cid:18) α g (cid:19) / Ψ / , (4.27)4 µ − µ − Normal ψ = 0 ψ = 0 MSF ψ = 0 ψ = 0 ASF ψ = 0 ψ = 0 FIG. 11: Mean field phase diagram in the µ - µ plane for afinite Feshbach resonance coupling α = 0 and g g < g . Incontrast with the α = 0 limit in Fig. 8(b), finite α eliminatesthe distinction between the ASF and AMSF phases. Fes-hbach resonance scattering also strongly modifies the MSF–ASF phase boundary, and for small chemical potentials drivesa segment of the N–ASF and MSF–ASF transitions first order(indicated by hatched curves), with the first order section ter-minated by two tricritical points. The point where the threephases meet, and the continuous N–MSF phase boundary ter-minates at the first order boundary, is a critical endpoint. Atlarge positive µ ’s the MSF–ASF transition retains its α = 0first order character, separated from the continuous section ofthis transition by another tricritical point. which, when inserted into H mf , Eq. (4.1), determines thefirst order MSF–ASF phase boundary, in a way detailedin Appendix B. V. DILUTE BEC LIMIT
Thus far, the phase diagram in the µ - µ plane, hasbeen studied by treating the atomic and molecular chem-ical potentials as independent tuning parameters. Asseen, within a mean field approximation, the tempera-ture then plays no apparent role.However, to make a direct contact with trapped de-generate atomic gas experiments, where it is the totalnumber of atoms N and the detuning ν that are varied,one needs to eliminate the chemical potentials µ σ in favorof the atom density n , detuning ν = 2 µ − µ , and tem-perature T . For an interacting system, this is a nontrivialchange of variables that can usually only be carried outperturbatively. However, in the dilute limit, appropriateto atomic gas systems, the transition out of the normal state can be treated by ignoring weak atomic interactions(with corrections in powers of na σ and γ ), thereby reduc-ing the problem to an easily calculable BEC limit. Thesystem then reduces to two independent ideal Bose gases,coupled only through the overall constraint of fixed den-sity n = n + 2 n , Eq. (3.1). A. Bulk N–ASF and N–MSF BEC transitions
In the noninteracting limit, for a bulk (uniform) systemthe free energy and atom density in d spatial dimensionsare easily calculated and are given by f = 1 βV X k ,σ =1 , ln h − e − β ( ε k σ − µ σ ) i = − β Λ dT X σ =1 , σ d/ g d +22 ( z σ ) (5.1) n = − X σ =1 , σ ∂f∂µ σ = 1 V X k ,σ =1 , σe β ( ε k σ − µ σ ) −
1= 1Λ dT X σ =1 , σ ( d +2) / g d ( z σ ) (5.2)where ε σ = ~ k / m σ (with m = m and m = 2 m ) arethe single particle energies, z σ = e βµ σ are the fugacities,Λ T = h/ √ πmk B T is the (atomic) thermal de Brogliewavelength, and g α ( z ) = 1Γ( α ) Z ∞ dx x α − z − e x − ∞ X n =1 z n n α (5.3)is the extended zeta function.For positive detuning, ν >
0, atoms (being less energet-ically costly than molecules) condense first at a criticalline µ = 0, where µ = 2 µ − ν = − ν <
0. The corre-sponding critical temperature T c ( ν ) for N–ASF transi-tion is easily determined by the fixed density condition n = Λ − dT c h ζ ( d/
2) + 2 ( d +2) / g d/ (cid:0) e − β c ν (cid:1)i , (5.4)with ζ ( α ) = g α (1). As usual, the transition exists onlyfor d >
2. In the far detuned limit, ν/k B T c ≫ z ≪ T ∞ c ≈ h πmk B (cid:20) nζ ( d/ (cid:21) /d , (5.5)that is approached exponentially as e − β c ν . In the oppo-site limit, 0 < ν/k B T ≪
1, the expansion g α ( e − x ) = Γ(1 − α ) x α − + ∞ X n =0 ζ ( α − n ) n ! ( − x ) n (5.6)5may be used to obtain T c T c − ≈ ( d +6) / Γ (cid:0) − d (cid:1) d ( d − (cid:2) ( d +2) / (cid:3) ζ ( d/ (cid:18) νk B T c (cid:19) d − (5.7)valid for the range 2 < d < T c = h πmk B ( n (cid:2) ( d +2) / (cid:3) ζ ( d/ ) /d (5.8)is the transition temperature at zero detuning, ν = 0corresponding to the critical endpoint in Fig. 2. Similarly, for negative detuning, ν <
0, molecules areenergetically less costly and therefore condense first. Thecorresponding N–MSF critical line, given by µ = 0, with µ = ν/ <
0, using the fixed density condition translatesinto a T c ( ν ), determined implicitly by n = Λ − dT c h ( d +2) / ζ ( d/
2) + g d/ ( e β c ν/ ) i . (5.9)In the far detuned limit the N–MSF transition tempera-ture approaches T ∞ c ≈ h πmk B (cid:20) n ζ ( d/ (cid:21) /d (5.10)exponentially as e − β c | ν | , while in the small detuninglimit | ν | /k B T ≪ T c according to T c T c − ≈ (6 − d ) / Γ (cid:0) − d (cid:1) d ( d − (cid:2) ( d +2) / (cid:3) ζ ( d/ (cid:18) | ν | k B T c (cid:19) d − . (5.11)The resulting ratios T ∞ c T ∞ c = 2 ( d +2) /d T ∞ c T c = h ( d +2) / i /d T ∞ c T c = h − ( d +2) / i /d T c ( ν ) − T c T c ( − ν ) − T c = 2 d , < νk B T c ≪ . (5.12) are noteworthy. The normalized transition temperatures, T cσ ( ν ) /T c , give the corresponding phase boundaries (afunction of ν/k B T c ) displayed in the phase diagram inFig. 2.The thermodynamics of this dilute Bose gas mixtureabove the transition temperature (i.e., inside the normalstate) can be obtained by using the atom number con-straints (5.2) to express the chemical potentials µ ≡ µ , µ = 2 µ − ν as functions of temperature and detuning. Inthe neighborhood (above) the N–ASF and N–MSF tran-sitions this can be done analytically using (5.2) and (5.6).To leading order, for 2 < d < | µ | ≪ ν one obtainsnear the N–ASF line: (cid:18) T c T (cid:19) d/ − ≈ Γ( − d )( β | µ | ) ( d − / ζ ( d/
2) + 2 ( d +2) / g d/ ( e − βν ) , (5.13)while in the neighborhood of the N–MSF line, defining δµ ≡ µ − ν/ ≪ | ν | , one obtains (cid:18) T c T (cid:19) d/ − ≈ d Γ( − d )( β | δµ | ) ( d − / ( d +2) / ζ ( d/
2) + g d/ ( e βν/ ) . (5.14)Finally, for ν = 0 one finds (cid:18) T c T (cid:19) d/ − ≈ (cid:0) d (cid:1) Γ( − d )( β | µ | ) ( d − / ζ ( d/ (cid:0) ( d +2) / (cid:1) , (5.15)Therefore, for σ = 0 , ,
2, corresponding to ν = 0, ν > ν <
0, respectively, one obtains | δµ σ | k B T cσ ≈ A σ (cid:20) TT cσ − (cid:21) / ( d − , (5.16)where T cσ are transition temperatures evaluated at thegiven values of ν , n , the chemical potential deviations aregiven by δµ = δµ = µ , δµ = δµ , and the amplitudesare A ( ν ) = ( d ( d − (cid:0) − d (cid:1) h ζ ( d/
2) + 2 ( d +2) / g d/ ( e − ν/k B T c ) i) / ( d − A ( ν ) = ( d ( d − d +2 Γ (cid:0) − d (cid:1) h ( d +2) / ζ ( d/
2) + g d/ ( e ν/ k B T c ) i) / ( d − A = ( d ( d − ζ ( d/ (cid:0) − d (cid:1) ( d +2) / d ) / ( d − . (5.17)6For d > / ( d −
2) sticks at unity, so that δµ varies linearly with the temperature deviation. For d = 3, using Γ(1 /
2) = √ π and ζ (3 / ≃ . T cσ ( ν ) are easily computed. For ν > µ = µ vanishes before the molecular one µ = 2 µ − ν and for T < T c ( ν ) an atomic condensate n ( T, ν ) = | Ψ | develops with n = n " − (cid:18) TT c (cid:19) d/ ζ ( d/
2) + 2 d +22 g d/ ( e − ν/k B T ) ζ ( d/
2) + 2 d +22 g d/ ( e − ν/k B T c ) , for ν > , T < T c ( ν ) , (5.18)as the gas transitions to the ASF in the BEC limit. Close to T c ( ν ), the atomic condensate (5.13) grows linearly withreduced temperature n ( T, ν ) ∼ n (cid:18) − TT c ( ν ) (cid:19) , for ν > , T → T − c ( ν ) , (5.19)consistent with the expected order parameter exponent β = 1 / ν < µ = 2 µ − ν vanishes before the atomic one µ = µ , and for T < T c ( ν ) a molecular condensate n ( T, ν ) = | Ψ | develops with n = 12 n " − (cid:18) TT c (cid:19) d/ d +22 ζ ( d/
2) + g d/ ( e ν/ k B T )2 d +22 ζ ( d/
2) + g d/ ( e ν/ k B T c ) , for ν < , T < T c ( ν ) , (5.20)as the gas undergoes a transition into a molecular BEC(MSF). Again, close to T c ( ν ), the molecular condensate(5.14) grows linearly with reduced temperature n ( T, ν ) ∼ n (cid:18) − TT c ( ν ) (cid:19) , for ν < , T → T − c ( ν ) , (5.21)consistent with the same order parameter exponent β =1 / | ν | /k B T ≫ T cσ ( ν )and n σ ( ν, T ) arising from the contribution of the sec-ondary, off-resonance, bosonic component that is gappedout for ν = 0. For example, for ν <
0, upon warming to-ward T c ( ν ), the molecular condensate is reduced due toboth the conventional mechanism of thermal excitationsof molecules out of the molecular condensate, as well asthe depairing of molecules into thermally excited bosonicatoms, with the latter special to a Feshbach-resonant sys-tem.Because of the suppression of T cσ ( ν ) near ν = 0 for T c < T < T ∞ c ( ν ), the gas is expected to undergo asequence of ASF → N → MSF transitions upon loweringof ν (see Fig. 2). For T < T c the transition is a directASF → MSF one, that for this noninteracting limit isfirst order. The condensate densities are undefined righton the critical line ν = 0, T < T c , and the noninteractingapproximation becomes particularly questionable there. B. N–ASF and N–MSF BEC transitions in a trap
The above results are straightforwardly extended tothe experimentally more relevant case of a harmonic trap.The modifications due to the trapping potential can all beincorporated through the change in the density of states.For an isotropic harmonic trap (easily extendable to ananisotropic trap) the single particle energy spectrum ε n = ~ ω ( n + n + . . . + n d ) (5.22)is linear in n = P di n i and exhibits a well-known degen-eracy that, for large quantum numbers of interest to us,in the macroscopic limit k B T cσ ≫ ~ ω , is given by D ( ε ) = 1( d − ε d − ( ~ ω ) d . (5.23)Note that ε n is actually σ independent, i.e., the trapfrequency ω is the same for atoms ( σ = 1) and molecules( σ = 2). This is a good approximation in the physicallyrelevant limit of the size d of the closed-channel moleculebeing much smaller than the trapped cloud size.In the thermodynamic limit the sum over single-particle states appearing in (5.1) and (5.2) can be re-placed by integration over energies ε weighted by abovedensity of states, giving N = (cid:18) k B T ~ ω (cid:19) d X σ σg d ( z σ ) . (5.24)7Paralleling the above calculations for the uniform system,from this one obtains all the relevant quantities for thetrapped system. Specifically, the transition temperatures T cσ ( ν ) are implicitly given by N = k B ˜ T c ~ ω ! d h ζ ( d ) + 2 g d (cid:16) e − ˜ β c ν (cid:17)i , (5.25) N = k B ˜ T c ~ ω ! d h ζ ( d ) + g d (cid:16) e − ˜ β c | ν | / (cid:17)i . (5.26)These can be solved in the asymptotic regimes of smalland large detuning, giving˜ T cσ ( ν ) ≈ ˜ T c h a σ | ν | k B ˜ T c i , | ν | ≪ k B ˜ T c ˜ T ∞ cσ h − b σ e −| ν | /σ ˜ T ∞ cσ i , | ν | ≫ k B ˜ T c , (5.27)with a σ = 2 ζ ( d − / σ dζ ( d ) and b σ = 2 /σ dζ ( d ). Thetransition temperatures ˜ T ∞ cσ in the limit of asymptoticallylarge positive ( σ = 1) and negative ( σ = 2) detuning( | ν | /k B ˜ T cσ ≫ T c ( ν = 0),are given by ˜ T ∞ cσ = ~ ω (cid:20) Nσζ ( d ) (cid:21) /d , (5.28)˜ T c = ~ ω (cid:20) N ζ ( d ) (cid:21) /d . (5.29)The latter is approached linearly with reduced detuningfrom either side, in any dimension d ≥ N σ : N = X σ =1 , σ " N σ + (cid:18) k B T ~ ω (cid:19) d g d ( z σ ) . (5.30)The analysis of these equations closely follows thatfor the bulk BEC of the previous subsection. Belowthe transition into the ASF and MSF phase one canstraightforwardly compute the number of atoms N σ = R d r | Ψ σ ( r ) | in the corresponding condensate. For ν > µ = µ vanishes before themolecular one µ = 2 µ − ν , and for T < ˜ T c ( ν ) the molec-ular condensate N = 0 and a finite atomic condensate N ( T, ν ) develops, given by N = N " − (cid:18) T ˜ T c (cid:19) d ζ ( d ) + 2 g d ( e − ν/k B T ) ζ ( d ) + 2 g d ( e − ν/k B ˜ T c ) , for ν > , T < ˜ T c ( ν ) . (5.31)For ν < µ = 2 µ − ν vanishes before the atomic one µ = µ , and for T < ˜ T c ( ν ) the atomic condensate N = 0 and a finite molec-ular condensate N ( T, ν ) develops, given by N = 12 N " − (cid:18) T ˜ T c (cid:19) d ζ ( d ) + g d ( e ν/ k B T )2 ζ ( d ) + g d ( e ν/ k B ˜ T c ) , for ν < , T < ˜ T c ( ν ) , (5.32)Just below the transition temperatures ˜ T cσ the conden-sate growth is of the expected linear in T form, charac-teristic of the order parameter exponent β = 1 /
2. Also,for a far detuned gas, | ν | /k B T ≫
1, the above resultsreduce to the standard single component BEC behavior.The advantage of a trapped system is that, as in thecase for an ordinary single-component trapped conden-sate that exhibits a striking narrow BEC peak, heretoo we expect ASF and MSF condensates in a trap todisplay clearly identifiable BEC peaks. As discussed inthe Introduction and illustrated in Fig. 1, provided thatatoms and molecules can be imaged separately, the ASFshould be easily identified by atomic and molecular BECpeaks, while the MSF is identified by the presence onlyof a molecular BEC peak. In a harmonic trap at lowtemperature, T ≪ ˜ T cσ , the density profile of the cloudis dominated by a narrow Gaussian σ -condensate peak,with the width given by the quantum oscillator length r σ = r ~ m σ ω . (5.33)This should be easily distinguishable from the high-temperature, classical Gaussian density profile (comingfrom the Boltzmann distribution) with the much widerwidth set by the thermal oscillator length r σT = s k B T m σ ω = r σ r k B T ~ ω ≫ r σ . (5.34)The full atomic (whether free or bound into molecules)density profile n ( r ) at arbitrary temperature is easily cal-culated for a noninteracting gas. It is given by n ( r ) = X σ σn σ ( r ) , (5.35)consisting of atomic and molecular contributions, in theBEC limit tied only by a common chemical potential µ determined by the overall particle number constraint(5.30). As derived and analyzed for a single Bose com-ponent in App. D, these in turn are given by n σ ( r ) = ∞ X n =0 | φ n σ ( r ) | e β ( ε n − µ σ ) − , (5.36)= ∞ X p =1 e pβµ σ ρ osc .σ ( r , r ; pβ ~ ω ) , where φ n σ ( r ) are harmonic oscillator eigenstates and ρ osc .σ ( r , r ; β ~ ω ) is the diagonal element of the single-particle density matrix for a harmonic oscillator with8mass m σ . In 3d it is given by ρ osc .σ ( r , r ; β ~ ω ) = (cid:18) m σ ω e β ~ ω π ~ sinh( β ~ ω ) (cid:19) / e − r /r σ ( β ) , (5.37)where r σ ( β ) = ~ m σ ω coth ( β ~ ω / , (5.38) ≈ ( ~ m σ ω , ~ ω /k B T ≫ , k B T m σ ω , ~ ω /k B T ≪ , (5.39)is the finite-temperature “oscillator length” that reducesto the quantum one p ~ / ( m σ ω ), Eq. (5.33), at low T ,and the classical (thermal) one, Eq. (5.34), at high T .The spatial profile of the σ -density, n σ ( r ), is deter-mined by the ratio of the chemical potential µ σ to thetrap level spacing ~ ω , with former in turn determined by the temperature through the total atom number con-straint. At high T ≫ T cσ (where the gas is nondegen-erate), such that 0 < − µ σ ≈ − k B T ln[ (cid:16) ~ ω k B T (cid:17) N ] ≈ k B T ln( T /T cσ ) ≫ k B T , the result is a purely classicalthermal (Boltzmann) distribution, n σ ( r ) ≈ (cid:18) k B Tπ ~ ω (cid:19) / r σ e − r /r σT −| µ σ | /k B T , T ≫ T c , (5.40)with only of order unity occupation of the lowest oscil-lator n = 0 state and a vanishing “condensate” density n ( r ) = π − / r − σ e − r /r σ −| µ σ | /k B T .As T is lowered further, approaching T cσ from above,the magnitude of the chemical potential drops below T (remaining negative) and the boson density profile devel-ops a small r non-Boltzmann peak structure even above T cσ : n σ ( r ) ≈ (cid:18) k B Tπ ~ ω (cid:19) / r σ g / h e − r /r σT −| µ σ | /k B T i , for T & T c , (5.41) ≈ (cid:18) k B Tπ ~ ω (cid:19) / r σ e − r /r σT −| µ σ | /k B T , r ≫ r σT ζ (3 / − π / (cid:16) r r σT + | µ σ | k B T (cid:17) / , r ≪ r σT . (5.42)The linear in r cusp is rounded on the length scale below r σ p µ σ / ~ ω .Finally at an even lower T < T cσ , | µ σ | drops below the level spacing, | µ σ | . ~ ω , and the density profile changesdramatically, developing a bimodal distribution n σ ( r ) = n σT ( r ) + n σ ( r ) (see Fig. 1), that consists of a broad (width r σT ) thermal part n σT ( r ) ≈ (cid:18) k B Tπ ~ ω (cid:19) / r σ ˜ g / (cid:18) e − r /r σT −| µ σ | /k B T , k B T ~ ω (cid:19) , for T < T cσ , (5.43)with a small r cusp (rounded by r σ ) and large r Gaussiantails, together with a narrow (width r σ ) condensate part n σ ( r ) ≈ N σ ( T ) π / r σ e − r /r σ . (5.44)In (5.43), ˜ g α ( x, p c ) = p c X p =1 x p p α , (5.45)has been defined, while in (5.44) N σ ( T ) ≈ ∞ X p = p c e − p | µ | /k B T ≈ e − ( p c − | µ | /k B T e | µ | /k B T − , (5.46)is the number of condensed bosons, given by (5.31) and(5.32) when the total atom number constraint is takeninto account. VI. ELEMENTARY EXCITATIONS
Having established the approximate nature of atomicand molecular superfluids, consider next the study oftheir excitations. On general grounds, as required by theGoldstone’s theorem, one expects one collective gapless (sound) mode in each of the ASF and MSF phases, asso-ciated with spontaneous breaking of global U (1) charge(phase-“rotation”) symmetry. In the MSF it is associ-ated with the phase θ of the molecular (two-atom) con-densate, Ψ , while in ASF it corresponds to in-phasefluctuations of the phases θ and θ of the atomic andmolecular condensates.In addition, there are three gapped excitations ineach of the superfluids. In the MSF these are asso-ciated with atom-like (squeezed by Feshbach resonancecoupling to the molecular condensate) quasiparticle ex-citations (accounting for two modes, ψ , ψ † ) and molec-9ular density fluctuations (fluctuations in the order pa-rameter magnitude | Ψ | ). In the ASF one gapped modecorresponds to out-of-phase fluctuations 2 θ − θ of theatomic and molecular condensates (gapped by the Fesh-bach resonance coupling α ), and two others are atomicand molecular condensate densities (fluctuations in theorder parameter magnitudes | Ψ | and | Ψ | ).As will be seen below, the MSF-to-ASF transition is ac-companied by closing of the gap for atom-like quasiparti-cle excitations. However, this mode remains gapless onlyat the MSF–ASF critical point, and is replaced by an-other gapped mode (associated with out-of-phase phasefluctuations of the two order parameters) that emergesinside the ASF. As discussed in Sec. III, this is consis-tent with the Goldstone theorem as (due to the Feshbachresonance coupling) it is only a discrete ( Z ) symmetrythat is being broken at the MSF-to-ASF transition andas such leads to no new gapless modes. A. Bogoliubov diagonalization
Bogoliubov theory provides an asymptotically exactdescription of the low energy excitations in a dilute Bosefluid, not too close to the transition lines. Focusing onquadratic fluctuations, it ignores interactions betweenquasiparticles and, among other things, misses the possi-bility for their decay. The method proceeds by expandingthe field operators about the mean field solution (equiv-alently, a coherent state of k = 0 fields labeled by Ψ σ ):ˆ ψ σ ( r ) = Ψ σ + ˆ φ σ ( r ) , (6.1)and keeping terms in the Hamiltonian only to quadraticorder in the small deviations ˆ φ σ . In the molecular super-fluid state Ψ = 0, so ˆ φ = ˆ ψ . Substituting (6.1) into(2.1) one obtainsˆ H = H mf + ˆ H + O ( ˆ φ σ , ˆ φ σ ) (6.2)in which H mf ≡ H [Ψ σ ] is the mean field approximation(4.1) to the ground state energy. The absence of termslinear in excitations ˆ φ σ is guaranteed by the conditionthat Ψ σ is an extremum of the mean field free energy ∂H mf /∂ Ψ ∗ σ = 0. To quadratic order, this is equiva-lent to the requirement h ˆ φ σ ( r ) i = 0. For a homo-geneous system [generalization to the trapped case maythen be accomplished through a local density approxima-tion (LDA)] the quadratic Hamiltonian, H governing thedynamics of fluctuations, can be represented in terms ofmomentum space operatorsˆ φ σ ( r ) = 1 √ V X k ˆ a k σ e i k · r , ˆ φ † σ ( r ) = 1 √ V X k ˆ a † k σ e − i k · r . (6.3) One obtainsˆ H = X k ,σ (cid:20) ˜ ε k σ ˆ a † k σ ˆ a k σ + 12 ( λ σ ˆ a k σ ˆ a − k σ + h.c. ) (cid:21) + X k (cid:16) t ˆ a † k ˆ a k + t ˆ a † k ˆ a †− k + h.c. (cid:17) , (6.4)where the coefficients are given by˜ ε k = ε k − µ + 2 g | Ψ | + g | Ψ | ˜ ε k = ε k − µ + 2 g | Ψ | + g | Ψ | λ = g Ψ − α Ψ , λ = g Ψ t = g Ψ Ψ ∗ − α Ψ ∗ , t = g Ψ Ψ , (6.5)with single particle energies ε k σ = ~ k / m σ .
1. MSF phase
Consider first the excitations in the molecular super-fluid, characterized by a finite Ψ and vanishing Ψ .As a result, the cross terms t and t vanish, and theatomic and molecular terms can be diagonalized inde-pendently. From (4.10), the mean field order parameteris given by Ψ = p µ /g = p (2 µ − ν ) /g (chosen realand positive for simplicity—more generally any phase θ can be absorbed into the operators via the redefinition a k σ → e − iσθ / a k σ ). It is straightforward to verify thatthe Bogoliubov canonical transformationˆ a k σ = u ∗ k σ ˆ γ k σ − v k σ ˆ γ †− k σ ˆ γ k σ = u k σ ˆ a k σ + v k σ ˆ a †− k σ (6.6)to new bosonic creation and annihilation operators ˆ γ † k σ ,ˆ γ k σ , with real, positive coefficients given by u k σ = 1 + v k σ = 12 (cid:18) ˜ ε k σ E k σ + 1 (cid:19) (6.7) E MSF k σ = q ˜ ε k σ − | λ σ | , (6.8)leads to the diagonal form δ ˆ H MSF = X k ,σ E MSF k σ (cid:16) ˆ γ † k σ ˆ γ k σ − v k σ (cid:17) . (6.9)The diagonalized Hamiltonian, δ ˆ H MSF , governing exci-tations in the MSF naturally separates into “atom-like”( σ = 1) and “molecule-like” ( σ = 2) contributions,with corresponding (explicitly positive) excitation ener-gies E MSF k σ and condensation energy δE MSFcond ≡ − X k ,σ E MSF k σ v k σ , = − X k ,σ
12 (˜ ε k σ − E MSF k σ ) . (6.10)The latter lowers the energy of the MSF below that givenby the mean field condensation energy value, H mf .0In the normal phase, Ψ = 0, one obtains λ σ = 0,˜ ε k σ = ε k σ − µ σ , yielding v k σ = 0 and u k σ = 1. One there-fore recovers the original atomic ( a k ) and the molecular( a k ) operators as true (to quadratic order) excitations inthe normal state, with corresponding free single-particlespectra ε k σ − µ σ .
2. ASF phase
It is clear from the structure of the Hamiltonian δ ˆ H ASF in the ASF phase (most notably the finite values of the t and t couplings), that in addition to the usual Bo-goliubov mixing between particles and holes, a true ex-citation is also a mixture of an atom and a molecule.Physically, this is a reflection of a coherent scattering (bythe Feshbach and atom-molecule density interactions) ofatoms and molecules mediated by their respective con-densates. The Bogoliubov theory for the ASF phase ishandled most simply by first converting from creationand annihilation operators to corresponding “position”and “momentum” operators (canonically conjugate “co-ordinates”, that are Fourier transforms of Hermitian fieldoperators): ˆ a k σ = 1 √ q k σ + i ˆ p k σ )ˆ a † k σ = 1 √ q − k σ − i ˆ p − k σ )ˆ q − k σ = ˆ q † k σ , ˆ p − k σ = ˆ p † k σ (6.11)with the only nonvanishing commutation relations being[ˆ q k σ , ˆ p − k ′ σ ′ ] = iδ kk ′ δ σσ ′ . (6.12)By substituting (6.12) into (6.4) one obtains δ ˆ H ASF = X k (cid:20) δ ˆ H ASF k −
12 (˜ ε k + ˜ ε k ) (cid:21) δ ˆ H ASF k ≡ ˆp † k P k ˆp k + 12 ˆq † k Q k ˆq k (6.13)in which the 2 × ˆq k = (cid:18) ˆ q k ˆ q k (cid:19) , ˆp k = (cid:18) ˆ p k ˆ p k (cid:19) P k = (cid:18) ˜ ε k − λ t − t t − t ˜ ε k − λ (cid:19) Q k = (cid:18) ˜ ε k + λ t + t t + t ˜ ε k + λ (cid:19) . (6.14)In deriving (6.14) the symmetry ε − k σ = ε k σ has beenused, and t , t , λ , λ have been taken to be all real (or,equivalently, their phases absorbed into redefinitions ofˆ a k σ ).One seeks a (real) linear transformation ˆp k = A k ˆP k , ˆq k = B k ˆQ k (6.15) which diagonalizes δ ˆ H ASF k . The canonical requirementthat the transformation preserve the commutation rela-tions (6.12), i.e., that[ ˆQ k , ˆP † k ′ ] = [ ˆq k , ˆp † k ′ ] = iδ kk ′ , (6.16)implies that B T k = A − k . (6.17)Thus, the transformation B k should simultaneously di-agonalize P − k and Q k . Without loss of generality this isequivalent to demanding that B T k P − k B k = , B T k Q k B k = E k , (6.18)in which E k = diag[( E ASF k ) , ( E ASF k ) ] is diagonal, con-taining the squares of the Bogoliubov energies (see be-low). It follows that B − k P k Q k B k = E k , (6.19)so that E k is obtained by diagonalizing P k Q k . Thesquared energies are therefore solutions to the eigenvalueequation det[( E ASF k σ ) − P k Q k ] = 0 . (6.20)The solutions to the resulting quadratic equation in( E ASF k σ ) are ( E ASF k σ ) = e k ± q d k + c (1) k c (2) k (6.21)in which the upper sign corresponds to σ = 1, the lowersign to σ = 2, and the various parameters are defined by P k Q k = e k + d k c (2) k c (1) k e k − d k ! e k = 12 (˜ ε k − λ + ˜ ε k − λ ) + t − t d k = 12 (˜ ε k − λ − ˜ ε k + λ ) (6.22) c (1) k = ( t − t )(˜ ε k + λ ) + ( t + t )(˜ ε k − λ ) c (2) k = ( t + t )(˜ ε k − λ ) + ( t − t )(˜ ε k + λ ) . It is easy to check that the MSF results (6.7) and (6.8)are recovered when t = t = 0.The columns b k σ of B k ≡ ( b k b k ) are the eigenvec-tors of P k Q k and take the form b k σ = 1 N k σ (cid:18) − c (2) k e k + d k − E k σ (cid:19) , (6.23)in which the normalization N k σ is chosen so that b T k σ P − k b k σ = 1.The quadratic Hamiltonian takes the form δ ˆ H ASF = 12 X k ,σ ( ˆ P − k σ ˆ P k σ + ( E MSF k σ ) ˆ Q − k σ ˆ Q k σ − ˜ ε k σ )= X k σ (cid:20) E ASF k σ ˆ γ † k σ ˆ γ k σ + 12 ( E ASF k σ − ˜ ε k σ ) (cid:21) (6.24)1in which the new bosonic raising and lowering operatorsare given byˆ γ k σ = 1 √ E / k σ ˆ Q k σ + iE / k σ ˆ P k σ ! ˆ γ † k σ = 1 √ E / k σ ˆ Q − k σ − iE / k σ ˆ P − k σ ! . (6.25)These may be reexpressed in terms of the original raisingand lowering operators viaˆ a k = 12 (cid:16) B k E − / k + B − T k E / k (cid:17) ˆ γ k + 12 (cid:16) B k E − / k − B − T k E / k (cid:17) ˆ γ †− k ˆ γ k = 12 (cid:16) E / k B − k + E − / k B T k (cid:17) ˆ a k + 12 (cid:16) E / k B − k − E − / k B T k (cid:17) ˆ a †− k (6.26)in which ˆ a k , ˆ a †− k , ˆ γ k , ˆ γ †− k are all column vectors definedin the natural way, consistent with (6.14). B. Acoustic and gapped modes
Consider now the Bogoliubov excitation spectra, (6.8)and (6.21) in more detail. It will be shown that in bothphases there is indeed one acoustic mode and one gappedmode (in addition to two other less interesting gappedmodes ), as required by general principles discussed inthe beginning of this section and in Sec. III. As previ-ously indicated, the MSF phase the acoustic mode cor-responds to long wavelength fluctuations in the phaseof Ψ , while the gapped mode is associated with pair-breaking fluctuations of molecules into two atom-like ex-citations, with spectral gap corresponding to a renormal-ized molecular binding energy. In the ASF phase theacoustic and gapped modes correspond to in-phase andout-of-phase fluctuations of Ψ and Ψ , respectively,with the gap in the latter governed by the Feshbach res-onance coupling α .
1. MSF phase
The MSF quasiparticle spectrum (6.8) appearing in(6.9) (and summarized in Fig. 3), may be written in theform E k σ = p ( ε k σ + ε σ − )( ε k σ + ε σ + ) (6.27)in which the (positive) energies ε σ ± are given by ε ± = − ν/ g − g / | Ψ | ± α | Ψ | ε = 2 g | Ψ | , ε − = 0 . (6.28) The molecule-like branch ( σ = 2) is gapless (consistentwith Goldstone’s theorem), having an acoustic spectrum E MSF k ≈ ~ c MSF2 k at small k with sound speed c MSF2 = | Ψ | p g /m , (6.29)corresponding to collective, long wavelength oscillationsof the molecular condensate. The spectrum crosses overto a particle-like E MSF k ≈ ε k , for kξ MSFcoh ≫
1, where ξ MSFcoh = ~ m c MSF2 (6.30)is a coherence length beyond which superfluid behaviorsets in: the collective superfluid response dominatesdisturbances with wavelength longer than ξ coh , while themicroscopic single molecule response dominates thosewith shorter wavelength. This length ξ MSFcoh ∝ / | Ψ | ∝ / √ µ diverges as the normal phase boundary, µ = 0,is approached.In contrast, the atomic-like branch has a gap E MSFgap = E MSF = √ ε ε − , (6.31)which closes with increasing ν precisely on the ASF–MSFtransition line, the latter being equivalent to the condi-tion ε − = 0. This leads to the critical detuning ν c = − ( g − g ) | Ψ | − α | Ψ | = 2 µ − g g (cid:16) α + 2 µg ± α p α + 4 µg (cid:17) → µ − g µ α , g → , (6.32)where the second line requires µ > − α / g , and fol-lows by substituting | Ψ | = (2 µ − ν ) /g and solving for ν . The existence of two solutions reflects the reentrantbehavior as a function of chemical potential seen in Fig.2. At low temperature and for weak interactions, the con-densate depletion is minimal, n = | Ψ | ≈ n/
2, andthe critical detuning for the quantum MSF–ASF transi-tion is given by ν c ( T = 0) ≈ −
12 ( g − g ) n − α √ n (6.33)The behavior of ν c ( T ) for high temperature [as well as thecorresponding temperature dependence of the condensate n ( T ) at fixed density n ], illustrated in Fig. 2, will bediscussed in Sec. VI D below.
2. ASF phase
In the ASF the extremum conditions (4.4) allow ˜ ε k σ to be reduced to the forms˜ ε k = ε k + λ + α ˜ ε k = ε k + λ + α , (6.34)2and from (6.5) one has t − t = α , (6.35)with the definitions, α = 2 α | Ψ | α = 12 α | Ψ | / | Ψ | α = α | Ψ | = √ α α . (6.36)Substituting (6.34)–(6.36) into (6.22) one obtains, e k + d k = − α ( t + t ) + ( ε k + α )( ε k + α + 2 λ ) e k − d k = − α ( t + t ) + ( ε k + α )( ε k + α + 2 λ ) c (1) k = ( ε k + α )( t + t ) − α ( ε k + α + 2 λ ) c (2) k = ( ε k + α )( t + t ) − α ( ε k + α + 2 λ ) . (6.37)At k = it is easy to verify that e + d = − r α α c (1) = − | Ψ || Ψ | c (1) e − d = − r α α c (2) = − | Ψ | | Ψ | c (2) , (6.38)and therefore that e = d + c (1) c (2) . (6.39)Substituting these results into (6.21) one obtains the ex-citation energies at zero momentum, i.e., the gaps: E ASF = √ e = p α ( α + 2 λ ) + α ( α + 2 λ ) − α ( t + t ) E ASF = 0 , (6.40)which confirms the existence of one gapped and one gap-less mode in the ASF state. From (6.5) and (6.36) onesees that α , α , α + 2 λ , and hence e , vanish on theMSF–ASF phase boundary where Ψ = 0. The gaptherefore closes on the transition line, as expected. Notealso that if α = 0 one has e = 0, and the atomic-like gapremains closed throughout the ASF phase, as expectedfrom the additional spontaneously broken U (1) symme-try (separate atom and molecule number conservation)and associated Goldstone modes, as discussed at the be-ginning of this section and in Sec. III.The small k (low-energy) behavior of the excitationspectra are now examined in the ASF phase, | Ψ | >
0, and in the neighborhood of the MSF–ASF transition,where | Ψ | →
0, but Ψ remains finite. To this end, the | Ψ | and k dependencies are isolated by writing e = 2 α | Ψ | (cid:18) g e + α | Ψ | | Ψ | (cid:19) d = 2 α | Ψ | (cid:18) g d − α | Ψ | | Ψ | (cid:19) c (1) = − α | Ψ | | Ψ | g (1) c c (2) = − α | Ψ || Ψ | (cid:18) g (2) c + α | Ψ | | Ψ | (cid:19) δe k ≡ e k − e = | Ψ | ε k (cid:18) γ e + | Ψ | | Ψ | δ e + 5 ε k | Ψ | (cid:19) δd k ≡ d k − d = | Ψ | ε k (cid:18) γ d + | Ψ | | Ψ | δ d + 3 ε k | Ψ | (cid:19) δc (1) k ≡ c (1) k − c (1) = | Ψ | ε k γ (1) c δc (2) k ≡ c (2) k − c (2) = | Ψ | ε k γ (2) c (6.41)in which the coefficients g e = ( g − g + g / | Ψ | + α/ g d = ( g − g / | Ψ | g (1) c = (2 g − g ) | Ψ | + α/ g e + g d g (2) c = ( g / − g ) | Ψ | + α/ g e − g d γ e = g | Ψ | / αδ e = g | Ψ | + α/ γ d = − g | Ψ | / αδ d = g | Ψ | − α/ γ (1) c = g | Ψ | − α/ γ (2) c = 2 g | Ψ | − α/ | Ψ | = 0 and k = 0. a. At the ASF–MSF critical point, | Ψ | = 0 : Onthe ASF–MSF transition line, | Ψ | = 0 (and also fora vanishing Feshbach resonance coupling, α = 0, whenthe order parameter phases are decoupled), the zero mo-mentum coefficients—the first four lines of (6.41)—vanishidentically and one obtains two gapless spectra( E crit k σ ) = δe k ± q δd k + δc (1) k δc (2) k , (6.43)which lead to two acoustic critical modes, E crit k σ ≈ ~ c crit σ k at small k . For | Ψ | = 0, δc (1) k , δc (2) k vanish and thesound speeds are given by( c crit σ ) = | Ψ | m ( γ e ± γ d ) = (cid:26) αm | Ψ | , σ = 1 g m | Ψ | , σ = 2 . (6.44) b. In the ASF phase, | Ψ | > : As found above,Eq. (6.40), in the ASF phase the spectrum of out-of-phaseexcitations, labeled by σ = 1, is gapped, while that for3 σ = 2 excitations, corresponding to in-phase fluctuationsof the two condensates is given by( E ASF k ) = e k − d k − c (1) k c (2) k E k (6.45) ≈ e (cid:20) ( e + d ) (cid:18) δe k − δd k + | Ψ | | Ψ | δc (2) k (cid:19) + ( e − d ) (cid:18) δe k + δd k + 2 | Ψ || Ψ | δc (1) k (cid:19)(cid:21) , in which (6.38) has been used. As expected from thegeneral symmetry arguments discussed in Sec. III andearlier in this section, the in-phase excitations are acous-tic, E ASF k ≈ ~ c ASF2 k at small k , with sound speed givenby( c ASF2 ) = | Ψ | ( e + d ) f (1) + ( e − d ) f (2) e m , (6.46)with the constants f ( σ )0 defined by f (1) ≡ g | Ψ | + ( g | Ψ | − α/ | Ψ | | Ψ | f (2) ≡ g | Ψ | − α + 2 g | Ψ | | Ψ | | Ψ | . (6.47)Note in passing, that, as expected, for a vanishing Fes-hbach resonance coupling, α = 0, both in-phase and out-phase modes become acoustic, with sound speeds2 m c σ = 12 g | Ψ | + g | Ψ | (6.48) ± s(cid:18) g | Ψ | − g | Ψ | (cid:19) + 2 g | Ψ | | Ψ | . that are real and positive for g g > g . c. Scaling form for small k and | Ψ | : It is easy tocheck that the | Ψ | → k → k , | Ψ | are both small, but have arbitrary ratio,is derived. By keeping only leading terms in | Ψ | and ε k , one obtains E ASF k σ = 2 α | Ψ | (cid:20) g e + γ e y (6.49) ± q ( g d + γ d y ) + ( g (1) c − γ (1) c y ) g (2) c (cid:21) in which the dimensionless scaling variable is y = | Ψ | ε k α | Ψ | . (6.50)It is easily checked that for large y (6.44) is recovered,while for small y (6.46) is recovered. C. MSF paired ground-state wave function
The zero temperature molecular superfluid groundstate is constructed by requiring that it be the quasi-particle vacuum:ˆ γ k σ | MSF i = 0 , for all k = 0 , σ. (6.51)The additional constraintˆ a | MSF i = √ V Ψ | MSF i , (6.52)where ˆ a = V − / R d r ˆ ψ ( r ), ensures that the MSF isa coherent state for the lowest single particle trap state k = and thereby has the correct amplitude Ψ corre-sponding to molecular superfluid order.Using the commutation relations[ˆ a, e λ ˆ a † ] = λe λ ˆ a † , [ˆ a, e λ ˆ a † ˆ b † ] = λb † e λ ˆ a † ˆ b † , (6.53)where ˆ a , ˆ b are any two independent harmonic oscillatoroperators, it follows that the state | MSF i = exp (cid:18) Ψ √ Vˆa † − X k =0 ,σ χ k σ ˆa † k σ ˆa †− k σ (cid:19) | i (6.54)indeed obeys (6.51) and (6.52) with the choice χ k σ = v k σ u k σ = ˜ ε k σ − E MSF k σ λ σ . (6.55)The factor of 1 / k and once for − k .The quantity χ k σ may be identified as the Fouriertransform of the atomic ( σ = 1) and molecular ( σ = 2)pair wavefunctions with zero center of mass momentum.The asymptotic long-distance behavior of its Fouriertransform χ σ ( r ), which is now computed, is governed bythe singularity of χ k σ nearest k = 0. Since χ k σ dependsonly on the magnitude k , one may use the Bessel functionidentity Z d Ω k e i k · r = 2 π d/ (cid:18) kr (cid:19) d − J d − ( kr ) (6.56)to perform the d − χ σ ( r ) = Z d k (2 π ) d e i k · r χ k σ , = Z ∞ k d − dk (4 π ) d/ χ k σ (cid:18) kr (cid:19) d − J d − ( kr ) . (6.57)Since the right hand side of (6.56) is an even function of k , the integration may be extended to the full real line,avoiding the branch cut along k < e iπ ( d − .Since k d − (2 /kr ) ( d − / J − ( d − / ( kr ) is analytic throughthe origin, and an odd function of k , its integral vanishes,and one may write χ σ ( r ) in the form χ σ ( r ) = Z ∞ + iη −∞ + iη k d − dk (4 π ) d/ χ k σ (cid:18) kr (cid:19) d − H (1) d − ( kr ) , (6.58)in which η is a positive infinitesimal and H (1) ν ( x ) is aHankel function of the first kind.From (6.27), one observes that E MSF k σ has finite branchcuts along the imaginary k = | k | axis over the inter-vals ± i ( k σ − , k σ + ), where k σ ± = √ mε σ ± / ~ . To evaluate χ σ ( r ) the integration contour is deformed into the upperhalf plane to run down, around the origin, and then backup the imaginary axis, avoiding the upper branch cut.Since the H (1) ν ( x ) decays exponentially in the upper-halfplane, one can close the contour and then shrink it aroundthe upper branch cut of E MSF k σ . Because the integrand isfinite near the branch points, the infinitesimal circularparts of the contour integral, and the complete integrals of the analytic parts of χ k σ , both vanish. The remainingparts on the left and right sides running along the branchcut double up, giving χ σ ( r ) = ~ π (2 π ) d/ λ σ m σ r − d Z k σ + k σ − dκκ d/ K d − ( κr ) × q ( κ − k σ − )( k σ + − κ ) (6.59)with K ν ( z ) = ( πi/ e iνπ/ H (1) ν ( iz ) the modified Besselfunction. For large k σ + r the integral is dominated by theregion near k σ − , and one may safely (with exponentialaccuracy) extend the upper limit to infinity and approx-imate the square root factor by the form ( q k σ − ( k σ + − k σ − ) p κ − k σ − , κ − k σ − k σ − r ≫ k σ + κ, k σ − r ≪ , (6.60)the lower relation being especially required for σ = 2where k − = 0. One therefore obtains χ σ ( r ) ≈ ~ λ σ m σ q k σ + − k σ − (cid:16) k σ − π (cid:17) d/ e − kσ − r r d +22 , k σ − r ≫ ~ k σ + λ σ m σ Γ( d +12 ) π d +12 r d +1 , k σ − r ≪ , (6.61)in which the asymptotic form K ν ( x ) ≈ p π/ xe − x , | x | ≫
1, has been used to obtain the first line, and the identity Z ∞ x µ K ν ( x ) dx = 2 µ − Γ (cid:18) µ + ν (cid:19) Γ (cid:18) µ − ν (cid:19) (6.62)to obtain the second.It thus follows that in the MSF phase the relativeatomic wavefunction decays exponentially according to χ ( r ) ∼ e − r/ξ σ , with a decay length ξ = 1 k − = ~ √ m ε − , (6.63)reflecting the confinement of (gapped) atomic excita-tions, and the corresponding absence of atomic long-range order inside the MSF. Since ε − ∼ ν − ν c , ξ ∼ ( ν − ν c ) − / has a square root divergence as the ASFphase boundary is approached. On the other hand, since k − = 0, the molecular wavefunction χ ( r ) ∼ /r d +1 hasa power law decay, reflecting the existence of molecularlong-range order inside the MSF.Note that the ground state (6.54), in addition to be-ing a molecular coherent state, is also an (atomic andmolecular) pair coherent state. It thus makes explicitthat within the molecular superfluid state, a molecular k = 0 condensation, Ψ = 0, is accompanied by a nonzero BCS-like atomic pairing at finite relative k , withan anomalous correlation function, h ˆ a k ˆ a − k i = − u k v k = α Ψ E k . (6.64)Exactly the same branch cut structure as described aboveapplies to the right hand side of (6.64), and its Fouriertransform, the BCS-type atom pair correlation function,falls off exponentially at the same rate e − r/ξ . The cor-relation length ξ (that is finite inside the MSF, but di-verges as the transition into ASF is approached) char-acterizes the size of the virtual cloud of atom pairs sur-rounding each closed-channel molecule (whose size, d ,characterized by the microscopic range of the interatomicpotential, remains finite throughout).On the other hand, the molecular anomalous pair cor-relation function h ˆ a k ˆ a − k i = − u k v k = g Ψ E k . (6.65)exhibits a 1 /k divergence near the origin [on top of theΨ V (2 π ) d δ ( k ) condensate contribution due to the long-range order], so that its Fourier transform approaches theΨ asymptote via a slow 1 /r d − power law decay. Thisis a signature of quantum fluctuations in the low energymolecular Goldstone mode.5 D. Thermodynamics
As is clear from (6.9) and (6.24), within the Bogoli-ubov approximation a superfluid (be it MSF or ASF)is a coherent state with excitations described by a gasof noninteracting bosonic Bogoliubov quasiparticles, ˆ γ k σ ,respectively given by (6.6) and (6.26). Thermodynamicsis therefore easily computed in a standard way.
1. MSF phase
The free energy density in the MSF consists of theground state condensate energy H mf = H [Ψ ], plus acontribution from the noninteracting Bogoliubov quasi-particles, governed by δ ˆ H MSF , Eq. (6.9). A standard freeboson computation gives f MSF [ h ] = − µ | Ψ | + 12 g | Ψ | − Re[ h ∗ Ψ ] + X σ Z d k (2 π ) d (cid:20) β ln(1 − e − βE MSF k σ ) + 12 ( E MSF k σ − ˜ ε k σ ) (cid:21) , (6.66)where a complex molecular “source field” h (that van-ishes for a physical system) has been included. As usual,derivatives of f MSF [ h ] with respect to h generate cor-relation functions of the molecular field. In interpret-ing this quantity, it is important to emphasize that Ψ here (in an unfortunate abuse of notation) is the meanfield order parameter, an explicit function of the Hamil-tonian parameters µ, ν, h , etc., that does not include anyfluctuation corrections. The leading Bogoliubov correc-tions are provided by the h derivatives of f MSF [ h ]. Forthe molecular condensate order parameter, corrected byquantum and thermal fluctuations this gives:Ψ ≡ Ψ + δ Ψ = − (cid:18) ∂f MSF ∂h ∗ (cid:19) h =0 , (6.67)in which h enters through its explicit appearance inthe first line of (6.66) as well as implicitly through Ψ .The extremum property of Ψ with respect to H mf [Ψ ]therefore gives δ Ψ = − (cid:18) ∂f MSF ∂ Ψ ∗ (cid:19) (cid:18) ∂ Ψ ∗ ∂h ∗ (cid:19) h =0 = − Ψ µ [ I d, ( µ, ν ) + 2 I d, ( µ, ν )] , (6.68)where the mean field longitudinal susceptibility is( ∂ Ψ ∗ /∂h ∗ ) h =0 = 1 / ( − µ + 6 g | Ψ | ) = 1 / µ . Con-sistency requires that the original forms (6.5) be used forthe Ψ dependence, and I d, = Z d k (2 π ) d (cid:20)(cid:18) n k + 12 (cid:19) g ˜ ε k − α E MSF k − g (cid:21) I d, = Z d k (2 π ) d (cid:20)(cid:18) n k + 12 (cid:19) g ˜ ε k − g | Ψ | E MSF k − g (cid:21) , (6.69) where n k σ = ( e βE k σ − − are the standard Bose occupa-tion factors for Bogoliubov quasiparticles. The numberdensity to this same order is n = − (cid:18) ∂f MSF ∂µ (cid:19) T,ν = 2 | Ψ | + δn MSF ( T, ν ) , (6.70)where the density of bosons not condensed into the lowest k = 0 single particle state (i.e., the condensate depletion)is given by δn MSF ( T, ν ) = X σ σ Z d k (2 π ) d (cid:2) v k σ + ( u k σ + v k σ ) n k σ (cid:3) , (6.71)The depletion density δn MSF ( T, ν ) comes from the ex-plicit µ -dependence in E MSF k σ and remains finite even atzero temperature due to the interaction-induced zero-point contribution v k σ . The remaining implicit µ -dependence entering through the condensate Ψ givesrise to the term | Ψ | in (6.70), in place of the mean fieldcondensate density | Ψ | .Evaluating (6.71) at T = 0 and ν = ν c one obtains6 δn MSF (0 , ν c ) = X σ σ Z d k (2 π ) d " ε k σ + ε σ + / p ε k σ ( ε k σ + ε σ + ) − = B d d/ (cid:18) πmh (cid:19) d/ ( ε d/ + 2 ( d +2) / ε d/ ) (6.72)in which ε ( ν c ) = 2 α | Ψ | , ε ( ν c ) = 2 g | Ψ | , and the coefficient is given by B d = Z ∞ dvv ( d − / " v + 1 / p v ( v + 1) − = 1 d √ π Γ (cid:18) d − (cid:19) Γ (cid:18) − d (cid:19) . (6.73)In d = 3 one finds B = 1 / ε d/ ∝ n d/ , this“correction” term becomes much larger than n close to the MSF–N transition line. This is a sign of the breakdownof the mean field description of criticality, and (6.73) ceases to valid in this nontrivial critical regime. For ν < ν c (i.e., inside MSF phase) one obtains: δn MSF (0 , ν ) = B d d/ (cid:18) πmh (cid:19) d/ n [1 + b d ( δ )] ε d/ + 2 ( d +2) / ε d/ o (6.74)in which δ = ε − ( ν ) /ε ( ν ), and b d ( δ ) = Z ∞ dv v ( d − / B d " v + (1 + δ ) / p ( v + δ )( v + 1) − v + 1 / p v ( v + 1) . (6.75)Of interest is the behavior of this integral near ν c , i.e.,for small δ . The singular behavior can be obtained byfirst computing the derivative db d dδ = − − δ B d Z ∞ v ( d − / dv ( v + δ ) / √ v + 1 . (6.76)For d < δ →
0, and one obtains db d dδ = − − δ B d [ β d,s δ ( d − / + β d, + O ( δ )] , (6.77)where the singular coefficient β d,s is obtained from thesmall v part of the (infrared divergent) integral by scalingout δ via the change of variable u = v/δ : β d,s = Z ∞ u ( d − / du ( u + 1) / = 2 √ π Γ (cid:18) d (cid:19) Γ (cid:18) − d (cid:19) , (6.78)The linear term is obtained by first subtracting this small v singular (in δ ) part of the integral, and then letting δ → β d, = Z ∞ v ( d − / dv (cid:18) √ v + 1 − (cid:19) = 1 √ π Γ (cid:18) d − (cid:19) Γ (cid:18) − d (cid:19) . (6.79)On the other hand, for 3 < d < db d (0) /dδ is finite, and one finds the leading term β d, simply by setting δ =0. Related to this, the singular term no longer diverges,and it is obtained by first subtracting the β d, (the δ = 0)term, and then again simply scaling δ out of the integral. One may verify that the final results for both coefficientsare identical to (6.78) and (6.79). Integrating (6.77) withrespect to δ , one finally obtains b d ( δ ) = − B d (cid:20) d − β d,s δ ( d − / + β d, δ (cid:21) [1 + O ( δ )] . (6.80)In d = 3 both β d,s and β d, separately diverge. Howeverthe sum is finite, giving rise to a logarithmic dependenceon δ : b ( δ ) = − δ { ln(1 /δ ) + 4 ln(2) − } [1 + O ( δ )] . (6.81)This same result also follows from a direct asymptoticevaluation of the integral (6.76) in d = 3.Defining a T = 0 critical exponent ˜ α via δn MSF ( T =0 , δ ) ∼ δ − ˜ α (i.e., the zero-temperature quantum transi-tion analog of a specific heat exponent), one finds˜ α = (cid:26) − d , d = 30 (log) , d = 3 . (6.82)This result will be modified by critical fluctuationssufficiently close to the MSF–ASF quantum phasetransitions. The resulting behavior of the condensatedepletion δn MSF ( T, ν ) is illustrated in Fig. 4.Before ending this subsection, the order of magnitudeof the MSF zero-temperature depletion (6.74) is exam-ined in light of the identification in Sec. II of the smallparameter γ , Eq. (2.14). In order for the fluctuationcorrection (6.74) to be accurate, it is necessary that it bemuch smaller than the mean field value n (and the same7should be true of the correction δ Ψ relative to Ψ ). Us-ing forms (6.28) for the energy gaps, one obtains(2 mε / ~ ) d/ n ≈ n ( d − / (4 mα/ ~ ) d/ , ∝ ( n r d ) ( d − / , ∝ γ d (4 − d ) / , (6.83)(2 mε / ~ ) d/ n , = n ( d − / (2 mg / ~ ) d/ , ∝ ( n a d ) ( d − / , (6.84)in which g ∝ ( ~ / m ) a d − and α ∝ ( ~ / m ) | r | ( d − / relate the Hamiltonian parameters to the molecular scat-tering length and effective range [see (2.7) and (2.8)] in d dimensions. In (6.83) it has been assumed that ǫ isof the same order of magnitude as it is on the MSF–ASFphase boundary, where ǫ = 2 α √ n .It is seen that the two terms in (6.74) are very differentin character. The second term, estimated via (6.84), isthe standard result for a monatomic Bose gas, and is (for d >
2) small in the dilute limit, n /d a ≪
1. The firstterm, estimated via (6.83), is small (for d <
4) only if γ ≪
1. However, this requires n /d r >>
1, which placesa lower bound on the density. The expansion about meanfield theory presented in this paper therefore requires asufficiently narrow Feshbach resonance (small α ) suchthat the separation of scales r ≫ a exists, and its va-lidity is limited to densities in the intermediate regime1 r d ≪ n ≪ a d . (6.85)This confirms, within an explicit perturbation calcula-tion, the claims made in Sec. II.
2. ASF phase
Next consider the ν > ν c case where both Ψ σ = 0.Computations in the ASF phase are most convenientlyperformed by taking h σ and Ψ σ real and positive at theoutset. Expressions for thermodynamic quantities arequite long and involved, and they will only be sketchedhere.The Bogoliubov free energy density in the ASF is givenby f ASF = H mf [Ψ , Ψ ]+ X σ Z d k (2 π ) d (cid:20) β ln(1 − e − βE ASF k σ )+ 12 ( E ASF k σ − ˜ ε k σ ) (cid:21) (6.86)where H mf [Ψ , Ψ ] takes the form of (4.1), but with, asin (6.66), additional ordering field terms − P σ Re[ h ∗ σ Ψ σ ]now included. The Bogoliubov corrections to the mean field order pa-rameter are then found in the formΨ σ ≡ Ψ σ + δ Ψ σ = − (cid:18) ∂f ASF ∂h σ (cid:19) h σ =0 (6.87)with δ Ψ σ = − X σ ′ (cid:18) ∂f ASF ∂ Ψ σ ′ (cid:19) (cid:18) ∂ Ψ σ ′ ∂h σ (cid:19) h σ =0 (6.88)in which, due to the mean field conditions (4.4), onlythe non-mean field part of f ASF actually contributes to(6.88).Self-consistency, via the mean field equations, but with h / h / h σ and µ -dependence of Ψ σ . The four (complex) equations for( ∂ Ψ σ ′ /∂h σ ) h σ =0 decouple into a pair of separate equa-tions for ( ∂ Ψ σ /∂h ) h = h =0 and ( ∂ Ψ σ /∂h ) h = h =0 : (cid:20) ( ∂ Ψ /∂h σ ) h σ =0 ( ∂ Ψ /∂h σ ) h σ =0 (cid:21) = (cid:18) A BC D (cid:19) − | σ i (6.89)where A = 4 g Ψ B = 2Ψ (2 g Ψ − α ) C = 2Ψ (2 g Ψ − α ) D = 4 g Ψ + α Ψ Ψ (6.90)and | i = (cid:18) (cid:19) and | i = (cid:18) (cid:19) .The free energy derivatives in (6.88), taken at constantvalues of the Hamiltonian parameters µ σ , g σ , α, g , aregiven by ∂f ASF ∂ Ψ σ = X σ Z d k (2 π ) d (cid:20)(cid:18) n k σ + 12 (cid:19) ∂E ASF k σ ∂ Ψ σ − ∂ ˜ ε k σ ∂ Ψ σ (cid:21) (6.91)with the two energies, and the parameters entering them,given by (6.5), (6.21), and (6.22). These will not be eval-uated any further here, except to note that, as in theMSF phase, at zero temperature the leading behavior ofthe integrand at small k is proportional to 1 /E k σ , whileat finite temperature it is proportional to 1 /E k σ . As ex-pected, the acoustic mode therefore generates divergentfluctuation corrections for d ≤ T = 0, and for d ≤ T > .The total density is given by n = − (cid:18) ∂f ASF ∂µ (cid:19) T,ν = Ψ + 2Ψ δ Ψ + 2Ψ + 4Ψ δ Ψ + δn ASF ( T, ν ) ≈ Ψ + 2Ψ + δn ASF ( T, ν ) (6.92)8where, in the second line, the two terms linear in δ Ψ σ subsume the implicit dependence of Ψ σ on µ . This re-sult follows from the fact that, via (4.4), ∂ Ψ σ /∂µ obeys(6.89), but with 2Ψ | i + 4Ψ | i replacing | σ i on theright hand side. The depletion δn ASF ( T, ν ) may be de-rived either from the derivative of the non-mean fieldpart of (6.86) with respect to the explicit µ -dependence(which appears only additively in ˜ ε k σ )—yielding a formidentical to the right hand side of (6.91), but with ∂/∂µ (performed at constant Ψ σ ) replacing ∂/∂ Ψ σ —or asthe total number of uncondensed particles, δn ASF ( T, ν ) = 1 V X k ,σ σ h ˆ a † k σ ˆ a k σ i , (6.93)in which (6.26) connects the ˆ a and ˆ γ operators. Usingeither approach, one obtains δn ASF ( T, ν ) = X σ Z d k (2 π ) d ( (cid:18) n k σ + 12 (cid:19) " ε k + λ + α + 2( λ + α )2 E k σ − ( − σ d k [ λ + α − λ + α )] + (3 t − t ) c (1) k + (3 t + t ) c (2) k E k σ q d k + c (1) k c (2) k − σ ) (6.94)The terms involving c ( σ ) k are of order t , t , t t , andtherefore vanish along the MSF–ASF transition line. Be-cause these expressions involve a number of parameters,such as scattering lengths and the Feshbach resonancecoupling, their final integrated expressions are not veryenlightening without additional (e.g., experimental) in-put. Thus these predictions are not explicitly evaluatedfurther. It is noted only that estimates similar to (6.85)(but now involving all three scattering lengths on theright hand side) may be derived for the range of validityof (6.94). E. Superfluid density
The superfluid (number) density n s is a measure ofthe stiffness Υ s of the order parameter against a long-wavelength spatial gradient in its phase θ ( r ), defined bythe corresponding change in the free energy∆ F s = 12 Υ s ( T ) Z d r |∇ ¯ θ | . (6.95)Expressing the free energy in terms of the superfluid ve-locity v s = ( ~ /m ) ∇ θ , one obtains ∆ F s = m n s R d r | v s | with the standard relation n s = m ~ Υ s . (6.96)For a two-component Bose (atomic and molecular)gas that is considered here, at long length scales thetwo phases are locked by the Feshbach coupling to be θ = 2 θ , i.e., θ σ = σθ . As discussed previously, this isobvious in the ASF state, where the gas is a superfluidwith respect to both atoms and molecules and out-of-phase fluctuations θ − θ / σ = 1) and molecules ( σ = 2), v sσ = ( ~ /m σ ) ∇ θ σ , relative to a stationary boundary arethe same.As outlined above n s is calculated by computing thefree energy change ∆ F s ( k ) in the presence of a uniformphase gradient θ ( r ) = k · r , corresponding to a super-fluid component with uniform velocity v = ~ k /m (andstationary normal component). To this end one imposesphase twist boundary conditions on the field operators:ˆ ψ σ ( r + L ˆn ) = e iσθ ˆ ψ σ ( r ) , (6.97)where L is the system length along a chosen direction ˆn of the phase gradient and θ ≡ θ ( L n ) = k · ˆ n L . Fromthis definition, one obtains Υ s ≡ lim L →∞ L θ ( f θ − f ) = (cid:18) ∂ f θ ∂k (cid:19) k =0 (6.98)where f θ is the free energy density in the presence of thetwist θ , and L → ∞ includes the thermodynamic limitand is to be taken here at fixed θ .Reexpressing the Hamiltonian in terms of periodic fieldoperators ˜ ψ σ = e − iσ k · r ˆ ψ σ , one obtainsˆ H [ ˆ ψ σ ] = ˆ H [ ˜ ψ σ ] + X σ (cid:18) ~ k m σ ˆ N σ + ~ k m · ˆP σ (cid:19) = ˆ H [ ˜ ψ σ ] + ~ k m ˆ N + v · ˆP , (6.99)where ˆP σ = − i ~ Z d r ˜ ψ † σ ( r ) ∇ ˜ ψ σ ( r )ˆ N σ = Z d r ˜ ψ † σ ( r ) ˜ ψ σ ( r ) (6.100)are momentum and number operators for component σ and ˆP ≡ P σ ˆP σ .9From the above form for ˆ H , and defining equations(6.96), (6.98) for n s and Υ s , one observes that the super-fluid density is also given by n s = m ∂j s ∂ ( ~ k ) (cid:12)(cid:12)(cid:12)(cid:12) k =0 , (6.101)or equivalently defined by the relationlim v → j s = n s v , (6.102)where the supercurrent density j s is the expectation valueof the (number) current density operatorˆ j s = 1 V ∂ ˆ H∂ ~ k , = 1 V (cid:18) v ˆ N + 1 m ˆP (cid:19) . (6.103)To compute n s one expands ˆ H [ ˜ ψ σ ] to quadratic orderin the fluctuations ˜ φ σ = ˜ ψ s − Ψ σ and diagonalizes itat a finite k . Because it is odd under k → − k thenew momentum term remains diagonal under the k = 0Bogoliubov transformation: ˆP σ = X k ~ k ˆ a † k σ ˆ a k σ = X k ~ k ˆ γ † k σ ˆ γ k σ . (6.104)Thus the Bogoliubov transformation at finite k is un-changed from that at k = 0, except for a shift in thechemical potential µ → ˜ µ = µ − ~ k / m . The spec-trum, however, does change, but in a simple way˜ E k σ = E k σ + v · ~ k , (6.105)that is in accord with a general requirement for aGalilean-invariant system. Computing the expectationvalue of ˆ j s in (6.103) and using (6.101) [or, equivalentlycomputing the free energy and using (6.98)] one finds n s ( T ) = n − n n ( T ) n n ( T ) = − d X σ σ Z d k (2 π ) d ε k σ dn k σ dE k σ , (6.106)where clearly n s ≤ n , i.e., the normal fluid density n n ≥
0. At zero temperature all excitations are expo-nentially suppressed and the normal fluid density n n van-ishes, giving n s ( T = 0) = n independent of interactions,as required by Galilean invariance. In the normal phase,where E k σ = ε k σ − µ σ , an integration by parts yields n n = n , and n s vanishes as expected for a normal fluid.In contrast, in the MSF phase, despite the absence ofatomic long-range order, at finite temperature there is anontrivial atomic contribution ( σ = 1) to the superfluiddensity (though not to the condensate). The correspond-ing reduction in n s is due to thermally excited, unpairedatoms with a gapped spectrum E MSF k (due to Feshbachcoupling to condensed molecules), that is not simply the free spectrum ε k of the normal state. In the weaklyinteracting limit, away from both T = 0 and T = T c , n s ( T ) /n ≈ − ( T /T c ) d/ is well approximated by theideal gas form, which in turn coincides with the con-densate fraction n ( T ) /n . As usual, sufficiently close T c this result must be modified by critical fluctuationswhich strongly modify (6.106). On the other hand,deviations near T = 0, where the reduction in n s is dom-inated by gapless molecular excitations that are sound-like with E MSF k ∼ √ ε k ε , are accurately described by(6.106), which implies that n n ( T ) ∼ T d +1 . The low tem-perature crossover from T d +1 to T d/ takes place whenthe temperature is high enough that excitation of thehigher energy quasiparticles with quadratic dispersion, E k σ ≈ ε k σ , dominate the thermodynamics. For spe-cific model parameters, the full detailed form of n s ( T )can be straightforwardly evaluated numerically. VII. MSF–ASF PHASE TRANSITION
As has already been seen in Sec. III, many of the prop-erties of the phase transitions appearing in the phase di-agram, Fig. 2 can be deduced based on the nature ofthe underlying symmetry that is spontaneously brokenin the MSF and ASF phases. In particular, there it wasargued that because the MSF exhibits a discrete residualˆ ψ → − ˆ ψ (global phase rotation by π ) symmetry associ-ated with the diatomic nature of the molecule, the MSF–ASF transition at the level of mean field theory is of theIsing type. However, it is important to stress that the Z symmetry that is broken at the MSF–ASF transitiondoes not necessarily imply that the critical properties ofthe transition (beyond a mean field approximation) areof Ising type. One condition for this is that, in additionto the local interactions of Ising symmetry, the derivativeterms associated with kinetic energy (spatial gradients)and (in quantum theory) the Berry phase (“ | ψ | ∂ t θ ” timederivative) terms must reduce to a standard rotationally-invariant (in Euclidean space) d + 1 dimensional gradi-ent term. The other condition is that additional fields(if any) coupled to the Ising order parameter must notmodify the Ising critical behavior, i.e., must be irrelevantin the renormalization group sense.Below these issues are explored in more detail. It willshown that although the first condition is indeed sat-isfied (i.e., for the quantum MSF–ASF transition thescalar Ising order parameter indeed has a standard gradi-ent d + 1-dimensional Lorentz-invariant “elasticity”), theexistence of the Goldstone mode (the phase of the molec-ular order parameter) that couples to the Ising field canhave nontrivial effects and (as was first pointed out byLee and Lee, based on an earlier study by Frey and Ba-lents in a different context ) likely drives the MSF–ASFtransition first order. For the extremely dilute gases ofexperimental interest, this first order behavior is weakand may only be visible very close to the transition.Focusing on a homogeneous trap (a box), the T = 00and finite T MSF–ASF transitions will now be studied inmore detail. This can be most easily done working withthe coherent-state action, S , Eq. (2.17), correspondingto the two-channel Hamiltonian (2.1). The MSF–ASFtransition will be studied from the MSF side, where themolecular field ψ = | ψ | e iθ exhibits massless Goldstonemode phase fluctuations in θ , and small, gapped fluc-tuations in the magnitude | ψ | about the molecular con-densate h ψ i = Ψ . Integrating out the latter leads to astandard superfluid hydrodynamic action S [ θ ] = 12 Υ s Z β ~ dτ Z d r (cid:2) c − ( ∂ τ θ ) + ( ∇ θ ) (cid:3) , (7.1) that controls the (acoustic) fluctuations of θ , withsound speed c MSF given by (6.29), and helicity modu-lus/superfluid density Υ s given by (6.96) and (6.106).The atomic contribution to the action, together withthe key Feshbach resonant atom-molecule coupling, isgiven by S = Z β ~ dτ Z d r (cid:20) ψ ∗ ~ ∂ τ ψ − ψ ∗ (cid:18) ~ m ∇ + µ (cid:19) ψ − α Re (cid:0) | Ψ | e − iθ ψ ψ (cid:1)(cid:21) + S nonlinear , (7.2)in which S nonlinear contains the conventional quartic scattering terms. As discussed earlier, the latter locks the moleculeand atom phase fluctuations such that low energy excitations are governed by θ = θ /
2. Thus it is convenient todefine “dressed” atomic fields ˜ ψ according to ψ ≡ e iθ / ˜ ψ , (7.3)which leads to S = Z β ~ dτ Z d r ( ˜ ψ ∗ ~ ∂ τ ˜ ψ + i | ˜ ψ | ~ ∂ τ θ + ~ m ψ ∗ "(cid:18) − i ∇ + 12 ∇ θ (cid:19) − µ ψ − α Re( | Ψ | ˜ ψ ˜ ψ ) ) + S nonlinear . (7.4)Straightforward analysis shows that near the MSF–ASF transition the minimal coupling to the induced gauge-like field ∇ θ above is irrelevant near a Gaussian fixed point. Dropping this subdominant contribution and writing˜ ψ = ˜ ψ R + i ˜ ψ I in terms of its real and imaginary parts one finds S = Z β ~ dτ Z d r (cid:20) i ψ R + ˜ ψ I ) ~ ∂ τ θ − i ˜ ψ I ~ ∂ τ ˜ ψ R − ˜ ψ R (cid:18) ~ m ∇ + µ R (cid:19) ˜ ψ R − ˜ ψ I (cid:18) ~ m ∇ + µ I (cid:19) ˜ ψ I (cid:21) + S nonlinear (7.5)where the shifted chemical potentials are given by µ R = µ + 2 α | Ψ | , µ I = µ − α | Ψ | . (7.6)This form of S makes it clear that in the presence of themolecular condensate, | Ψ | >
0, positive α reduces the O (2) = U (1) symmetry down to Z , and with µ R > µ I results in ˜ ψ R reaching criticality before ˜ ψ I . Because thecanonically conjugate field ˜ ψ I remains “massive” (non-critical) at the MSF–ASF critical point [defined by wherethe coefficient µ R of ˜ ψ R vanishes, consistent with (1.10)],it can be safely integrated out and leads to a d + 1-dimensional (Lorentz-invariant) action which is even inthe scalar order parameter φ ≡ ˜ ψ R , and whose relevantpart is given by S eff [ θ , φ ] = S SF [ θ ] + S Ising [ φ ] + S int [ θ , φ ] , (7.7) in which S SF [ θ ] = 12 Υ s Z β ~ dτ Z d r (cid:2) c − ( ∂ τ θ ) + ( ∇ θ ) (cid:3) S Ising [ φ ] = Z β ~ dτ Z d r (cid:20) K τ ( ∂ τ φ ) + ~ m ( ∇ φ ) − µ R φ + gφ (cid:21) S int [ θ , φ ] = i Z β ~ dτ Z d r φ ~ ∂ τ θ , (7.8)represent separate superfluid hydrodynamic and Ising ac-tions, together with a Berry phase-like term that cou-ples them. The coefficient K τ ≈ / ( α | Ψ | ) to lowestorder in 1 /α , and the leading φ nonlinearity comes from S nonlinear .1Thus as advertised, if the coupling of the Ising orderparameter φ to the molecular Goldstone mode θ is ne-glected, the T = 0 MSF–ASF transition (near µ R = 0) isindeed in the ( d + 1)-dimensional Ising universality class;at finite T it crosses over to the d -dimensional Ising tran-sition. The Ising transition is well studied, and leads tothe following predictions. At T = 0, for d = 3, upto logarithmic corrections, the mean field theory derivedabove remains an accurate description. On the otherhand in d = 2, the MSF–ASF exponents are nontrivialbut are well-known. For example, standard scaling argu-ments predict: n ∼ | ν − ν c | β I , E (1)gap ∼ | ν − ν c | z I ν I , (7.9)where β I ≈ . z I = 1, and ν I ≈ .
63 are 3d clas-sical Ising exponents. These, together with the rel-evance of T at this quantum critical point, also im-ply a universal shape of the MSF–ASF phase boundary ν c ( n, T ) ∼ ν c ( n,
0) + a T /ν I , as shown in Fig. 2. Onemay hope that when long-lived molecular condensatesare produced, nontrivial behavior of E (1)gap ( ν ) and the fullexcitation spectra may be observed in Ramsey fringes ,and in Bragg and RF spectroscopy experiments .However, as first emphasized and studied by Leeand Lee the existence of θ fluctuations can mod-ify this conclusion sufficiently close to the MSF–ASFtransition—intuitively this follows from the fact that ifone integrates out the superfluid fluctuations, a long-range power law φ - φ interaction (highly anistropic inspace-imaginary time) is generated. Indeed, at T = 0 and d +1 < φ ∂ τ θ coupling term is relevant around theGaussian fixed point, scaling like b (3 − d ) / with increasingrenormalization length scale b , and thus competes withthe Ising φ nonlinearity. The resulting theory embod-ied in S eff has, in fact, a form very similar to that of anIsing model on a compressible lattice, with θ playingthe role of a phonon ~u and the θ − φ coupling scalingsimilarly to the magneto-elastic coupling φ ∇ · ~u . Thelatter model (as well as its Heisenberg generalizations)have been extensively studied. The conclusion of thatwork is that for α I > α I the specific heat expo-nent of the uncoupled model) the magneto-elastic cou-pling leads to runaway flows that has traditionally beeninterpreted as a signature of a fluctuation-driven first or-der transition. For α I < d -dimensional Isingspecific heat exponent is positive for d > one thusconcludes that here too, sufficiently close to the MSF–ASF critical point, the transition is driven first order. Itshould be emphasized that many results in this paper,namely those that refer to thermodynamic and elemen-tary excitation properties of the different phases , not tooclose to the transition lines, remain valid and are unaf-fected in any way by this issue. VIII. BOSE-BCS MODEL
As is shown in this section, the analysis of the Boseatom-molecule system via the two-channel model, pre-sented in the previous subsections can be complementedby Bose-BCS variational approach of the one-channelmodel (2.15). Similar analyses have been presentedpreviously, but it is worth presenting, and general-izing them somewhat, here in a form that can be com-pared to the results of the two-species model (2.1) thathas so far been the focus of this paper. A descriptionentirely in terms of Bose atomic constituents (single-channel model) lends further physical insight into themicroscopic nature of the phases and phase transitions,and facilitates comparisons with BEC–BCS crossover inFermi systems.
Furthermore, the atom-only model should be more appropriate for describingthe case of a broad resonance.The phase diagram of the system is explored as a func-tion of the gas parameter n / a , where n is the atomicdensity and a is atomic scattering length. As discussed inSec. II, experimentally a is controlled by magnetic field-tuned proximity to the Feshbach resonance (diverging atthe resonance) as well as by the atom specific backgroundscattering length, a bg . However, within the one-channelmodel (2.15), this is encoded into the tunable pseudo-potential amplitude g , characterizing atom-atom micro-scopic interaction and connected to experiments via theatomic scattering length, a ( g ). Focusing on a dilutegas, the gas parameter away from the Feshbach resonance(where a → a bg ≈ d , with d the microscopic range ofthe interatomic potential) is taken to be small.For detuning far below the resonance, the attractiveinteraction is strong enough to lead to a deep two-bodymolecular bound state, that corresponds to the appear-ance of a dilute gas of strongly bound compact molecules.In addition, for a large atomic scattering length the sys-tem is known to exhibit Efimov states of trimers, leav-ing the questions of the stability and the nature of thecondensed state open. However, for a sufficiently largerepulsive three-body interaction, and/or away from theFeshbach resonance (short scattering length), the systemis expected to be stable. In this case, at low energiesit will be governed by an effective low energy s-wavemolecule-molecule scattering length a m , correspondingto a repulsive interaction characterized by a molecu-lar pseudo-potential g . Thus, the appropriate effectiveHamiltonian is given by (2.1) with all terms containingthe atomic field ˆ ψ dropped. The theory is identical toan atomic theory of a dilute gas of composite bosons withmass m = 2 m , and chemical potential µ = 2 µ − E b ,where E b is the binding energy. Thus a Bogoliubov anal-ysis provides an essentially exact description. In par-ticular, at T = 0 the system is a vacuum for µ < µ > = p µ /g , density n = n = | Ψ | and lowenergy acoustic excitation spectrum E ( k ) = c ~ k with c = p n g /m = ~ p µ /m .2In the opposite limit of repulsive interactions, nomolecules are present and the system is, conversely, de-scribed by the Hamiltonian (2.1) with all terms contain-ing the molecular field ˆ ψ set to zero. The phenomenol-ogy is again that of a dilute, single component (this timeatomic) Bose gas as described above, with constituentsof mass m , chemical potential µ and interaction g .The focus here is on the interesting intervening re-gion around the transition between these two atomicand molecular superfluid phases. Thus, the behaviorof the single species fluid is explored from the conven-tional g > g < g < H = Z d r (cid:26) − ˆ ψ † ( r ) (cid:18) ~ m ∇ + µ (cid:19) ˆ ψ ( r )+ 12 g ˆ ψ † ( r ) ˆ ψ ( r ) + 16 w ˆ ψ † ( r ) ˆ ψ ( r ) (cid:27) , (8.1)where for simplicity the subscript “1” on the atomic fieldshas been dropped. Though g can have either sign, andthe new physics of primary interest here enters for g < w remains positive to ensure thermodynamic stability.The relation of (8.1) to the more general two-channelmodel (that reduces to it in the wide resonance limit)was summarized in Sec. II, and discussed in detail inRef. 15. A. Variational mean field approximation
In the dilute limit, and away from any phase tran-sitions, the variational approach to be presented, es-sentially equivalent to the Bogoliubov approximation,provides asymptotically exact results. In a standardtreatment the approach relies on the inequality F ≤ F v ≡ F MF + h ˆ H − ˆ H MF i MF , (8.2)between the true free energy F = − k B T ln Tr[ e − β ˆ H ](where ˆ H is the system’s full interacting Hamiltonian)and the variational free energy F v defined in terms of anarbitrary Hamiltonian H MF and its corresponding freeenergy F MF = − k B T ln Tr[ e − β ˆ H MF ]. Here h·i MF is thethermodynamic average with respect to ˆ H MF . Since F v is an upper-bound for F , Eq. (8.2) guarantees that thebetter the choice of ˆ H MF the closer one can approximatethe true free energy with F v . On the other hand to takeadvantage of the variational method one needs to pick a simple enough H MF that the thermodynamic averagesappearing in F v may be calculated explicitly. Thus, H MF is chosen here to be a quadratic Hamiltonianˆ H MF = X k (cid:20) N k ˆ a † k ˆ a k + 12 P k (cid:16) ˆ a † k ˆ a †− k + H . c . (cid:17)(cid:21) − Q √ V (ˆ a + ˆ a † ) , (8.3)with variational parameters N k , P k , Q (chosen real, byabsorbing any extra phase factors into the Bose operatorsif necessary) to be selected to minimize F v .The linear term (required to deal properly with thepossibility of an atomic condensate) is removed via a zeromomentum shift˜ ψ ( r ) = ˆ ψ ( r ) − ψ , ˜ a k = ˆ a k − √ V ψ δ k , , (8.4)with ψ = Q N + P ) . (8.5)Following this, the Bogoliubov transformation˜ a k = u k ˆ γ k − v k ˆ γ †− k ˜ a † k = u k ˆ γ † k − v k ˆ γ − k (8.6)with the choices u k = 1 + v k = 12 (cid:18) N k E k + 1 (cid:19) E k ≡ q N k − P k , (8.7)leads to the diagonal quadratic formˆ H MF = X k (cid:20) E k ˆ γ † k ˆ γ k + 12 ( N k − E k ) (cid:21) + [( N + P ) ψ − Q ψ ] V. (8.8)The two-point averages are easily computed, and aregiven by h ˜ a † k ˜ a k i MF = N k E k (cid:18) n k + 12 (cid:19) −
12 (8.9) h ˜ a † k ˜ a †− k i MF = h ˜ a − k ˜ a k i MF = − P k E k (cid:18) n k + 12 (cid:19) , in which n k = ( e βE k − − is again the Bose occupationfactor. The atomic number density and molecular orderparameter defined by ˆ H MF are therefore given by n MF = ψ + ˜ n MF , Φ MF = ψ + ˜Φ MF , in which˜ n MF ≡ h ˜ ψ † ˜ ψ i MF = Z d k (2 π ) d (cid:20) N k E k (cid:18) n k + 12 (cid:19) − (cid:21) (8.10)˜Φ MF ≡ h ˜ ψ ˜ ψ i MF = − Z d k (2 π ) d P k E k (cid:18) n k + 12 (cid:19) . (8.11)3Using (8.4) to shift the operators in ˆ H , together withWick’s theorem and Eqs. (8.10), (8.11), the variationalfree energy density (8.2) takes the form f v = Z d k (2 π ) d (cid:26) k B T [ n k ln( n k ) − (1 + n k ) ln (1 + n k )]+ ǫ k N k E k (cid:18) n k + 12 (cid:19) − ǫ k (cid:27) + F (˜ n MF , ˜Φ MF , ψ ) , (8.12)in which F ≡ f − ˜ µψ + 12 ˜ gψ + 16 wψ f = − µ ˜ n MF + 12 g (2˜ n + ˜Φ )+ 12 w ˜ n MF (2˜ n + 3 ˜Φ )˜ µ = µ − g (2˜ n MF + ˜Φ MF ) − w (2˜ n + 2˜ n MF ˜Φ MF + ˜Φ )˜ g = g + w (3˜ n MF + 2 ˜Φ MF ) . (8.13)In minimizing (8.12) one may treat the ratio R k = P k /N k , n k , and ψ as independent variables. The deriva-tive with respect to R k yields the simple result R k = γ MF ǫ k − µ MF , (8.14)in which µ MF ≡ − ∂ F ∂ ˜ n MF = µ − g ˜ n MF − w (2˜ n + ˜Φ ) − [2 g + 3 w (2˜ n MF + ˜Φ MF )] ψ − wψ , (8.15) γ MF ≡ ∂ F ∂ ˜Φ MF = ( g + 3 w ˜ n MF ) ˜Φ MF + [ g + 3 w (˜ n MF + ˜Φ MF )] ψ + wψ . (8.16)All k -dependence therefore resides in the energy denom-inator of (8.14).The derivative with respect to n k yields − β ln (cid:18) n k n k (cid:19) = E k = ( ǫ k − µ MF ) N k − γ MF P k E k , (8.17)which, upon substitution of (8.7) and (8.14), yields thesingle particle excitation spectrum E k = q ( ǫ k − µ MF ) − γ , (8.18)with an energy gap E gap ≡ E k = = q µ − γ . (8.19) One identifies in addition, N k = ǫ k − µ MF , P k = γ MF . (8.20)By substituting these results into (8.10) and (8.11), onefinally obtains the self-consistency conditions,˜ n MF = Z d k (2 π ) d (cid:20) ǫ k − µ MF E k (cid:18) n k + 12 (cid:19) − (cid:21) (8.21)˜Φ MF = − γ MF Z d k (2 π ) d E k (cid:18) n k + 12 (cid:19) . (8.22)In the normal phase, ψ = 0, Φ MF = 0, Eq. (8.22)is automatically satisfied, and (8.21) determines n MF .In the MSF phase, ψ = 0 but Φ MF = 0, giving γ MF = (3 g + w ˜ n MF ) ˜Φ MF and leading to a self-consistent(“gap”-like) equation (8.22), that together with (8.21),determines ˜Φ MF and ˜ n MF . Since the integral in (8.22) ispositive, clearly, there is a nontrivial solution, Φ MF = 0only if γ MF <
0, i.e., for sufficiently attractive atomicinteractions ( g sufficiently negative).Finally, the derivative with respect to ψ yields0 = − ˜ µ + ˜ gψ + 12 wψ , (8.23)that, together with (8.21) and (8.22), self-consistentlydetermines ψ inside the ASF phase. The limit ψ → µ c = 0. At this point (and only at thispoint) µ MF = γ MF , and the energy gap (8.19) there-fore vanishes at the ASF–MSF transition. This is consis-tent with the results of the two-channel (atom-molecule)model [see (6.31) and (6.40)], where in both the ASFand MSF phases the atomic branch E k (that corre-sponds to spectrum E k , above) remains gapped exceptat the MSF–ASF transition point. Because, in the one-channel model, molecular excitations do not explicitlyappear in the Hamiltonian [though molecular superfluidorder clearly does appear, via anomalous averages (8.11)],neither does the gapless spectrum of the corresponding(molecular) superfluid mode. However, as will be seen inthe discussion in Sec. VIII B, the presence of these gap-less molecular modes will appear in the calculation of thesuperfluid density.With the above substitutions, the free energy density(8.12) simplifies to f v = Z d k (2 π ) d (cid:26) k B T ln (cid:0) − e − βE k (cid:1) + γ E k n k + ǫ k − µ MF (cid:20) ǫ k − µ MF E k − (cid:21)(cid:27) + F (˜ n MF , ˜Φ MF , ψ ) + µ MF ˜ n MF . (8.24)The similarity of (8.24) to the two-species form (6.66) isevident.The extremum conditions ( ∂f v /∂ ˜Φ MF ) ˜ n MF = 0,( ∂f v /∂ ˜ n MF ) ˜Φ MF = 0, in which the derivatives include4all dependence in E k , µ MF , γ MF , yield precisely the con-straints (8.21), (8.22). As a consequence, one also hasthe relation n MF = − ( ∂f v /∂µ MF ) n MF , Φ MF , in which thederivative includes only the dependence from the combi-nation ǫ k − µ MF (appearing especially in E k ).Equations (8.21), (8.22) and (8.23) are the fundamen-tal results of this section, providing a set of closed re-lations to be solved for n MF , Φ MF and ψ as functionsof µ, T . The stabilizing three-body repulsion w plays noessential role here: the equations remain perfectly welldefined for w = 0. This is because the form (8.3) forthe variational Hamiltonian already precludes the typeof real space system collapse against which w is intendedto stabilize. It should be kept in mind, however, that itis only in the presence of w that the type of superfluid-order considered here would actually occur. Thus, w > w effectively disappears fromthe calculation. As observed in experiments, where w isgenerally quite small and system collapse does eventu-ally occur, such states are expected to be dynamicallymetastable in the dilute limit even when they do not de-scribe true equilibrium. B. Superfluid density
One can infer the existence of gapless molecular exci-tations in the single species model from the existence ofa nonzero superfluid (number) density, n s = ( m/ ~ )Υ s defined (as before) in terms of the change in the free en-ergy, ∆ F s , (6.98) associated with imposition of twistedboundary conditions on ˆ ψ ( r ).As in (6.97)–(6.99), one expresses ˆ H in terms of theperiodic field operator ˜ ψ ( r ) = e − i k · r ψ ( r ), with the resultˆ H [ ˆ ψ ] = ˆ H [ ˜ ψ ] + ε ˆ N + v · ˆP (8.25)where ε = ~ k / m , v = ~ k /m , and ˆ N , ˆP are givenby (6.100) with ˜ ψ replacing ˜ ψ σ and dropping σ summa-tions. In the presence of twisted boundary conditions onegeneralizes the variational Hamiltonian to the formˆ H MF = X k (cid:20) ( N k + M k )˜ a † k ˜ a k + 12 P k (cid:16) ˜ a † k ˜ a †− k + H . c . (cid:17)(cid:21) − Q √ V (cid:16) ˜ a + ˜ a † (cid:17) , (8.26)in which ˜ a k is the Fourier transform of ˜ ψ . All variationalcoefficients depend on the twist wavevector k , but N k , P k are even functions of k as before, while M k is an odd function of k . In fact, as will seen shortly, M k = ~ v · k ,but this is not assumed at the outset. The shifted Bogoli-ubov transformation (8.4)–(8.7) is performed in an iden-tical fashion, with ˜ a k replacing ˆ a k , and u k , v k dependingonly on the even functions N k , P k . As in (6.104), because the odd part is invariant under the transformation, X k M k ˜ a † k ˜ a k = X k M k ˆ γ † k ˆ γ k , (8.27)the Bogoliubov transformation is independent of k . Af-ter the diagonalization, ˆ H MF takes on the following form:ˆ H MF = X k (cid:20) ( E k + M k )ˆ γ † k ˆ γ k + 12 ( N k − E k ) (cid:21) + [ N − P ] ψ − Q ψ ] V. (8.28)The atom number and molecular order parameter densi-ties follow in a form identical to (8.10) and (8.11), butwith the Bose occupation factor given by n k = 1 e β ( E k + M k ) − , (8.29)that includes the odd function M k . The variational freeenergy (8.2) follows in the form (8.12), but (i) with asingle additional term∆ F v ≡ Z d k (2 π ) d v · ~ k n k (8.30)arising from the v · ˆP term in (8.25), and (ii) with µ replaced by µ − ε everywhere, arising from the ε ˆ N termin (8.25).One performs the variational minimization as before,treating n k , R k = P k /N k , and ψ as independent vari-ational parameters. Minimization over R k yields (8.14),but again with µ replaced by µ − ε everywhere in (8.16).Minimization over n k yields E k + M k = q ( ǫ k − µ MF ) − γ + v · ~ k , (8.31)and one immediately recovers (8.18) for E k , and M k = ~ v · k , (8.32)as promised. Minimization over ψ recovers (8.23), againwith µ replaced by µ − ε everywhere.The superfluid density, n s , Eq. (6.96) is proportionalto the second derivative of f v with respect to k , Eq.(6.98). Since there is an implicit k -dependence throughall terms in the free energy, the computation of Υ s ap-pears overwhelming at first sight. However, two observa-tions simplify it enormously: (1) All single k -derivativesof even functions of k are odd functions of k , and hencevanish at k = 0, and (2) the variational property impliesthat all single derivatives of the free energy with respectto R k , n k of ψ vanish identically. Thus, for example,(1) implies that ( ∂ ˜ n MF /∂k ) k =0 = ( ∂ ˜Φ MF /∂k ) k =0 =( ∂ψ /∂k ) k =0 = 0, and hence that cross termssuch as ( ∂ f v /∂ ˜ n MF ∂ ˜Φ MF )( ∂ ˜ n MF /∂k )( ∂ ˜Φ MF /∂k ) van-ish in the limit k →
0. Similarly, (2) impliesthat ( ∂f v /∂n k )( ∂ n k /∂k ) = ( ∂f v /∂R k )( ∂ R k /∂k ) =( ∂f v /∂ψ )( ∂ ψ /∂k ) = 0.5The result is that there are only two contributions tothe superfluid density. The first comes from the µ − ε combination, and yields a term − ∂f v ∂µ ∂ ε ∂k = ~ m n MF , (8.33)where n MF = − ∂f v /∂µ = ψ + ˜ n MF is the number den-sity [the variational conditions again imply that the onlycontributions to the µ -derivative come from the explicitdependence in F —see (8.13)]. The other contribution,interpreted as the normal fluid density, comes from theterm (8.30) and together these give: n s = m ~ Υ s = n MF − n n n n = − lim k → Z d k (2 π ) d (ˆ k · k ) ∂n k ∂k = − d Z d k (2 π ) d ǫ k ∂n k ∂E k , (8.34)in which ˆ k is the unit vector along k , and k has beenset to zero inside n k in the last line. The resemblance to(6.106) is clear. As promised, the superfluid density is fi-nite even though E k is gapped in both the ASF and MSFphases, indirectly indicating the presence of the gaplessmolecular excitations. C. Solutions to the variational equations
In what follows µ MF will be treated as the indepen-dent control parameter, and γ MF viewed as fixed. If onewishes, one may use solutions to (8.16) and (8.17) ob-tained in this way, together with (8.12), to solve in theend for the behavior as a function of µ at fixed g, w . Forsimplicity only T = 0 where n k ≡
1. Onset of molecular superfluidity: vacuum-MSFtransition
In the MSF phase the order parameter constraint(8.22) reduces to1 g MF = − Z d k (2 π ) d E k (cid:18) n k + 12 (cid:19) , (8.35)in which g MF ≡ g + 3 w ˜ n MF , and µ MF is given by the sec-ond line of (8.15). The onset of molecular superfluidityat T = 0 takes place at the value µ = µ at which parti-cles first begin to enter the system, i.e., at zero density.Letting n MF , Φ MF →
0, (8.35) yields − g = Z d k (2 π ) d ǫ k − µ (8.36)In App. C an equation is obtained for the ground stateenergy E of a single molecule in the weak-binding limit where the molecular size is much larger than the diam-eter d of the attractive potential. Comparing (8.36) to(C3) (in the case where ˜ v ( k ) ≡ k Λ ∼ π/d is required in this case to regularize theintegral for d ≥
2) one sees that µ = − E/
2. Thereforethe onset of molecular superfluidity indeed occurs pre-cisely when the chemical potential rises above the molec-ular binding energy per particle. This confirms that H MF correctly captures this limit.
2. MSF–ASF phase boundary and the closing of the singleparticle gap
More generally, the MSF phase single particle spec-trum (8.18) has a gap E gap = E k =0 = q µ − g Φ , (8.37)representing the minimum energy required to create asingle atom excitation. As µ increases from µ this gap shrinks and vanishesat a critical value µ c such that µ c MF = g MF Φ c MF . Thispoint identifies the MSF-to-ASF transition, and at T = 0(8.35) reduces to − g MF = Z d d k (2 π ) d p ǫ k ( ǫ k − µ c MF ) (8.38)Subtracting (8.35) (at T = 0) from (8.38) one obtains0 = Z d d k (2 π ) d " p ǫ k ( ǫ k − µ c MF ) − E k , (8.39)in which the integral is now fully convergent for 2 < d <
4, and the short-scale uv-cutoff may be dropped. The lat-ter is now effectively subsumed into a nonuniversal valueof the critical chemical potential, µ c MF , while critical be-havior, which depends only on deviations of µ MF fromthis value, remains universal.The density integral (8.21) is already convergent for d <
4, and one obtains the critical value n c MF = (cid:18) − mµ c MF ~ (cid:19) d/ I d (8.40)which exhibits a simple power-law relation between n c MF and µ c MF . The substitution u = − ~ k / m A µ c MF hasbeen used to define a dimensionless constant I d ≡ Z d d u (2 π ) d " u + 1 p u ( u + 2) − , (8.41)The quantity r M = p mµ MF / ~ may be interpretedas the background molecular diameter. Thus (8.40)shows that the condition for closing of the atomic gapcorresponds to a criterion of the mean atomic separa-tion r A = n /d MF reaching r M . The MSF–ASF transition6therefore takes place in a regime in which pairs beginto strongly overlap, which is the condition under whichatoms can begin to hop from one molecule to another asthey themselves become delocalized in a sea of extendedpairs (see Fig. 12).
3. Critical behavior near the ASF phase boundary
The approach to ASF–MSF critical point may be an-alyzed by considering small deviations τ = µ c MF − µ MF µ c MF ,f = | Φ c MF | − | Φ MF | | Φ c MF | ,ρ = n c MF − n MF n c MF (8.42) from criticality, all of which will turn out to be positivein the MSF phase. Thus, in fact, although µ increasesas the density increases, µ MF decreases due to the extraterms in (8.15). In terms of these one obtains from (8.21)and (8.35):0 = J ( τ, f ) ≡ Z d d u (2 π ) d " p u ( u + 2) − p ( u + 1 − τ ) − f ρ = R ( τ, f ) ≡ I d Z d d u (2 π ) d " u + 1 p u ( u + 2) − u + 1 − τ p ( u + 1 − τ ) − f (8.43)The detailed analysis of (8.43) is relegated to AppendixE. The results may be summarized as follows.The critical behavior is found to change dramaticallyat dimension d = 3, which is to be expected becausethe upper-critical dimension for a quantum Ising model,controlling the zero-temperature MSF–ASF transition is d Iuc = 3. For d > f ( τ ) = τ h f (1) d + f ( c ) d τ ( d − / + O ( τ ) i ρ ( τ ) = τ h ρ (1) d τ + ρ ( c ) d τ ( d − / + O ( τ ) i , (8.44)where the d -dependent coefficients are given in (E5),(E6), and one may expect H MF to provide an asymp-totically exact description of the MSF–ASF transition.For d < f ( τ ) = τ h f (1) d + f ( c ) d τ (3 − d ) / ( d − + O ( τ ) i ρ ( τ ) = τ h ρ (1) d + ρ ( c ) d τ (3 − d ) / ( d − + O ( τ ) i , (8.45)[see (E12), (E13)], but this approximation must breakdown sufficiently close to the transition point, determinedby a Ginzburg criterion that can be worked out in a stan-dard way. For a dilute gas, relevant to the atomic gasexperiments considered here, the size of the Ginzburg re-gion should be be very small, and therefore it is unlikely that the resulting asymptotic critical behavior (which,as discussed in Sec. VII, is likely actually a fluctuation-driven first order transition) can be observed. The ap-parent exponent singularity at d = 1 is a signature of thelower-critical dimension below which the ordered phase(ASF), and therefore the transition to it, is destabilizedby quantum fluctuations.In d = 3 there are logarithmic corrections: f ( τ ) = τ ( f (1)3 + f ( c )3 ln( τ /τ ) + O (cid:20) ln ln( τ /τ )ln ( τ /τ ) (cid:21)) ρ ( τ ) = τ ( ρ (1)3 + ρ ( c )3 ln( τ /τ ) + O (cid:20) ln ln( τ /τ )ln ( τ /τ ) (cid:21)) , (8.46)with amplitudes given in (E17) and (E18).In all cases, there is a leading analytic dependence,linear in τ , followed by a subleading singular contribution τ − ˜ α , with “quantum specific heat” exponent˜ α = − d − , d >
30 (log) , d = 3 − − dd − , d < . (8.47)Note that ˜ α here differs from that in (6.82) which, al-though computed for a slightly different quantity, never-theless reflects the same universal energy singularity. Thechange is not due to a change in the universality class, but7 x x’ φ ( , ) −1/3 d ~ n(a) (c)(b) FIG. 12: Schematic illustration of the single species MSF–ASF transition, accompanying the BEC–BCS crossover. (a) Forsufficiently strongly attractive interactions, the extent of the molecular wavefunction φ ( x , x ′ ) is much smaller than the inter-molecular separation d , and is essentially determined only by two-body physics. Interactions between molecules are weak, andonly rarely do they overlap sufficiently to exchange atoms. At low temperatures the molecules Bose condense, but coherentintermolecular hopping that would lead to atomic Bose condensation is suppressed. (b) The MSF–ASF transition takes placewhen the size of the molecular wavefunction (which is strongly renormalized by many body effects) becomes comparable to d and molecules begin to significantly overlap. Coherent atomic hopping occurs over ever larger distances as the overlap increases,and the divergence of this distance signifies the MSF-to-ASF transition. (c) Deep in the ASF phase, the size of the molecularwavefunction is much larger than d , and essentially loses its physical meaning: coherent hopping of atoms over the entire systemleads to atomic Bose condensation, and one can no longer identify any given pair of atoms with a particular molecule. Instead,the physics is described by a BCS-type many body wavefunction, embodied in the Hamiltonian (8.3), which maintains strongnonlocal, pair correlations in the absence of identifiable molecules. rather to the extra constraint embodied in the first lineof (8.43), which essentially reflects the existence of only asingle species. It is well known that such constraints leadto a “Fisher renormalization” α → − α/ (1 − α ) whenever α >
0, while leaving it unchanged if α < If one were to enforce the total density con-straint n = 2 n + n , the condensate depletion (6.72)would also display the Fisher-renormalized exponent.One may use a similar analysis to extend these resultsinto the ASF phase, and to positive temperatures. How-ever, rather than exploring (difficult to probe) criticalbehavior, the aim here is mainly to demonstrate the ex-istence of the same three phases (N, MSF, ASF) illus-trated in Fig. 2; the role of the detuning ν is played hereby the s-wave interaction parameter g , or more properlythe corresponding dimensionless measure of scatteringlength, i.e., the gas parameter n / a . Despite the com-plete loss of molecular identity in the broad resonance(single-channel) model as the ASF phase is approached,the topology of the phase diagram and critical behavior(accounting appropriately for constraints) is the same asfor the two-channel model. IX. TOPOLOGICAL EXCITATIONS
In the previous sections a description of a (s-wave)resonantly interacting atomic Bose gas was presented, formulated in terms the atomic and molecular super-fluid (condensate) order parameters Ψ and Ψ , andcorresponding fluctuations in the two ordered, ASF andMSF states were studied, and characterized the associ-ated T = 0 and finite T phase transitions.In this section, a complementary description of thistwo-component (atoms and diatomic molecules) Bose gasis presented, in terms of topological excitations in theMSF and ASF phases. These will be shown below to bevortices and domain walls. Descriptions of phases andphase transitions in terms of topological excitations hasa long and successful history, with ordinary vortices insuperfluids and superconductors, dislocations and discli-nations in crystalline solids, and domain walls in Ising fer-romagnets being only a few most prominent examples. The importance of this description is two-fold. Firstly,topological defects are true nonlinear excitations of thesystem and thus are essential for a full characterizationof the response of an ordered state to an external per-turbation. For example, a rotated neutral superfluid (ora superconductor in the presence of a sufficiently strongmagnetic field) responds by nucleating quantized vorticesthat carry discrete units of fluid’s angular momentum(magnetic flux). Secondly, fluctuation-induced (quantumor thermal) topological defects provide a dual charac-terization of phases and transitions between them thatcomplements their description in terms of a Landau-typeorder-parameter. For example, a superfluid-to-normaltransition can be understood through a dual model of8fluctuation-induced proliferation of vortices, with the su-perfluid state acting as a vortex vacuum (or an insulator)and the normal state as a vortex condensate. In addition to simply playing a complementary role,such dual vortex “disorder parameter” descriptions arealso a powerful way to characterize subtly ordered phasesthat do not allow a direct Landau order parameter de-scription. The most prominent examples of this are2d ordered phases with a continuous symmetry thatare (usually ) “forbidden” by the Hohenberg-Mermin-Wagner theorem to exhibit true long-range orderand thus cannot be characterized by a condensate or-der parameter. Such “ordered” phases (e.g., a 2d su-perfluid or a 2d crystalline solid) are in fact disordered,only distinguished from the short-ranged (exponentially)correlated fully disordered states by a quasi-long-ranged(QLR) order with correlation functions falling off as apower-law. Descriptions of such QLR-ordered phases andtheir transition to fully disordered states is best done interms of a proliferation of topological defects, e.g., vor-tices in 2d superfluids. In higher dimensions, a descrip-tion in terms of a proliferation of topological defects canalso be more effective, as for example found in disorder-ing of a 3d type-II superconductors by proliferation ofvortex loops.
Such dual topological defect descriptions, in addi-tion to providing added physical insight, provide impor-tant complementary computational tools for the study-ing these phenomena. With this motivation, topologicaldefects in the ASF and MSF will now be considered.
A. Atomic superfluid
Since the fully ordered ASF state has two nonzero or-der parameters Ψ , Ψ , there are interesting featuresof the topological excitations generated by the (Fesh-bach) coupling between them. The thermodynamics ofthe state can be conveniently and equivalently describedin terms of the local magnitudes and phases of its two(atomic, σ = 1, and molecular, σ = 2) coherent-statefields ψ σ = √ n σ e iθ σ . (9.1) In terms of these, the real-time coherent-state action cor-responding to the Hamiltonian (2.1) takes the form S = S + S + S (9.2) S = Z dtd r (cid:20) ~ n ∂ t θ + ~ m n ( ∇ θ ) − µn + 12 g n (cid:21) S = Z dtd r (cid:20) ~ n ∂ t θ + ~ m n ( ∇ θ ) − (2 µ − ν ) n + 12 g n (cid:21) S = Z dtd r h g n n − αn n / cos(2 θ − θ ) i , where terms involving ∇ n σ , that are less important thanthe finite compressibility g σ terms, have been dropped.From the action S the phase diagrams of Sec. IV andBogoliubov modes of Sec. VI can be straightforwardlyreproduced in terms of the canonically conjugate densi-ties n σ and phases θ σ .Note now that the mean field equations of motionfor the phases θ σ , namely the Euler-Lagrange equations δSδθ σ = 0, are given by ∂ t n + ~ m ∇ · ( n ∇ θ ) = 2 J ~ sin(2 θ − θ ) , (9.3) ∂ t n + ~ m ∇ · ( n ∇ θ ) = − J ~ sin(2 θ − θ ) , (9.4)where the internal Josephson coupling between atomicand molecular superfluids, J = αn √ n , (9.5)is proportional to the Feshbach resonance amplitude α .These can be combined to derived the total boson number n = n + 2 n conservation (continuity) equation ∂ t n + ∇ · j = 0 , (9.6)where the total number current j = j + 2 j naturallyconsists of the atomic and molecular contributions, j = ~ m n ∇ θ , (9.7) j = ~ m n ∇ θ . (9.8)Observe that, due to the atom-molecule Feshbachresonant interconversion captured by the “current” J sin(2 θ − θ ) on the right hand sides of (9.3) and (9.4),as expected, n and n are not independently conserved.The microscopic action S in (9.2) is not completelygeneric. A more general model (that can be obtainedeither based on symmetry or by incorporating quantumand thermal fluctuations) includes an atomic-molecularcurrent-current interaction of the form δS = 12 K |∇ (2 θ − θ ) | , (9.9)9arising from coarse-graining of the action S in the pres-ence of the Feshbach resonance cosine nonlinearity. Theform of this term ensures that total atom conservationembodied in the continuity equation (9.6) remains satis-fied.Fluctuations lead to corrections to the mean field equa-tions of motion. At the hydrodynamic level, in whichonly the dynamics of slow, large scale distortions of thefields are considered, these corrections may be embodiedsimply in renormalization of the terms appearing in S .Most significantly, in this limit phase fluctuations domi-nate, and fluctuations in n σ may be subsumed into renor-malized stiffness coefficients. Thus, the squared phasegradient terms undergo the replacement ~ m σ n σ |∇ θ σ | → K σ |∇ θ σ | , K σ ≡ ~ m n sσ , (9.10) which replaces the fluctuating number density n σ by theatomic and molecular superfluid (number) densities, n sσ ,to be distinguished from the corresponding, quite dis-tinct (see Sec. VI) condensate fractions n σ . As in asingle-component superfluid at T = 0, Galilean invari-ance enforces the condition n s + 2 n s = n, for T = 0 , (9.11)namely that the total superfluid density equals the totaldensity.With this preface, the focus will now be on the finitetemperature classical limit, ignoring quantum dynamicsthat are left to future investigation. The model to bestudied is defined by the “hydrodynamic” energy density E = 12 K ( ∇ θ ) + 12 K ( ∇ θ ) + 12 K |∇ (2 θ − θ ) | − J cos(2 θ − θ ) , (9.12)and is valid in any region where the phases θ σ are welldefined, but must be supplemented by core energies inregions where an order parameter magnitude vanishes.The four coefficients are all renormalized hydrodynamicparameters that depend on the chemical potentials, andother microscopic parameters, and have their own non-trivial critical behavior. The detailed knowledge oftheir exact values (some of which have been computedin the dilute limit in earlier sections of this paper) is notrequired to understand general features of topological ex-citations.
1. Vortices in the ASF
In the absence of the atom-molecule couplings, K and J , the superfluid admits independent atomic andmolecular vortices—pictured in Fig. 13. Focusing forsimplicity on d = 2, these are point defects in the atomicand molecular superfluid order parameters, around whichtheir respective phases wind by an integer-multiple of 2 π as the point is encircled. As usual, this quantization con-dition arises from the single-valuedness of the superfluidorder parameter away from the vortex core (located atposition r σ ): I r σ ∇ θ σ · d r = 2 πp σ , (9.13)with “charge” p σ .Imposing this topological constraint and minimizingthe energy E at K = J = 0, one obtains indepen-dent atomic and molecular superfluid velocities around the corresponding vortices v = p ~ m ˆ ϕr , v = p ~ m ˆ ϕr , (9.14)with integer charges p σ . Equivalently, the superfluidphases θ σ are given simply by integer multiples of theazimuthal coordinate angles ϕ σ (measured with respectto an origin chosen at the vortex core positions r σ ), with θ σ = p σ ϕ σ . As usual in 2d, vortex energies grow loga-rithmically with system size L , E ( v ) σ = p σ K σ π ln( L/ξ σ ) , (9.15)where ξ σ is the vortex core size set by the correspondingcoherence lengths.In the presence of the inter-superfluid couplings K , J no general solution is available. However, considerableinsight can be obtained by analyzing limiting regimes.It is clear from the energy density E , Eq. (9.12), thatto avoid extensive (scaling with system size) energy costproportional to J (and/or K ), the two phases are onaverage locked together according to θ = 2 θ . (9.16)Hence the energy is minimized when positions of theatomic and molecular vortices coincide, and their topo-logical charges (winding numbers) are related by p =2 p . Thus an elementary p = 1 vortex in the atomicorder parameter will be accompanied by a spatially co-incident molecular vortex of topological charge that isdouble its elementary value, p = 2.In contrast, an elementary p = 1 molecular vortex isenergetically significantly more costly due to incompat-ibility of the atomic order parameter single-valuedness0 (a) (b) FIG. 13: (a) A unit vortex in the order parameter field Ψ ( r )represented as a 2 π -rotation in the vector field M ( r ). (b) Inthe Ψ complex plane, a unit vortex in Ψ ( r ) is representedby a π -rotation in the double headed vector field Q ( r ). Oneidentifies the Feshbach resonance coupling − α Re[Ψ ∗ Ψ ] asthe energetic tendency for spatial alignment of M ( r ) and Q ( r ). Details of the mapping between these two representa-tions are provided in Appendix A. constraint and the Feshbach resonance constraint (9.16).It is clear that energetically there are two competing,least costly, ways to accommodate this frustration. One,illustrated in Fig. 14(a), is with a spatially coincidenthalf-integer ( p = 1 /
2) atomic π vortex, that requiresthat Ψ vanish (atomic component of the gas is normal)along a ray emanating from the location of the vortexcore, and across which θ exhibits a π jump discontinu-ity. The cost of such a defect clearly scales linearly withthe length, L of the defect ray (more generally, as L d − in d dimensions) and is dominated by the loss of the con-densation energy along the linear defect.Another competing possibility, illustrated in Fig.14(b), is an atomic p = 1 vortex localized on the elemen-tary molecular vortex, but (in contrast to the noninter-acting case where the vortex is isotropic with θ ( ϕ ) = ϕ )the atomic phase winding is highly anisotropic, with θ ( ϕ ) ≈ θ ( ϕ ) / , ≈ ϕ/ , (9.17)outside a narrow domain wall strip. The atomic phasemakes up the remaining π deficit angle (required bysingle-valuedness of the atomic order parameter), byrapidly winding across the domain wall of width set by J and a combination of phase stiffnesses K σ .It is clear that the first scenario is the limiting case ofthe second configuration, with large J and small K σ , suchthat the wall width is microscopic (formally smaller than ξ σ ) and the corresponding energy comparable to conden-sation energy, thus driving the discontinuity ray normal.The resulting energy in both cases clearly grows linearly(as L d − in d dimensions) with the length of the domainwall ray, and the energy scale is set by the minimum ofthe condensation energy or line tension, with the lattergiven by the geometrical mean of J and a combinationof the K σ (see below). FIG. 14: A schematic illustration of a 2 π (elementary unit)molecular vortex that induces a π (half-unit) atomic vortex,that in turn induces a domain-wall ray. In (a) the wall width, ξ is smaller than the coherence length and the energy cost perunit of wall length exceeds that of the condensation energy,thus leading to a “normal” (Ψ = 0) domain wall. In (b) thesuperfluid stiffness is large and Feshbach resonance is narrow(small α ) leading to a wide domain wall (width ξ exceedingthe coherence length), with an interface that is in the ASFstate and Ψ only slightly suppressed below its bulk value.The wall structure is illustrated in more detail in Fig. 15.
2. Domain walls in the ASF
It has been argued that the domain wall with energylinear in its length L (more generally, growing as its sur-face area L d − in d dimensions) is another type of a topo-logical excitation in the ASF. Although in the previoussubsection it emerged as a necessary string component ofa p = 1 molecular vortex, the existence of a domain wallexcitation can be understood on more general grounds.Quite similar to domain walls in an Ising ferromagnet,here too it is a defect that separates ordered domainsassociated with two physically distinct configurations ofthe Ising order parameter in the ASF that spontaneouslybreaks the Z symmetry of the MSF state. In terms ofphases θ σ the two domains correspond to two solutions θ ( n )1 = θ / nπ, n = 0 , , (9.18)of the constraint in (9.16), that are associated with twovalues of the atomic order parameter Ψ (0 , = e iθ (0 , = ± e iθ / , pictorially illustrated in Fig. 15.A detailed solution for a domain wall can be straight-forwardly worked out. To this end, the key (internal)Josephson nonlinearity associated with the Feshbach res-onance is isolated by a convenient change of phase vari-ables to new phase fields θ and φ : θ = 12 (2 θ + θ ) (9.19) φ = 12 (2 θ − θ ) , (9.20)corresponding to the in-phase and out-of-phase fluctua-tions of θ = ( θ + φ ) / θ = θ − φ phases.1 FIG. 15: Details of the domain wall structure, separatingΨ (0 , = ± θ (0)1 = θ / θ (1)1 = θ / π , respectively (illustrated here for K θφ /K θ = 1). Across the wall the atomic condensate phase, θ , winds by π relative to the molecular condensate phase(double-headed arrow angle), θ . In terms of θ and φ , the energy density is given by E = 12 K θ ( ∇ θ ) + 12 K φ ( ∇ φ ) − K θφ ∇ θ · ∇ φ − J cos(2 φ ) , (9.21)with K θ = 14 K + K (9.22) K φ = 14 K + K + 4 K (9.23) K θφ = − K + K . (9.24)The corresponding saddle-point equations δ E /δθ = δ E /δφ = 0 are given by − K θ ∇ θ + K θφ ∇ φ = 0 , (9.25) − K φ ∇ φ + K θφ ∇ θ + 2 J sin 2 φ = 0 . (9.26)Eliminating θ via (9.25) reduces (9.26) to the well-studiedsine-Gordon equation for φ , − K ∇ φ + 2 J sin 2 φ = 0 , (9.27)where K = K φ − K θφ /K θ . (9.28)For a straight domain wall oriented along x , definedby the boundary conditions φ dw ( y → −∞ ) = 0 and φ dw ( y → + ∞ ) = π , the solution is given by φ dw ( y ) = 2 arctan (cid:16) e y/ξ (cid:17) , (9.29)illustrated in Fig. 15. The domain wall width is given by ξ = p K/J. (9.30) The associated θ dw ( y ) and corresponding θ dwσ ( y ) solu-tions can now be easily obtained from (9.25), (9.19), and(9.20): θ dw ( y ) = K θφ K θ φ dw ( y ) (9.31) θ dw = 12 (cid:18) K θφ K θ + 1 (cid:19) φ dw ( y ) , (9.32) θ dw2 = (cid:18) K θφ K θ − (cid:19) φ dw ( y ) . (9.33)Using the above expressions inside the energy den-sity E , Eq. (9.21), and taking advantage of the Euler-Lagrange equations (9.25), (9.26) one finds that the do-main wall energy is given by E dw = Z dxdy K ( ∇ φ dw ) , (9.34)= Z dxdy J [1 − cos(2 φ dw )] , (9.35)= σ dw L x , (9.36)with domain wall line tension (energy per unit of length), σ dw = 4 √ JK. (9.37)
3. Point-to-“dumbbell” atomic vortex transition
As is clear from the discussion in Sec. IX A 1, in theASF a 2 π ( p = 1) atomic elementary vortex is drivenby the Feshbach resonance (internal Josephson) coupling J to be accompanied by a 4 π ( p = 2) molecular doublevortex. In the limit where J ≫ K , corresponding to abroad Feshbach resonance, and deep in the ASF state, thetwo superfluids are strongly coupled, and will behave as asingle-component conventional atomic superfluid. Thusthe 2 π atomic and 4 π molecular vortices must spatiallycoincide. This leads to an isotropic topological defectwith energy (measured relative to the background energy − J of the uniform state) given in 2d by E (point) v = E (point) c + πK ln( L/ξ ) + 4 πK ln( L/ξ ) , (9.38)where E (point) c = E (2 π )1 c + E (4 π )2 c (9.39)consists of the atomic (2 π ) and molecular (4 π ) vortexcore energies.Such a concentric, isotropic vortex configuration mini-mizes the Feshbach resonance energy. However, becauseit involves a 4 π ( p = 2) molecular vortex that is doublethe elementary charge, it raises the large-scale part ofthe kinetic energy by 2 πK ln( L/ξ ) over the energy of atopologically equivalent vortex configuration consistingof two elementary 2 π ( p = 1) molecular vortices—see(9.15). As will be shown below, this can drive the split-ting of the 4 π double molecular vortex into its elementary22 π constituents. This is driven by the fact that, in the ab-sence of the atomic component, two elementary vorticesrepel via a potential V (2 π − π )2 ( R ) = − πK ln( R/ξ ),where R is the separation. On the other hand, as shownin Sec. IX A 1, in the ASF phase an elementary 2 π molec-ular vortex is driven by the internal Josephson coupling J to be accompanied by a ( p = 1 / π vortex and a domain-wall string defect. Thus, in theASF state the logarithmic repulsion of two 2 π molecularvortices is arrested by the confining (Josephson coupling)domain-wall energy that, according to (9.36), grows lin-early with separation R . Hence, the energy E ( point ) v of apoint vortex (consisting of coincident 2 π atomic and 4 π molecular vortices) must be compared to a topologicallyequivalent “dumbbell” configuration split by a separation R into two units, each consisting of coinciding π atomicand 2 π molecular vortices—see Fig. 5. The energy of the dumbbell configuration is estimatedas E (dmbl) v ≈ E (dmbl) c + πK ln L/R + 4 πK ln L/R − πK ln( R/ξ ) − πK ln( R/ξ ) + σ dw R (9.40) ≈ E (dmbl) c + πK ln( L/ξ ) + 4 πK ln( L/ξ ) − πK ln( R/ξ ) − πK ln( R/ξ ) + σ dw R, (9.41)where the core energy is E (dmbl) c = 2 h E ( π )1 c + E (2 π )2 c i . (9.42)The first two logarithmic terms in (9.40) estimate theenergy associated with the total 2 π atomic and total 4 π molecular topological charges outside the dumbbell ofsize R . The second two logarithmic terms in (9.40) givethe energy of two π atomic vortices and two 2 π molecularvortices, including their repulsive interaction at separa-tion R . Finally, the last term accounts for the energyof the domain wall in the atomic superfluid. Accuracy ofthe estimate requires that the system size be much largerthan the dumbbell size, which in turn must be larger thanthe microscopic scale: L ≫ R ≫ ξ , ξ .To estimate the optimum size R of the dumbbellvortex configuration, one minimizes E (dmbl) v ( R ) over R ,yielding R ≈ π K + 4 K √ JK , (9.43) ≈ π ~ q mαn / r n n , (9.44)where (9.37) has been used for σ dw , and the estimate inthe last line is made by ignoring depletion effects, validin the dilute, weakly interacting limit.It is clear that there is a range of parameters K σ and J such that the energy E ( dumbbell ) v ( R ), Eq. (9.41), of the dumbbell vortex is lower than that, Eq. (9.38), of thepoint vortex configuration. In this case, a 2d system willundergo a transition to a state in which the 2 π atomicvortices are in a dumbbell π − π configuration (locked to a2 π − π molecular vortex pair). Because there is no sym-metry change associated with a transition into a statewhere vortex dumbbells are randomly oriented, one ex-pects this transition to generically be first order. Sincethe energy balance between the two competing states in-volves core energies, a more detailed microscopic anal-ysis (that is not pursued here) is necessary to pinpointthe location of the transition. However, it is clear fromthe structure of E ( dumbbell ) v ( R ) and R , that such transi-tion takes place for a sufficiently large domain-wall width p K/J ≫ ξ σ , where the line tension is small and same-sign vortex repulsion is large. One thus expects such atransition in narrow (small α ) Feshbach resonance sys-tems. B. ASF-MSF as a confinement-deconfinementtransition
Observe that the atomic condensate density n van-ishes as the detuning ν decreases towards a critical value ν c . Thus the 2 π dumbbell length R ≈ r K α πn / n / → ∞ , for ν → ν c , (9.45)diverges along with the associated domain-wall width, ξ . Therefore, the ASF–MSF transition in d = 2 canbe complementarily described as a 2 π molecular vortexdeconfinement transition. While 2 π molecular vorticesare confined by a linear potential inside the ASF state, inthe MSF state this confining potential (in 2d) is replacedby a much weaker logarithmic potential, that binds each2 π molecular vortex to its oppositely charged partner. C. Molecular superfluid
Since molecular 2 π vortices only appear as neutraldipoles in the MSF phase, the state is characterizedby long-range order in the molecular order parameter,Ψ ∼ h e iθ i (however, as usual in 2d, at finite T ,Ψ itself vanishes, while the molecular helicity modu-lus, or superfluid density, n s remains finite). On theother hand, a deconfined domain wall (across which theatomic phase θ jumps by π ) leads to a vanishing of theatomic order parameter, Ψ ∼ h e iθ i = 0. The MSFstate exhibits ordinary molecular 2 π point vortices, alongwith the atomic and molecular Bogoliubov quasiparti-cles discussed in Sec. VI. It is easy to see that, analo-gous to the conventional BCS superconductor, here tooan atomic Bogoliubov quasiparticle (that is gapped) ac-quires a phase of π upon encircling a molecular 2 π vortex.Thus, these two excitations interact strongly with each3other, with statistical-like interactions that can be cap-tured by a Chern-Simons field theory. X. SUMMARY AND CONCLUSIONS
In this paper the thermodynamics of a resonant atomicBose gas has been studied. Working within a two-channelmodel, formulated in terms of bosonic atoms and theirdiatomic molecules, the complete phase diagram of thesystem has been worked out as a function of temperatureand detuning, the properties of the phases, and the na-ture of quantum and classical phase transitions betweenthem, studied. This analysis was supplemented by a vari-ational calculation on a one-channel model, whose salientresults appear in Sec. I B.A most notable feature is the appearance of two dis-tinct superfluid phases, ASF and MSF, separated by anIsing type transition. These are distinguished by therespective presence and absence of atomic off-diagonallong-range order, atomic (gapped and gapless) Bogoli-ubov spectra, and the nature of topological excitations.In addition to a distinction based on the atomic mo-mentum occupation number, these phases can be dis-tinguished through the domain wall excitations in theASF (which separate regions in which the atomic phasealigns with the molecular phase in the two possible dif-ferent ways), characteristic of the broken discrete Isingsymmetry.
Acknowledgments
L.R. thanks Victor Gurarie and Subir Sachdev for dis-cussions, and the Kavli Institute for Theoretical Physicsfor its hospitality during the “Strongly Correlated Phasesin Condensed Matter and Degenerate Atomic” workshop,supported in part by the NSF under grant No. PHY05-51164. The authors acknowledge financial support by theNSF under grant No. DMR-0321848 (LR, JP), NIST un-der the NRC fellowship (JP), and NASA under contractNo. NNC04CB23C (PBW).
APPENDIX A: DETAILS OF CONNECTION TOPOLAR-NEMATIC ORDERING
In this Appendix a connection, stated in Sec.III, be-tween the two complex scalar order parameters, charac-terizing phases of a resonant atomic Bose gas, to thoseof a vector-tensor model of polar, nematic liquid crystalsis elaborated.In a thermodynamic description, the order parametersare derived from the full free energy density f AM viaderivatives with respect to their conjugate fields:Ψ σ ( r ) = − (cid:18) ∂f AM ∂h ∗ σ (cid:19) h σ =0 = h ˆ ψ σ ( r ) i . (A1) This prescription is unambiguous for the two-speciesmodel (2.1), but in the one-species model (2.15) or (8.1),Ψ remains to be defined. One expects a molecular com-posite operator of the formˆ ψ † m ( r ) = Z d r φ ( r ) ˆ ψ † ( r + r /
2) ˆ ψ † ( r − r / . (A2)to play the role of ˆ ψ , where φ ( r ) is the molecular wave-function. Therefore, to investigate molecular superfluidordering, one is motivated to look at the anomalous cor-relation function Φ ( r , r ′ ) = h ˆ ψ ( r ) ˆ ψ ( r ′ ) i . (A3)Since the fundamental object is a two-point function,one generally lacks a unique definition of the one-pointquantity Ψ ( r ). However, if the molecular size is muchsmaller than their separation, in the spirit of the coarsegraining picture of the molecular operator ˆ ψ , one coulddefine Ψ ( r ) = Z d r ′ φ ( r − r ′ )Φ ( r , r ′ ) (A4)in which the two-particle molecular wavefunction φ isused to weigh the local volume average.A complex-scalar atomic superfluid order parameter isclearly isomorphic to a 2d vector order parameter M = √ , ImΨ ) whose components are the averages M , ( r ) = h ˆ Q ( r ) i , M , ( r ) = h ˆ P ( r ) i (A5)of the corresponding conjugate Hermitian operators,ˆ Q ( r ) = 1 √ h ˆ ψ ( r ) + ˆ ψ † ( r ) i ˆ P ( r ) = 1 i √ h ˆ ψ ( r ) − ˆ ψ † ( r ) i (A6)obeying the commutation relation [ ˆ Q ( r ) , ˆ P ( r ′ )] = iδ ( r − r ′ ). Phase symmetric combinations of Ψ (i.e., prod-ucts of its complex conjugate with itself) correspond torotation invariant combinations (i.e., dot products) of M . Similarly, one defines the 2d conjugate field vec-tor H = √ (Re h , Im h ).Since the molecular order parameter is fundamentallya two-point correlation function, one is motivated to ex-amine the tensor Q ( r , r ′ ) = (cid:20) h ˆ Q ( r ) ˆ Q ( r ′ ) i h ˆ Q ( r ) ˆ P ( r ′ ) ih ˆ P ( r ) ˆ Q ( r ′ ) i h ˆ P ( r ) ˆ P ( r ′ ) i (cid:21) (A7) ≡ Q s ( r , r ′ ) + Q a ( r , r ′ ) + 12 Tr[ Q ( r , r ′ )] Q s ( r , r ′ ) = 12 (cid:8) Q ( r , r ′ ) + Q T ( r , r ′ ) − Tr[ Q ( r , r ′ )] (cid:9) = (cid:20) ReΦ ( r , r ′ ) ImΦ ( r , r ′ )ImΦ ( r , r ′ ) − ReΦ ( r , r ′ ) (cid:21) . (A8)4and is therefore seen to be precisely equivalent to theanomalous correlation function Φ ( r , r ′ ). The antisym-metric part Q a ( r , r ′ ) = 12 (cid:2) Q ( r , r ′ ) − Q T ( r , r ′ ) (cid:3) = (cid:20) − Im G ( r , r ′ )Im G ( r , r ′ ) 0 (cid:21) . (A9)and the trace Tr[ Q ( r , r ′ )] = Re G ( r , r ′ ) (A10)are given by the real and imaginary parts of the usual(non-anomalous) two point correlation function G ( r , r ′ ) = h ˆ ψ ( r ) ˆ ψ † ( r ′ ) i , (A11)and are therefore phase invariant scalars. The informa-tion about the molecular order parameter therefore liesentirely in Q s . One similarly defines the conjugate fieldtensor H = 12 (cid:18) Re h Im h Im h − Re h (cid:19) . (A12)By substituting Q s ( r , r ′ ) into (A4) one obtains a def-inition of the local tensor order parameter Q ( r ). Thefollowing identities now follow:12 M = | Ψ | H · M = Re[ h ∗ Ψ ]12 Tr (cid:0) Q (cid:1) = − det [ Q ] = | Ψ | , Tr( H Q ) = Re[ h ∗ Ψ ]12 M T Q M = Re (cid:2) Ψ ∗ Ψ (cid:3) . (A13)This demonstrates that the phase invariant combinationsin the Ginzburg-Landau free energy correspond to 2d ro-tation invariant combinations in the vector-tensor repre-sentation.A symmetric traceless tensor order parameter is famil-iar from a theory of nematic liquid crystals, encoding theheadless-arrow nature of the nematic state of anisotropicmolecules. Using the “dictionary”, Eqs.(A13), in thevector-tensor representation, M , Q , the mean fieldHamiltonian (4.1) takes the form H mf = − µ M + g M − H · M − µ Tr (cid:0) Q (cid:1) + 18 g (cid:2) Tr (cid:0) Q (cid:1)(cid:3) − Tr( H Q )+ 14 g M Tr (cid:0) Q (cid:1) − α M T Q M , (A14)representing the theory of an interacting vector and ne-matic order parameters. In the current scalar superfluidcontext both have two components, but (A14) is clearly not limited to this case. The conjugate field H is theanalogue of a magnetic field, while H is the analogue ofa nematic liquid crystal polarization field.The “double-headedness” of the nematic order param-eter is exhibited via the eigenvector decomposition Q ( r ) = q [ ˆn ( r ) ˆn ( r ) − ˆm ( r ) ˆm ( r )] , (A15)in which q ≥ ˆm , ˆn are or-thonormal unit eigenvectors characterizing the nematicorder of the MSF. Clearly Q is invariant under sign re-versal of the unit vectors. Thus, although q and ˆn com-pletely define Q , both ˆn and − ˆn characterize the samestate.To make contact with the complex molecular or-der parameter, let ˆn = [cos( θ n ) , sin( θ n )], ˆm =[ − sin( θ n ) , cos( θ n )]. From (A8) one obtainsΨ = q [(ˆ n − ˆ m ) + i (ˆ n ˆ n − ˆ m ˆ m )]= q e iθ n . (A16)Thus the eigenvalue q = | Ψ | is the MSF order param-eter magnitude. Although θ n and θ n + π are equivalent, θ ≡ θ n is uniquely defined.Inserting (A15) into (A14), the Feshbach resonancephase coupling ( α ) term takes the form − αq (cid:20) ( M · ˆn ) − M (cid:21) = − αq M (cid:20) cos ( θ n − θ ) − (cid:21) = − αq M cos( θ − θ ) , (A17)where the representation M = M [cos( θ ) , sin( θ )] hasbeen used. The coupling is again clearly invariant under θ n → θ n + π , and represents an alignment between thenematic and polar order parameters, familiar from thetheory of polar nematic liquid crystals. APPENDIX B: DETAILED ANALYSIS OF MEANFIELD PHASE DIAGRAM
In this Appendix a more detailed analysis of the meanfield phase diagram, summarized in Sec. IV B, especiallyFigs. 10 and 11, is presented. To this end, it is convenientto introduce the following scaled quantities: r = − g α µ , r = − g α µ , γ = g g g (B1)¯ ψ = √ g g α Ψ , ¯ ψ = g α Ψ , ¯ H = g g α H Substituting these into the mean field Hamiltonian (4.1),one obtains the dimensionless form¯ H = r | ¯ ψ | + 12 | ¯ ψ | + r | ¯ ψ | + 12 γ | ¯ ψ | + | ¯ ψ | | ¯ ψ | − Re[ ¯ ψ ∗ ¯ ψ ] (B2)5The scaling has succeeded in reducing the problem to itsessentials, removing α entirely, and subsuming all inter-action strengths into the single parameter γ . The onlyfree parameters are the two scaled chemical potentials, r σ . One may always choose the phase of ¯ ψ so that it isreal and non-negative. It is then clear from the last termin (B2) that ¯ H is minimized by taking ¯ ψ to also be realand non-negative, consistent with (4.3). With this inputthe scaled extremum equations take the form0 = ¯ ψ (cid:0) r − ¯ ψ + ¯ ψ + ¯ ψ (cid:1) (B3)0 = −
12 ¯ ψ + ¯ ψ (cid:0) r + γ ¯ ψ + ψ (cid:1) , (B4)and lead to the phase diagrams for γ > γ <
1. ASF free energy in terms of ¯ ψ : One begins by expressing the energy density in termsof ¯ ψ alone by using (B3) and (B4) to eliminate ¯ ψ . Inthe ASF phase (the transition to which, from the N andMSF phases, is the main focus), where ¯ ψ = 0, (B3) gives¯ ψ = ¯ ψ − ¯ ψ − r , (B5)which clearly requires that the right hand side be posi-tive. The resulting condition, r ≤ ¯ ψ − ¯ ψ , (B6)will be important in what follows.Substituting (B3) into the first line of (B4), a cubicsaddle-point equation is obtained purely in terms of ¯ ψ ∂ ¯ H ASF ∂ ¯ ψ ≡ r + 2 t ¯ ψ + 3 ¯ ψ + 2( γ −
1) ¯ ψ , (B7)and the corresponding energy density¯ H ASF = − r + r ¯ ψ + t ¯ ψ + ¯ ψ + 12 ( γ −
1) ¯ ψ , (B8)in which the parameter t = r − r − (B9)has been defined.The apparent instability of ¯ H ASF at large | ¯ ψ | for γ < ψ to afinite interval max { , (1 − √ − r ) } ≤ ¯ ψ ≤ (1 + √ − r ) for any given r . Similarly, even for γ > ψ (e.g.,if r ≥ ψ are ofphysical interest.
2. Continuous transitions:
It follows from (B4) that if ¯ ψ = 0, then either ¯ ψ = 0(normal phase) or ¯ ψ = − r /γ (MSF phase, existingonly for r < H = r ¯ ψ + γ ¯ ψ , showing that the N–MSF transitionmust be continuous, and take place at r = 0.Substituting ¯ ψ = 0 into (B5), one concludes that any continuous N–ASF transition must take place along theline r = 0 (with r ≥ r <
0, it is easy tocheck that ¯ H is locally unstable to nonzero ¯ ψ . Hence,if the transition is first order, representing some globalinstability, it must occur for r > ψ = − r /γ into (B5), onefinds that any continuous ASF–MSF transition must takeplace along the curve r = r c ( r ) ≡ p | r | /γ + r /γ. (B10)For r < r c it is easy to check that ¯ H is locally unstableto nonzero ¯ ψ . Hence, any first order transition mustoccur for r > r c .
3. First order N–ASF transition:
Consider now the possible existence of a first orderN–ASF transition. For t ≥ r , are positive. A positive¯ ψ root therefore exists only for r < r → − ), and is clearly unique—see the upper panel inFig. 16. The continuous transition at r = 0 thereforetakes place for t >
0, i.e., r > .On the other hand, as illustrated in the lower panelof Fig. 16, for t <
0, ¯ H ASF has a local minimum atsome positive value of ¯ ψ . Thus, for a range of r ≥
0, there will be two non-negative roots, the smaller ofwhich (vanishing at r = 0) corresponds to a maximum of¯ H ASF . It is the larger root, existing for a range of positive r , that corresponds to the minimum of ¯ H ASF associatedwith the ASF state. For this range of positive r thetransition from the normal state to this ASF minimummust therefore be first order. These considerations holdfor both γ > γ ≤ q/ + ( p/ (B11)where p = 1 γ − (cid:20) t − γ − (cid:21) q = 12( γ − (cid:20) r − t γ − γ − (cid:21) . (B12)For ∆ >
0, ¯ H ′ ASF has only one real root, while for ∆ < As ∆ → − two of the rootsmerge, and subsequently annihilate as ∆ changes sign.6
00 Order parameter ψ F r ee ene r g y de r i v a t i v e d f AS F / d ψ t > 000 Order parameter ψ F r ee ene r g y de r i v a t i v e d f AS F / d ψ t < 0 FIG. 16: Graphical illustration of the solution to (B7) for γ > t > t < r = 0. Curves above(below) these correspond to r > r < t > r < r → − , and then crosses to unphysical negativevalues. As discussed in the text, this signifies a second orderN–ASF transition at r = 0. For t < r for which there are two positive roots. However,the one (triangles) that vanishes at r = 0 is a local maximumof the free energy (B8), and therefore unstable. The physicalroot (stars) is the larger one, which never reaches the origin,but annihilates with the unstable one at the positive valueat which the cubic discriminant ∆, Eq. (B11), vanishes. Thetransition must therefore be first order, taking place at someintermediate point where the free energy itself vanishes, but ∆is still positive. For γ ≤ γ < γ = 1 it disappears entirely. In either case,the evolution of the other two roots remains as described. There are two branches to the ∆ = 0 curve (shown asdashed lines in Figs. 17 and 18) given by r ± ( t ) = ± (cid:2) − ( γ − t (cid:3) / − γ − t γ − , (B13)of which r corresponds to the merging and subsequentdisappearance of the two roots of interest. The twobranches meet and terminate at a cusp (denoted by astar in Figs. 17 and 18) at the point t = 34( γ − , r = 14( γ − ⇒ r = γ (2 γ − γ − , (B14)where all three roots coincide. The first order transitionmust therefore take place at a point r c ( t ) in the interval0 < r c < r where the energy density (B8) itself van-ishes. For small t one finds r = t { O [( γ − t ] } (the leading quadratic form being exact to all orders for γ = 1), and the very small deviation r − r c = − ( t ) [1 + O ( t )].The point r = 0, r = , at which the transitionline turns from second order to first is a tricritical point,labeled C AN in Figs. 17 and 18. Numerical results for r c are shown using γ = 2 and γ = 0 .
2, respectively.
4. First order MSF–ASF transition:
Along the putative second order line (B10), the ex-tremum condition (B7) may be factored in the form¯ H ′ ASF = 2( γ − ψ − ¯ ψ )( ¯ ψ − ¯ ψ )( ¯ ψ − ¯ ψ − ) (B15)in which ¯ ψ = p | r | /γ is the MSF state value, and theother two roots are given by¯ ψ ± = 14( γ − (cid:26) − − γ −
1) ¯ ψ (B16) ± q [2( γ −
1) ¯ ψ + 1] + 8 γ (cid:27) . The argument of the square root is positive, so all threeroots are real, and a picture similar to the lower panel ofFig. 16 obtains. a. The case γ ≥ : In order that ¯ ψ be the physicalroot, the condition ¯ ψ ≥ ¯ ψ is required, leading to theinequality ¯ ψ ≥ √ γ + 1) , (B17)which remains well defined for γ = 1. Thus, for r < r T ≡ − γ √ γ ) (B18)the transition is indeed second order. For r > r T ,¯ ψ becomes the physical root: the transition turns firstorder, and takes place when the two energy densitiesmatch: ¯ H ASF = ¯ H MSF = − r / γ . The point r = r T , r = r T ≡ (2 √ γ + 1) / √ γ + 1) is a second tricriticalpoint, labeled C AM in Fig. 17. At r T , (B15) has a co-incident pair of roots, and the line r and the transitionline must therefore osculate—see Fig. 17. b. The case γ < : For γ <
1, ¯ ψ must be theintermediate root: ψ − ≤ ¯ ψ ≤ ¯ ψ , leading to theinequality 12(1 + √ γ ) ≤ ¯ ψ ≤ − √ γ ) . (B19)Thus, for r T − ≡ − γ − √ γ ) < r < r T + ≡ − γ √ γ ) (B20)the transition is indeed second order. For r > r T + ,¯ ψ becomes the physical root, while for r < r T − , ¯ ψ − −0.2 −0.1 0 0.1 0.2 0.3−2.5−2−1.5−1−0.500.511.5 r r C AM EC AN ASF NMSF
Mean field phase diagram: global ( γ = 2) r C AM E ASF N MSF
Mean field phase diagram: detail ( γ = 2) FIG. 17: Mean-field phase diagram for γ ≥
1, in the scaled coordinates defined by (B1), with second order phase boundariesgiven by the thinner solid lines, and first order phase boundaries the thicker solid line. The right plot shows an expandeddetail near the labeled points E and C AM . The value γ = 2 is used in the numerical computation here, but the basic structureremains unchanged for other values. The second order N–ASF line along the positive r axis encounters a tricritical point C AN at r = 1 /
2, below which the transition turns first order and the line bends into positive r . The second order N–MSF transitionline along the positive r axis terminates on this first order line at a critical endpoint E . The first order line continues below E , now separating the ASF and MSF phases, but turns second order at another tricritical point C AM , and follows the linedefined by (B10). For large r , r this line asymptotes to r = γr , which agrees with (4.11) in unscaled units. Its unphysicalcontinuation toward the origin is shown by the dash-dotted line. The dashed line is the curve ∆ = 0, and therefore provides abound on the ASF phase boundary. All three lines osculate at the point C AM . The N–ASF and dashed lines also osculate at C AN . For γ = 0 the cusp together with the left hand branch of the ∆ = 0 line is pushed off to infinity, but C AN , E and C AM remain finite and well defined. becomes the physical root: in either case the transitionturns first order, and takes place when the ASF and MSFenergies match. The point r = r T ± , r = r T ± ≡ [2( √ γ ± − / √ γ ± are both tricritical points,labeled C AM ± in Fig. 18. At both points (B15) has acoincident pair of roots, and at both points the (two dif-ferent branches of the) ∆ = 0 curve and the transitioncurve must therefore osculate, as illustrated in Fig. 18.Notice that for γ → − , C AM + remains well defined, andcoincides with C AM , while C AM − is pushed to infinity.The semi-infinite second order line is therefore recoveredin this limit, consistent with the γ ≥
5. Critical endpoint for the N–MSF transition:
Since ¯ H MSF → r →
0, the N–ASF and MSF–ASFfirst order transition curves must meet at r = 0. Thesecond order N–MSF curve also terminates at this point,which therefore represents a critical endpoint, labeled E in Figs. 17 and 18.Figures 17 and 18 show complete phase diagrams, com-puted for the cases γ = 2 and γ = 0 .
2, respectively,and correspond to the two phase diagram topologies il-lustrated in Figs. 10 and 11 of the main body of the pa-per. Here the first order transition curves are computednumerically. The osculation with the ∆ = 0 line at thevarious tricritical points is also shown.
APPENDIX C: THE TWO-BODY MOLECULARBINDING PROBLEM
The center of mass Schr¨odinger equation for two par-ticles of mass m A interacting via an attractive potential v ( r ) with microscopic range d is given by − ~ m A ∇ ψ ( r ) + v ( r ) ψ ( r ) = Eψ ( r ) . (C1)Of interest is the regime where the particle is very weaklybound, with binding energy E < | E | ≪ | v | .The wavefunction ψ of such a weakly bound state willextend a distance much greater than d outside the po-tential. A good approximation is then to treat ψ ≈ ψ (0)as essentially constant over the potential region r < d (this notion may be made rigorous using effective s-wavescattering parameters). A Fourier analysis of (C1) thenyields ˆ ψ ( k ) = m A | v | ~ ˜ v ( k ) k + κ ψ (0) (C2)where ˆ v ( k ) ≡ v ˜ v ( k ) is the Fourier transform of the po-tential, v = ˆ v (0) < κ = m A | E | / ~ . The self-consistency condition de-termining E is therefore ~ m A | v | = Z d d k (2 π ) d ˜ v ( k ) k + κ . (C3)8 −0.5 0 0.5−0.500.51 r r C AM+ C AM− EC AN ASF NMSF
Mean field phase diagram: global ( γ = 0.2) r C AM+ C AM− E ASF N MSF
Mean field phase diagram: detail ( γ = 0.2) FIG. 18: Mean field phase diagram for γ <
1, in the scaled coordinates defined by (B1), with second order phase boundariesgiven by the thinner solid lines, and first order phase boundaries the thicker solid line. The right plot shows an expanded detailnear the labeled points E , C AM + and C AM − . The value γ = 0 . γ >
1. The secondorder N–ASF line along the positive r axis encounters a tricritical point C AN at r = 1 /
2, below which the transition turnsfirst order and the line bends into positive r . The second order N–MSF transition line along the positive r axis terminates onthis first order line at a critical endpoint E . The first order line continues below E , now separating the ASF and MSF phases,but turns second order at another tricritical point, now labeled C AM + , and follows the line defined by (B10). However, nowthe second order ASF–MSF line turns first order again at a new tricritical point C AM − . The unphysical continuation of thisline above C AM + and below C AM − is shown by the dash-dotted line. When γ → − , the latter is pushed out to infinity, whilethe former remains finite. For large r , r the first order line asymptotes to r = √ γr , which agrees with (4.14) in unscaledunits. The dashed line is the curve ∆ = 0, and therefore provides a bound on the ASF phase boundary. All three lines (thoughtwo different branches of the ∆ = 0 line) osculate at the points C AM + and C AM − . The N–ASF and dashed lines also osculateat C AN . For d ≤ E = 0. There-fore a bound state solution exists for arbitrarily weakpotential. For d > | v ,c | , below which the potential fails to bind, given by ~ m A | v ,c | = Z d d k (2 π ) d ˜ v ( k ) k , (C4)and one may write ~ m A (cid:18) | v ,c | − | v | (cid:19) = κ Z d d k (2 π ) d ˜ v ( k ) k ( k + κ ) . (C5)Close to the critical point the integral is dominated bythe small k region, k ≪ π/a . One may then ignore the k -dependence of ˜ v (the integral remains convergent for2 < d < κ = (cid:20) ~ m A C d (cid:18) | v ,c | − | v | (cid:19)(cid:21) d − , (C6)where C d = Z d d u (2 π ) d u ( u + 1) , (C7)in particular C = 1 / π . One therefore obtains the powerlaw relationship | E | ∼ ( | v | − | v ,c | ) / ( d − describing thevanishing of the binding energy upon approach to thecritical point. APPENDIX D: THERMAL AND BOSECONDENSATE DENSITY PROFILES IN AHARMONIC TRAP
The most direct probe of a trapped degenerate atomicgas is through its spatial density profile, obtained from afreely expanding cloud. In this Appendix, the details ofthe density profile calculations for a trapped Bose gas arepresented. For the weakly interacting case, the profile iswell approximated by the noninteracting expression n ( r ) = ∞ X n = | φ n ( r ) | n n , (D1)with the occupation number n n , for a Bose gas given bythe Bose-Einstein distribution n n = h ˆ a † n ˆ a n i = 1 e β ( ε n − µ ) − . (D2)The single-particle spectrum ε n and normalized wave-functions φ n ( r ) are solutions of the single particleSchr¨odinger equation ˆ hφ n = [ − ( ~ / m ) ∇ + V ( r )) φ n = ε n φ n appropriate to a trapping potential V ( r ). For a 3dharmonic potential V ( r ) = mω r − ~ ω (that for9simplicity is taken to be isotropic) ε n = ~ ω ( n x + n y + n z ) ≡ ~ ω n, (D3) φ n ( r ) = Y i = x,y,z (cid:18) √ π r n i n i ! (cid:19) / H n i (ˆ r i ) e − ˆ r i / , (D4)with H n ( x ) the n th Hermite polynomial, a function of thenormalized coordinates ˆ r i = r i /r ( i = x, y, z ), expressedin units of the quantum oscillator length r = p ~ /mω .The chemical potential, µ < N = Z d r n ( r ) = ∞ X n = e β ( ε n − µ ) − . (D5)In this noninteracting limit, at T = 0 all atoms gointo the lowest single-particle state φ , forming a Bose-Einstein condensate, with n T =0 ( r ) = Nπ / r e − r /r . (D6)At finite T , a fraction of atoms is thermally excited tohigher single-particle states, and n ( r ) = h r | e β (ˆ h − µ ) − | r i = ∞ X p =1 e βµp h r | e − βp ˆ h | r i = ∞ X p =1 e βµp ρ osc ( r , r ; pβ ) (D7)is expressible purely in terms of diagonal elements of thesingle-particle density matrix for a harmonic oscillator ρ osc ( r , r ′ , β ) = h r | e − β ˆ h | r ′ i , (D8)governed by a single-particle Hamiltonian ˆ h . The densitymatrix can be found by solving a diffusion equation in aharmonic potential (or equivalently obtained from an-alytic continuation of the harmonic oscillator evolutionoperator) with the “initial” condition of ρ osc ( r , r ′ ; β =0) = δ ( r − r ′ ) obvious from (D8). For d = 3 one has ρ osc ( r , r ; β ) = (cid:20) mω e β ~ ω π ~ sinh( β ~ ω ) (cid:21) / e − r /r ( β ) , (D9)where r ( β ) = ~ mω coth ( β ~ ω / ≈ ( ~ mω , ~ ω /k B T ≫ k B T mω , ~ ω /k B T ≪ , (D10)is the finite-temperature “oscillator length” that reducesto the quantum one r = p ~ /mω at low T and the classical (thermal) one, r T = p k B T /mω (defined by mω r T = k B T ) at high T .The p sum in n ( r ), Eq. (D7), can be evaluated analyt-ically in various limits. For ~ ω /k B T ≪ T , where (D6) holds] the p sum natu-rally breaks up into two parts with n ( r ) = n T ( r ) + n ( r ).The two contributions correspond, respectively, to ranges1 ≤ p < p c = k B T / ~ ω (“thermal”) and p c ≤ p < ∞ (“quantum”) with p c determined by p c β ~ ω = 1. Thethermal range is characterized by p such that pβ ~ ω < r ( β ) and the corresponding density matrix ρ osc ( r , r ; pβ ) can be approximated by their thermal clas-sical forms, giving n T ( r ) ≈ (cid:18) k B T π ~ ω (cid:19) / r p c − X p =1 p / e − p ( r /r T + | µ | /k B T ) ≡ (cid:18) k B T π ~ ω (cid:19) / r ˜ g / (cid:18) e − r /r T −| µ | /k B T , k B T ~ ω (cid:19) . (D11)where a “cutoff” extended zeta function,˜ g α ( x, p c ) = p c − X p =1 x p p α , (D12)has been defined.The quantum density contribution n ( r ) is character-ized by values of p such that pβ ~ ω >
1, and thusby a zero-temperature oscillator length r ( pβ ) ≈ r and the density matrix is given by ρ osc ( r , r ; pβ ~ ω ) ≈ π − / r − e − r /r . The resulting sum is then easily com-puted, yielding n ( r ) ≈ N ( T ) π / r e − r /r , (D13)with amplitude factor N ( T ) ≈ ∞ X p = p c e − p | µ | /k B T = e − ( p c − | µ | /k B T e | µ | /k B T − e − ( p c − | µ | /k B T ≈ e −| µ | / ~ ω .Eliminating p c = k B T / ~ ω , it is clear that n T ( r ) and n ( r ) depend strongly on the ratio of the chemical po-tential µ to the trap level spacing ~ ω , with the formerdetermined by the temperature through the total atomnumber constraint. For | µ | / ~ ω ≫ T > T c ), the sum in the expression for n T ( r ), Eq.(D11), can be extended to infinity, introducing only ex-ponentially small error O ( e − µ/ ~ ω ) ≪
1. The resultingthermal density n T ( r ) is then given by the extended zetafunction n T ( r ) ≈ (cid:18) k B T π ~ ω (cid:19) / r g / (cid:16) e − r /r T −| µ | /k B T (cid:17) , for T > T c , (D15)0As expected for T > T c , the condensate spatial distribu-tion is still given by the Gaussian expression (D13), butwith an exponentially small condensate N ( T ) ≈ e −| µ | / ~ ω ≈ , for T > T c , (D16)At high T ≫ T c = ~ ω ( N/ζ (3)) / (wherethe gas is nondegenerate), such that 0 < | µ | ≈− k B T ln[ (cid:16) ~ ω k B T (cid:17) N ] ≈ k B T ln( T /T c ) ≫ k B T , the ther-mal (and therefore total) density (D15) reduces to a pureGaussian with thermal width r T , reflecting the high-temperature Boltzmann statistics n ( r ) ≈ (cid:18) k B T π ~ ω (cid:19) / r e − r /r T −| µ | /k B T , = Nπ / r T e − r /r T , for T ≫ T c (D17) in which the relation N = ( k B T / ~ ω ) e | µ | /k B T has beenused to satisfy the particle number constraint.As T is lowered, approaching T c from above, the mag-nitude of the chemical potential drops below T (remain-ing negative), and (while the condensate fraction remainsvanishingly small) the boson density profile (D15) devel-ops a non-Boltzmann peak structure even above T c : n ( r ) ≈ (cid:18) k B T π ~ ω (cid:19) / r e − r /r T −| µ | /k B T , r ≫ r T ζ (3 / − π / (cid:16) r r T + | µ | k B T (cid:17) / , r ≪ r T , (D18)retaining a Gaussian falloff at large r/r T . The small- r cusp in n ( r ), that develops as T c is approached fromabove, is rounded on the cutoff length r c ( T ), which for T > T c is given by r c ( T ) ≈ r µ ≡ r p | µ | / ~ ω , with r ≪ r c ( T ) ≪ r T .From the prefactor in n ( r ) it is clear that this thermalform cannot persist to low temperatures, as the volumeunder n ( r ) drops with T . Thus, for even lower tem-perature, T < T c ≈ ~ ωN / , to accommodate all N particles | µ | is forced to drop below the level spacing, | µ | . ~ ω . In this region, the high p terms in (D11),estimated to be O ( e − p c | µ | /k B T ) = O ( e −| µ | / ~ ω ), are nolonger exponentially small and the sum can no longer, ingeneral, be extended to infinity. However, for p c ≫ r c ( T ), thethermal density n T ( r ), Eq. (D11), is still well approxi-mated by the extended zeta function in Eq. (D15), butwith µ/k B T ≈ r cusp smoothed out on alength scale r c ( T < T c ) ≈ r T / √ p c = 2 r [following fromthe condition p c ( r/r T ) = O (1)].Correspondingly, the condensate contribution n ( r )begins to grow for T < T c through the growth of the coef-ficient N ( T ), Eq. D14. In fact for T < T c , | µ | /k B T ≪ | µ | / ~ ω ≪
1, one has N ( T < T c ) ≈ k B T / | µ | ≫ r ) are precisely what is required to makeup for those that cannot fit into the thermal distribu-tion n T ( r ). Thus for T < T c the total atom densityprofile n ( r ) changes dramatically, developing an easilyidentifiable bimodal distribution n ( r ) = n T ( r ) + n ( r ), illustrated in Fig. 1.As discussed in Sec. V B, this analysis easily general-izes to the two-component Bose gas (bosonic atoms andmolecules) that is the focus of this paper. APPENDIX E: DETAILS OF BOSE-BCSMSF–ASF CRITICALITY
In this appendix, the critical behavior in the vicinityof the ASF transition line is derived as a function of di-mension d via an analysis of the integral equations (8.43)for small deviations ρ, τ, f .For 3 < d < J ≡ − ∂ τ J (0 ,
0) = Z d d u (2 π ) d u + 1[ u ( u + 2)] / J ≡ ∂ f J (0 ,
0) = 12 Z d d u (2 π ) d u ( u + 2)] / R ≡ − ∂ τ R (0 ,
0) = J /I d R ≡ ∂ f R (0 ,
0) = J / I d . (E1)Higher order derivatives, however, lead to divergent in-tegrals at small u . The subtracted integral has leading1behavior δJ ( τ, f ) ≡ J ( τ, f ) + J τ − J f ≈ j c ( s ) j c ( s ) ≡ √ Z d d u (2 π ) d (cid:18) u − s u − √ u + s (cid:19) = j c (1) s d − , (E2)where s = (1 − τ ) − (1 − f )2(1 − τ ) ≈ f − τ > . (E3)With the subtractions, the integral j c (1) converges atboth large and small u . The vanishing of J requires now0 = J f − J τ + j c (1) s d − + O ( f , τ ) , (E4)and leads to f = J J τ − j c (1) J (cid:18) J J − (cid:19) d − τ d − + O ( τ ) . (E5)It is easy to see from (E1) that J > J and that thissolution is consistent with the requirement that s > ρ = R f − R τ + j c (1)2 I d s d − + O ( f , τ )= J I d (cid:18) J J − (cid:19) τ − j c (1)2 I d (cid:18) J J − (cid:19) d +12 τ d − + O ( τ ) . (E6)For d < u indicatingsingular dependence on τ and f . This infrared singularityhas been isolated by writing J ( τ, f ) = J c ( τ, f ) + δJ ( τ, f ) R ( τ, f ) = R c ( τ, f ) + δR ( τ, f ) (E7)where J c ( τ, f ) = 1 √ − τ ˜ j c ( s ) R c ( τ, f ) = √ − τ I d ˜ j c ( s )˜ j c ( s ) ≡ √ Z d d u (2 π ) d (cid:18) u − √ u + s (cid:19) (E8)where ˜ j c ( s ) now requires only the single subtraction forconvergence at large u in this lower dimension. It canbe checked that for 1 < d < δJ and δR are finite at the criticalpoint, and so have leading linear dependence on τ and f .The corresponding coefficients are˜ J = Z d d u (2 π ) d (cid:26) u + 1[ u ( u + 2)] / − √ u (cid:27) ˜ J = 12 Z d d u (2 π ) d (cid:26) u ( u + 2)] / − √ u (cid:27) ˜ R = ˜ J /I d , ˜ R = ˜ J / I d . (E9) One obtains again ˜ j c ( s ) = ˜ j c (1) s d − , (E10)in which ˜ j c (1) is finite for 1 < d < d = 1is the lower critical dimension, below which the phasetransition ceases to exist, and so one expects special be-havior here). This dominates the linear behavior, andthe constraint J ( τ, f ) = 0 becomes0 = ˜ j c (1) s d − − ˜ J τ + ˜ J f + O ( τ , f ) , (E11)which, to leading order, has the solution s = 0, and hence f = 2 τ + 2 " ˜ J − J ˜ j c (1) d − τ d − + O ( τ ) . (E12)From (E9) it is easy to see that ˜ J > J , so that (E12)is again consistent with s >
0. Using (E7) and (E8), thedensity deviation is ρ = ˜ R f − ˜ R τ + ˜ j c (1)2 I d s d − + O ( τ , f ) (E13)= ˜ J − J I d τ + " ˜ J − J ˜ j c (1) d − τ d − + O ( τ ) . The singular corrections to the linear behavior may beidentified with the energy exponent − α , where theGaussian specific heat exponent is α = − dd − (not to beconfused with the Feshbach resonance coupling). In a fulltheory this form would be replaced by the exact Ising ex-ponent.At the upper critical dimension d = 3 itself there willbe logarithmic corrections. Since d = 3 is the physicallyimportant dimension, it is worth presenting the resultsfor this case as well. One may now express the results interms of elliptic integrals (see Ref. 48, pp. 232, 235, 905): J ( τ, f ) = p − τ + √ − f π E ( q ) − √ π R ( τ, f ) = p − τ + √ − f π I (cid:2) − (1 − τ ) E ( q )+ (1 − τ − p − f ) K ( q ) (cid:3) + 16 √ π I (E14)where q = 2 √ − f − τ + √ − f . (E15)At small τ, f , hence q →
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Quantum Mechanics:non-relativistic theory , Permagon, New York, 1977. At low energies the same scattering amplitude structurearises for two atoms interacting through a potential witha large potential barrier separating a short-range attrac-tive minimum from the rapidly decaying repulsive partat infinity. For a sufficiently deep minimum, a (negativeenergy) bound state (molecule) exists. For a sufficientlyshallow minimum, a (positive energy) quasi-bound stateexists with a finite lifetime (resonance). However, thereis also an intermediate range in which there is no boundstate, but the real part of the energy pole is negative, andthe metastable molecule interpretation fails. D. E. Sheehy and L. Radzihovsky, Annals of Physics ,1790 (2007). From a renormalization group point of view, one may alsothink of the two species model as a partially renormal-ized Hamiltonian in which all fluctuations below the scaleof the molecular diameter have been integrated out. Itwill be seen later that the dissociation amplitude α will“dress” the ψ † particles with a virtual cloud of ψ † pairsthat diverges in size as resonance is approached. Thus,such a model captures all further renormalization, includ-ing critical fluctuations. In particular, the multi-body pa-rameters in ˆ H , Eq. 2.1 should all be treated as constantsthroughout the neighborhood of the Feshbach resonance. Since the operators ˆ ψ , ˆ ψ † commute with the operatorsˆ ψ , ˆ ψ † , the standard derivation of the coherent state for-mulation goes through essentially without change. See,e.g., J. W. Negele and H. Orland, Quantum Many-ParticleSystems (Addison-Wesley, 1988). Such an order-parameter classification of the phases bythe types of broken symmetries is incomplete. There aresystems that exhibit (so-called) “topological” phase tran-sitions, that do not break any global symmetry and there-fore are not associated with any local order parameter.Probably the best known example of this is the Kosterlitz-Thouless transition in a two-dimensional XY-model [M.Kosterlitz and D. J. Thouless, J. Phys. C , 1181 (1973)],where the ordered phase, while exhibiting a finite supe-fluid density and power-law correlations (that distinguishit from the fully disordered paramagnetic state) does not break any global symmetries, and is not characterized bya local order parameter. J. Zinn-Justin,
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The physics of liquid crys-tals, 2nd edition (Oxford, 1995). In 1d, for attractive interactions, the energy functionalalso admits bright soliton solutions that (for a range of pa-rameters) minimize the energy: V. Gurarie, unpublished.Focusing here on sufficiently large nonresonant repulsiveinteractions, these inhomogeneous solutions we will be ne-glected. P. M. Chaikin and T. C. Lubensky,
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1. For the dilute Bose fluid, char-acterized by a σ / Λ T ≪
1, interactions become impor-tant only when ξ/ Λ T ≫ (Λ T /a σ ) ( d − / (4 − d ) . Similarly,one expects shifts in the critical line of order δµ ∼ g σ n ,i.e., δµ/k B T ∼ ( a σ / Λ T ) d − . Since, for the ideal gas, | µ | /k B T ∼ t / ( d − , this corresponds to relative shifts δt ∼ ( a σ / Λ T ) ( d − / in the critical temperature. For adetailed analysis of the crossover between ideal and in-teracting superfluid criticality, see P. B. Weichman, M.Rasolt, M. E. Fisher and M. J. Stephen, Phys. Rev. B , 4632 (1986), as well as Ref. 77 below. See, e.g., R. M. Ziff, G. E. Uhlenbeck and M. Kac, Phys.Rep. , 169 (1977). See, e.g., A. Erd´elyi,
Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1, Secs. 1.10 and1.11. The ν – T phase diagram, Fig. 2, corresponds to the µ - µ phase diagram, Fig. 10 with a constant density constraintapplied. As demonstrated in Sec.IV B, the coupling be-tween the atomic and critical molecular order parametersdrives the transition first order (even in mean field the-ory), leading this point where the three phases meet tobe a critical endpoint. The analysis in Sec. V neverthe-less remains valid within any of the phases, with T c stillsetting a valid reference temperature. The nonlinear relation (5.16) between the canonical andgrand canonical variables is an example of the very generalfeature of Fisher renormalization: see M. E. Fisher, Phys.Rev. , 257 (1968). This relation will also be alteredby interaction effects very close to the critical line: seefootnote 68 above. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.Wieman, and E. A. Cornell, Science , 198 (1995). K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. vanDruten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys.Rev. Lett. , 3969 (1995). Whether there are three or one gapped modes is a some-what a semantic question, which comes down to one’s def-inition of a mode. For example a one-dimensional quan-tum harmonic oscillator is said to describe one type ofquasiparticle. However, from the point of view thermo-dynamics and symmetry, it is convenient to count this system as having two modes, corresponding to its coor-dinate and momentum, q , p , or equivalently creation andannihilation operators, a , a † , associated with a particleand a hole and the fact that it is governed by two (firstorder) Heisenberg equations. The cubic terms in (6.2) lead to nonzero h ˆ φ σ i at higher or-der, corresponding to corrections to the mean field value ofΨ σ . Such corrections can also be derived from the lead-ing (quadratic) corrections to the free energy—see Sec.VI D 1. More detailed discussion of how the extremum conditionfor the order parameter mixes different orders in pertur-bation theory about the weakly interacting limit, and howto ensure consistency order by order for any given quan-tity, may be found in: P. B. Weichman, Phys. Rev. B ,8739 (1988). In the presence of a trapping potential, the spectrumshould be observably discrete for excitation wavelengthscomparable to the system size, and will depend on thetrap shape. In future work, it might be interesting to revisit the usualFeynman argument that predicts a downturn (roton min-imum) in the energy spectrum in the neighborhood of thepeak in the structure factor S ( k ). For the dense fluid thereis a strong peak at k ∼ n / , the inverse of the mean inter-particle spacing, reflecting the oscillation of the density-density correlation function on this scale. For the dilute,weakly interacting fluid this “incipient crystal” structuredisappears. However, since the gas parameter na can belarge near resonance, interesting structure in S ( k ) couldreturn, and in turn generate interesting structure in E k σ at higher momenta. For d ≥ k ) cutoff, whichphysically arises from the vanishing of the scattering co-efficients at large k . P. C. Hohenberg, Phys. Rev. , 383 (1967). N. D. Mermin and H. Wagner, Phys. Rev. Lett. , 1133(1966). M. E. Fisher, M. N. Barber, and D. Jasnow, Phys. Rev.A , 1111 (1973). In the “hard-spin” ψ σ = e iθ σ description, the Feshbach in-terconversion term reduces to α cos(2 θ − θ ), locking theatomic phase to the phase of the molecular condensate upto π , corresponding to the unbroken Z symmetry of theMSF. In terms of a vector order parameter, the MSF ischaracterized by a finite quadrupole moment ( l = 2 angu-lar harmonic, akin to the nematic, “headless arrow” orderin liquid crystals), and ASF by a vector order parameter( l = 1 angular harmonic). In this language, in the ASFboth the “nematic director” (headless arrow, ψ ) and thevector ( ψ ) are aligned, with their directions locked to-gether. In contrast, in the MSF, while the nematic direc-tor is ordered, the vector order is disordered by virtue ofonly being locked to the nematic director up to an angle π . Clearly this unlocking corresponds to a restoration ofthe Z Ising symmetry of the MSF. Further discussion ofthis, and related, issues can be found in Appendix A. See, e.g., V. N. Popov,
Functional Integrals in QuantumField Theory and Statistical Physics, (D. Riedel, Dor-drecht, 1983), especially Chapter 6. The “diamagnetic” nonlinearity, ( ∇ θ ) | ψ | in Eq. (7.4)is clearly subdominant to |∇ θ | . To see that the “para- magnetic” term, ∇ θ · ψ ∗ ∇ ψ is irrelevant it is easiest tointegrate out ∇ θ , obtaining a local current-current in-teraction, that (because of the two extra powers of thegradient) at long wavelengths is subdominant to the localquartic coupling. D. J. Bergman and B. I. Halperin, Phys. Rev. B , 2145(1976). The connection to a compressible Ising model can be madeeven closer by a duality transformation. See for exampleRef. 51. B. I. Halperin, T. C. Lubensky, and S. K. Ma, Phys. Rev.Lett. , 292 (1974). M. E. Fisher, Rev. Mod. Phys. , 597 (1974). M. Girardeau and R. Arnowitt, Phys. Rev. , 755(1959). R. P. Feynman,
Statistical Mechanics, Second Edition ,(Perseus Books Group 1998). Despite this particle gap, it can be shown that the re-quired Goldstone mode appears in the spectrum of thetwo particle bound states: A. Coniglio and M. Marinaro,Il Nuovo Cimento
XLVIII B , 262 (1967). The single particle energy spectrum should indeed begapped in the MSF phase. The fact that it is gappedeven in the ASF phase is an artifact of the variationalcalculation, and can be cured by more sophisticatedtreatments. In particular, the usual Bogoliubov ap-proximation for g > M. E. Fisher, Phys. Rev. , 257 (1968). C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. , 1556(1981). For 2d systems that do spontaneously break a continuoussymmetry see, for example, L. Radzihovsky,
Anisotropicand Heterogeneous Polymerized Membranes , in “Statisti-cal Mechanics of Membranes and Surfaces,” (World Scien-tific, edited by D. R. Nelson, T. Piran and S. Weinberg),as well as Ref. 67, and references therein. M. P. A. Fisher and D.-H. Lee, Phys. Rev. B , 2756(1989). In a different context, this was independently discussed inF. R. Klinkhamer and G. E. Volovik, JETP Lett. , 343(2004). The existence of highly anisotropic dumbbell 2 π atomicvortices allows for the possibility of liquid crystal nor- mal phases. These can, in principle, appear when theanisotropic 2 π vortices proliferate but exhibit some formof (e.g., smectic or nematic) spatial order, that distin-guishes the resulting liquid-crystal normal state from afully disordered normal state, where vortices form a ho-mogeneous, isotropic liquid. The physics of the two-component (atom and moleculetwo-channel) superfluid model characterized by phases θ and θ coupled by the internal Josephson coupling (arisingfrom Feshbach resonance) is quite closely related to theextended xy-model H extxy = − X h i,j i { J cos( θ i − θ j ) + J cos[2( θ i − θ j )] } , analyzed by G. Grinstein and D. H. Lee [Phys. Rev. Lett. , 541 (1985)]. When J /J is sufficiently large this lattermodel exhibits fully ordered polar phase where ψ = e iθ is ordered, a nematic phase where only ψ = e i θ is or-dered (but ψ is not), and a fully disordered, isotropicphase. These are isomorphic to the ASF, MSF and normalphases, respectively, and (as shown by Grinstein and Lee)the polar and nematic phases are separated by an Ising de-confinement transition characterized by the roughenningand loss of line tension of the domain wall across which θ jumps by π ; L. Radzihovsky and J. Park, unpublished. L. Balents, M. P. A. Fisher, and C. Nayak, Int. J. Mod.Phys. B , 1033 (1998); Phys. Rev. B , 1654 (1999). Similarly, in the microscopic model underlying H AM , inwhich molecules are formed of bosons with different inter-nal “atomic spin” quantum numbers, the correspondinganomalous average is Φ mm ′ ( x , x ′ ) = h ˆ ϕ m ( x ) ˆ ϕ m ′ ( x ′ ) i , inwhich ˆ ϕ m is Bose field operator for atoms with spin m .For example, if free bosons have spin m a , and must pairwith spin m b to form a molecule, then the atomic orderparameter is Ψ = h ˆ ϕ m a i , while the molecular order pa-rameter Ψ must be constructed from Φ m a m b . The parameters p and q defined in (B11) and (B12) areobtained by expressing the cubic equation f ′ ASF = 0 in theform y + py + q = 0 via the shift y = ¯ ψ + 1 / γ − u ± = ( − q/ ± √ ∆) / , both defined to be real for ∆ ≥ <
0. Then the exact rootsare given by y l = e i πl/ u + + e − i πl/ u − , l = 0 , ,
2. For∆ = 0 one obtains y = − q/ / , y = y = ( q/ / . R. P. Feynman and A. R. Hibbs,