Pairing and the spin susceptibility of the polarized unitary Fermi gas in the normal phase
PPairing and the spin susceptibility of the polarizedunitary Fermi gas in the normal phase
Lukas Rammelmüller,
1, 2
Yaqi Hou, Joaquín E. Drut, and Jens Braun
4, 5, 6 Arnold Sommerfeld Center for Theoretical Physics (ASC),University of Munich, Theresienstr. 37, 80333 München, Germany Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27599, USA Institut für Kernphysik (Theoriezentrum), Technische Universität Darmstadt, D-64289 Darmstadt, Germany FAIR, Facility for Antiproton and Ion Research in Europe GmbH, Planckstraße 1, D-64291 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI, Planckstraße 1, D-64291 Darmstadt, Germany
We theoretically study the pairing behavior of the unitary Fermi gas in the normal phase. Ouranalysis is based on the static spin susceptibility, which characterizes the response to an externalmagnetic field. We obtain this quantity by means of the complex Langevin approach and compareour calculations to available literature data in the spin-balanced case. Furthermore, we presentresults for polarized systems, where we complement and expand our analysis at high temperaturewith high-order virial expansion results. The implications of our findings for the phase diagram ofthe spin-polarized unitary Fermi gas are discussed, in the context of the state of the art.
I. INTRODUCTION
Pair formation in fermionic quantum matter is at theheart of a wide range of physical phenomena and per-vades physics at vastly different lengths scales, rang-ing from ultracold quantum gases to neutron stars. Inspin- systems with weak attractive interactions, for ex-ample, pairing of spin-up and spin-down particles at di-ametrically opposite points of the Fermi surface (i.e.,Cooper pairing) typically leads to a superfluid phase atlow enough temperatures, accompanied by the openingof an energy gap in the quasiparticle spectrum. Similarconclusions hold for strongly correlated systems, wherepairing and superfluidity have been addressed thoroughlyin theory [1–19] and experiment [20–30]. A central openquestion in this regard is whether signatures of strongpairing fluctuations survive in the normal, non-superfluidhigh-temperature phase, which is often referred to as a“pseudogap regime”, obtaining its name from a (poten-tially strong) suppression of the single-particle density ofstates around the Fermi surface.An intriguing system in the above sense is the unitaryFermi gas (UFG), a spin- system with a zero-range at-tractive interaction tuned to the threshold of two-bodybound-state formation [31]. This strongly correlated sys-tem features non-relativistic conformal invariance [32]and universal properties [33, 34] that have attracted re-searchers from several areas across physics. The thermo-dynamics of this system has been investigated via a broadrange of theoretical methods [31] and many of its prop-erties have been measured experimentally (see e.g. [35]).However, a number of open questions remain, in par-ticular regarding the appearance or not of the above-mentioned pseudogap [27, 36].There is consensus on the qualitative features of a“pseudogap phase”, which is expected to leave an im-print on quantities such as odd-even staggering of the en-ergy per particle and suppression of the spin susceptibil- ity above the critical temperature T c . However, there isno long-range order parameter associated with the emer-gence of a such a behavior. An unambiguous definitionof this “phase” therefore does not exist. The case of theUFG is especially challenging in this regard, as it is ina strongly correlated non-ordered (i.e., non-superfluid)phase above the critical temperature T c (see Ref. [37] fora discussion and Refs. [12, 17] for recent reviews). More-over, as argued in Ref. [38], the UFG is not described by(normal) Fermi liquid theory at temperatures between T c and the Fermi temperature T F .Consequently, the debate remains active due to thevarying opinions as to what the magnitude of the varioussignals for a “pseudogap phase” should be (e.g., how muchsuppression in the density of states or the susceptibilitytruly constitutes a pseudogap). This issue is further com-plicated by the lack of a small expansion parameter or ef-fective theory that would allow to study this regime in asystematic manner. Thus, to face this challenge and shedlight on the pairing properties of the UFG in the normalphase, one must use nonperturbative methods that startfrom the microscopic degrees of freedom, i.e., ab initio approaches.In the superfluid, low-temperature phase, pairing isloosely speaking parameterized by the pairing gap. How-ever, at temperatures above the phase transition, the gapis necessarily zero in the long-range limit and thereforethis quantity does not allow us to quantify pairing effectsunambiguously. Several quantities have instead been pro-posed for that purpose. A prominent example is the spec-tral function A ( k , ω ) which should be suppressed aroundthe Fermi energy in the presence of extensive pairing fluc-tuations. While, e.g., T-matrix studies report a substan-tial suppression of spectral weight [1, 39], self-consistentLuttinger-Ward (LW) studies have found almost no sig-nature of a pseudogap in this quantity [40]. Notably,the density of states ρ ( ω ) = (cid:82) d k (2 π ) A ( k , ω ) was alsocomputed via an auxiliary-field quantum Monte Carlo a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b (AFQMC) approach combined with a numerical analyticcontinuation to real frequencies [9]. There, a distinctsuppression was found. However, those results were ob-tained with a spherical cutoff in momentum space whichcould lead to unphysical results (see Ref. [16, 17] for adiscussion).As an alternative probe to study pairing above the su-perfluid transition, one may focus on thermodynamic re-sponse functions, such as the magnetic (or spin) suscep-tibility χ : χ ≡ (cid:18) ∂M∂h (cid:19) T,V,µ = β (cid:0) (cid:104) M (cid:105) − (cid:104) M (cid:105) (cid:1) , (1)where M = N ↑ − N ↓ denotes the magnetization of thesystem. In the context of cuprate superconductors, itwas suggested that the pairing gap would suppress χ ,making it a good candidate to study pairing in the normalphase [41, 42]. Similar conclusions hold for the specificheat, see, e.g., Ref. [16, 17].The spin susceptibility allows for a clear definition ofa spin-gap temperature T s based on the following argu-ments. The spin susceptibility vanishes as T → asa consequence of the presence of a pairing gap in thequasiparticle spectrum of a fermionic superfluid (or su-perconductor). At high temperatures, on the other hand,the system enters the Boltzmann regime and decays fol-lowing Curie’s law χ = CT − , where C is the material-specific Curie constant. Therefore, χ necessarily devel-ops a maximum at some intermediate temperature, whichdefines the aforementioned spin-gap temperature T s (seeRef. [10] for a discussion). Note that the suppression of χ in the superfluid phase was dubbed “anomalous” sincethe noninteracting case approaches the Pauli susceptibil-ity χ P = nε F monotonically as T → (where n and ε F are the density and Fermi energy of the noninteractingcase, respectively). The spin-gap temperature T s may beinterpreted as the highest temperature at which pairingcorrelations may strongly influence the dynamics of thesystem. For T > T s , the spin susceptibility is dominatedby thermal fluctuations and approaches the noninteract-ing limit.In this work, we use the complex Langevin (CL) ap-proach [43, 44] to characterize the spin susceptibility andspin-gap temperature of the polarized UFG and gaininsight into pair formation above T c . Previous non-perturbative ab initio investigations [16, 19] have alsocalculated the susceptibility as well as odd-even stag-gering and the single-particle spectrum, but focused onthe unpolarized case. Here, we review and comprehen-sively compare extant susceptibility data for the unpo-larized system and extend the presentation to the polar-ized case, showing results for several temperatures andpolarizations. In addition, we validate our findings athigh temperature using a Padé-resummed virial expan-sion (PRVE) analysis [45, 46]. Finally, we set our find-ings in context by examining implications for the phase-diagram of the spin-polarized Fermi gas. II. MODEL
We consider a spin- Fermi gas with a short-range at-tractive interaction between spin-up and spin-down parti-cles. The Hamiltonian describing this system (in naturalunits (cid:126) = k B = m = 1 ) reads ˆ H = (cid:88) s = ↑ , ↓ (cid:90) d x ˆ ψ † s ( x ) (cid:18) − (cid:126) ∇ m (cid:19) ˆ ψ s ( x ) − g (cid:90) d x ˆ n ↑ ( x )ˆ n ↓ ( x ) , (2)where g is the bare coupling, ˆ ψ † s ( x ) and ˆ ψ s ( x ) create andannihilate fermions of spin s at position x , respectively,and ˆ n s ( x ) = ˆ ψ † s ( x ) ˆ ψ s ( x ) is the density operator.Formally, this model is a simple non-relativistic inter-acting field theory which has been widely studied andtherefore many of its gross features are well known. Asa function of the coupling strength, in particular, thismodel realizes the so-called BCS-BEC crossover [31, 47],given by a smooth evolution from a gas of weakly attrac-tive fermions (BCS regime) to a gas of weakly repulsivecomposite bosons (BEC regime), typically parameterizedby the dimensionless parameter ( k F a s ) − . Here, k F de-notes the Fermi momentum which is determined by thetotal particle density n via k F = (3 π n ) / and a s denotesthe s -wave scattering length. When a s diverges, the sys-tem is at the threshold of two-body bound-state forma-tion and the scattering cross-section assumes its maximalvalue (only bounded by unitarity of the scattering ma-trix) implying strong interactions. Since a s drops out ofthe problem in that limit, the density and the temper-ature are the only physical parameters determining thephysical behavior. The system is then said to exhibit universality in the sense that the microscopic details ofthe interaction become irrelevant.The leading instability of the model is toward a super-fluid state below a (coupling-dependent) critical temper-ature T c throughout the entire BCS-BEC crossover. Inthe limit of weak attraction, mean-field theory is appli-cable and the situation is well described by BCS theory.In that regime, pair formation (i.e., the appearance ofa pairing gap) and pair condensation (i.e., the appear-ance of off-diagonal long-range order and superfluidity)happen simultaneously, i.e., the so-called pairing tem-perature T ∗ is identical to T c . On the other side of theresonance associated with the limit of an infinite s -wavescattering length, the interatomic potential supports atwo-body bound state, such that the relevant degrees offreedom are bound ↑↓ -molecules that form at T ∗ (cid:29) T c and condense below T c to form a superfluid. While thesituation appears to be comparatively clear in the afore-mentioned two limits, the situation at unitarity above T c has remained ambiguous. In fact, the question of whetherstrong pairing correlations survive above T c is intrinsi-cally challenging due to strong interactions in the uni-tary limit. Non-perturbative methods, either stochasticapproaches or semi-analytic methods (including resum-mation techniques) are therefore ultimately required toanalyze the UFG even above the phase transition. III. MANY-BODY METHODS
In this section we briefly introduce the two theoreticalapproaches that we use to analyze the spin susceptibilityof the polarized unitary Fermi gas in the normal phase.
A. Lattice field theory
To evaluate the thermodynamics of the UFG in a non-perturbative fashion, we use the CL approach. The par-tition function is given by Z = Tr (cid:104) e − β ( ˆ H − µ ↑ ˆ N ↑ − µ ↓ ˆ N ↓ ) (cid:105) , (3)where β = 1 /T is the inverse temperature. Moreover,we have introduced the chemical potential µ s and theparticle-number operator ˆ N s for the spin projection s .With this as a starting point, we regularize our Hamilto-nian by putting the problem on a spatial lattice of extent V = N x × N x × N x , where we set the lattice spacing (cid:96) = 1 for all directions. We also employ a Suzuki-Trotterdecomposition to discretize the imaginary time directioninto N τ = β/ ∆ τ slices, leading to a (3 + 1) dimensionalspacetime lattice of extent N x × N τ . Crucially, we cannow employ a Hubbard-Stratonovich transformation torewrite the quartic interaction term in Eq. (2) as a one-body operator coupled to a bosonic auxiliary field φ . Thisallows us to integrate out the fermions and thus expressthe above partition function as Z = (cid:90) D φ det ( M ↑ [ φ ] M ↓ [ φ ]) ≡ (cid:90) D φ e − S [ φ ] , (4)where D φ = (cid:81) τ, i dφ τ, i is the field integration measure.Note that the product runs over all space-time latticepoints. This integral is amenable to Monte Carlo (MC)importance sampling free from the sign-problem when det M ↑ [ φ ] = det M ↓ [ φ ] , i.e., in the SU(2) symmetriccase. In the case of unequal spin populations, which is thecase for unequal chemical potentials in the grand canon-ical ensemble, we may resort to the CL approach. Thelatter is able to deliver accurate results despite the pos-sibly severe sign problem, as recently demonstrated for avariety of non-relativistic fermionic systems [48–51]. Fur-ther details on the lattice approach to ultracold fermionsas well as on CL can be found in Refs. [43, 44, 52, 53].To tune the system to the unitary point, i.e., the limitof infinite s -wave scattering length, we choose the barecoupling constant g in Eq. (2) such that the lowest eigen-value of the lattice two-body problem matches the eigen-value obtained from the exact solution in a continuous fi-nite volume, which is given by Lüscher’s formula [54, 55].In this work, we use periodic boxes of linear extent N x = 11 , which was previously found adequate for the study of the normal phase of the UFG [50]. The inversetemperature is set to β = 8 . , such that the thermalwavelength λ T = √ πβ ≈ . falls in the appropriatewindow (cid:96) (cid:28) λ T (cid:28) N x for physics in the continuum limit.The temporal discretization was set to τ = 0 . (cid:96) . Notethat this choice of lattice parameters was previously usedin several studies and found to be suitable [50, 56–59].We fix the adaptive target discretization of the Langevintime to values in the range ∆ t L = 0 . to . and ex-trapolate (linearly) to the limit ∆ t L → . In our im-plementation of CL, we also included a regulator termto suppress uncontrolled excursions of the solver into thecomplex plane. The strength of this regulator term canbe quantified in terms of a parameter ξ which we setto ξ = 0 . , see Refs. [43, 48–51] for a detailed discussion.With respect to the spin susceptibility, we found that theresults are independent of the regulator within reason-able bounds. All results presented below reflect averagesof approximately decorrelated samples, which givesa statistical uncertainty on the order of − %. B. Virial expansion
The dilute limit associated with the limit of vanishingfugacity z = e βµ → (with µ = ( µ ↑ + µ ↓ ) / ) can beaddressed using the virial expansion, where the ratio ofthe interacting partition function Z to its noninteractingvalue Z is written as a series in powers of z . Morespecifically, for an unpolarized system we have ln( Z / Z ) = Q ∞ (cid:88) n =1 ∆ b n z n , (5)Here, ∆ b n is the interaction-induced change in the n -thvirial coefficient and Q = 2 V /λ T is the single-particlepartition function. While ∆ b has been known forarbitrary coupling strengths since the 1930’s [60] (seealso [61]), results for ∆ b only became available in thiscentury [62–67], those for ∆ b (only at unitarity) withinthe last decade [68–72], and, more recently, calculationsfrom weak coupling to unitarity were extended up to ∆ b [45, 46]. At a given order n , ∆ b n accounts for theinteraction effects of the n -particle subspace of the Fockspace. For the spin- / system of interest here, each ∆ b n can be further broken down into polarized contributions ∆ b mj , with m + j = n , coming from the subspace with m spin-up particles and j spin-down particles. In Ref. [45],the ∆ b mj have been calculated, which are essential forour purpose here as they are needed to calculate the prop-erties of polarized matter. Indeed, in this case, Eq. (5)becomes ln( Z / Z ) = Q ∞ (cid:88) n =2 (cid:88) m,j> m + j = n ∆ b mj z m ↑ z j ↓ , (6)where z s = e βµ s and s = ↑ , ↓ . Using the knowledge of ∆ b mj and differentiating with respect to βµ s , we obtainexpressions for the density, polarization, compressibility,and spin susceptibility within the virial expansion. Inparticular, the interaction effect on the individual spinparticle numbers is given by ∆ N s = Q ∞ (cid:88) n =2 (cid:88) m,j> m + j = n ( mδ s ↑ + jδ s ↓ ) ∆ b mj z m ↑ z j ↓ . (7)From this, one obtains the particle number of the speciesassociated with spin s as N s = N ( z s ) + ∆ N s , where N ( z s ) is the particle number for a noninteracting spin-less system at fugacity z s . Using the latter expressions,the total particle number in the interacting system is N = N ↑ + N ↓ , the magnetization is M = N ↑ − N ↓ , andthe relative polarization is p = M/N . In the same way,the interaction effect on the susceptibility is given by ∆ χ = λ T π Q ∞ (cid:88) n =3 (cid:88) m + j = n ( m − j ) ∆ b mj z m ↑ z j ↓ . (8)Using results up to fifth order, it is possible and bene-ficial to sum the virial expansion using the Padé resum-mation method, as explained in Ref. [46]. Hence, foreach physical quantity, we calculate the coefficients of arational function of z , F ( z ) = P M ( z ) Q K ( z ) = p + p z + · · · + p M z M q z + · · · + q K z K , (9)for each fixed value of βh , where h = ( µ ↑ − µ ↓ ) / with M + K = n max . Here, n max = 5 is the maximum avail-able order in the virial expansion. Note that the resum-mation F ( z ) is defined such that the virial expansion isreproduced by expanding F ( z ) up to order n max in z . IV. NUMERICAL RESULTS FOR THE SPINSUSCEPTIBILITY
In this section we present our results for the spin sus-ceptibility as defined in Eq. (1). To put this quan-tity in dimensionless form, we use the Pauli suscepti-bility χ P = 3 n/ (2 ε F ) , which reflects the spin suscep-tibility of a noninteracting Fermi gas at T = 0 where ε F = T F = k / , k F = (3 π n ) / and n is the to-tal density of the interacting system at a given value of µ = ( µ ↑ + µ ↓ ) / and h = ( µ ↑ − µ ↓ ) / . The spin-balancedlimit corresponds to βh = 0 , whereas values of βh > reflect spin-polarized systems.In the top panel of Fig. 1, we show our results forthe spin susceptibility as a function of T /T F for βh = 0 (blue squares). We find that, across all studied tem-peratures, the spin susceptibility is lower than that ofthe ideal Fermi gas (gray dashed-dotted line). At hightemperature, we find excellent agreement with the third-order virial expansion (dotted line). At even higher tem-perature, our results tend to the expected ∼ /T decayaccording to Curie’s law. In addition to our CL results, a variety of otherdeterminations of the spin susceptibility of the bal-anced UFG is shown in Fig. 1. Apart from the resultsfrom the Nozières-Schmitt-Rink (NSR) formalism [11],all theoretical approaches are found to be in agree-ment in the regime above T /T F (cid:38) . In the regime T c /T F < T /T F (cid:46) , curves obtained from the NSR for-malism [11] as well as from T-matrix approaches [6, 10]predict a substantial suppression of χ . Interestingly, alsoan AFQMC study in the grand-canonical ensemble [9](yellow diamonds) predicts a substantial suppression of χ below T /T F ≈ . , which differs from a more recentAFQMC study at fixed particle number [16]. This dis-crepancy was traced back to the lattice momentum cutoffwhich appears to leave a significant imprint in the results:while the earlier study relied on a spherical cutoff in mo-mentum space, the latter considered the full Brillouinzone [16, 17]. This was confirmed by the recent AFQMCstudy of Ref. [19]. Our CL results agree very well withthe latter AFQMC studies as well as with results fromstudies based on the LW formalism [7, 73] across the en-tire temperature range, see Fig. 1 for details.The only experimental value [74] shown in Fig. 1 (de-picted by the red circle in the top panel) differs byroughly two standard deviations from the AFQMC re-sults. Very recently, new experimental data for the spinsusceptibility of the UFG became available [30]. In thecorresponding study, consistency with a mean-field ap-proach was found, suggesting Fermi-liquid-type behav-ior. However, the presented values are trap-averaged suchthat a straightforward comparison to bulk-properties aspresented here is not possible (see App. A for a compar-ison of the integrated susceptibility). Finally, all theo-retical values are at odds with a previous experimentaldetermination of the spin susceptibility [24] (not shownin the plot). This discrepancy is likely due to trap aver-aging in the analysis of the experimental data (see alsoRef. [73]).Among the aforementioned studies, the two recentAFQMC determinations, our CL values, and the LW re-sults all display a very mild variation (on the order ofa few percent at most) of χ/χ P as a function of T /T F in a region above T c , before decreasing monotonically athigh enough temperature. In other words, χ displays avery broad maximum centered at the spin-gap temper-ature T s > T c . Notably, these results clearly deviatefrom those for the noninteracting gas, which is a signalof strong interaction effects, even above the superfluidtransition. Specifically, in the region around T s we have χ/χ P (cid:38) . for the noninteracting system (for all thepolarizations we considered; see below).In the bottom panel of Fig. 1, we show the spin sus-ceptibility for polarized systems with | βh | ≤ . . At veryhigh temperature T > T F , the curves for the polarizedsystems approach those from the virial expansion (dottedlines, color-coding as for CL) and eventually agree withthe noninteracting limit, as in the balanced case. More-over, at temperatures T > T c , we find excellent agree- . .. . / /. .. . / = .= .= .= .= .= . FIG. 1. Spin susceptibility χ in units of the Pauli sus-ceptibility χ P as a function of T /T F . The blue shaded areashows the superfluid phase (with T c from experiment [26]),the gray shaded area marks the maximum of the balancedcurve (blue squares) and the dash-dotted line represents thenoninteracting normalized susceptibility. (Top) CL values(squares) for the balanced UFG are compared to and ex-perimental value [74] (red circle) as well as theoretical re-sults from LW [7] (green triangles), T-matrix [6] (dark dashedline), extended T-matrix [10] (red solid line), NSR [11] (darksolid line), spherical cutoff AFQMC [9] (yellow diamonds) andtwo AFQMC studies [16, 17] (dark diamonds) and [19] (pur-ple octagons). (Bottom) CL results for the susceptibility fornonzero Zeeman field βh along with the corresponding valuesfrom the PRVE (thin solid lines). For comparison, the barethird-order VE is shown (dotted lines). ment of our CL results at large βh with those from thePRVE (thin solid lines, color-coding as for CL) up tofifth order, which is indeed not unexpected as interac-tion effects are more suppressed in that region. Saidagreement deteriorates as the unpolarized limit is ap-proached, where correlations are strongest. For βh > ,we find that the functional form of χ/χ P as a function of T /T F is qualitatively the same as for the balanced system.With increasing asymmetry βh , we observe a mild shiftof the spin susceptibility toward larger values. As men-tioned above, this is not unexpected as the system tendsto evolve slowly towards the noninteracting gas with in-creasing polarization. However, even for βh = 2 . , thespin susceptibility of the interacting system still clearlydiffers from the one in the noninteracting limit.Within the accuracy of our study, the aforementionedbroad maximum in χ as a function of the temperature evolves to a plateau-like feature with increasing imbal-ance βh . Below the plateau we find that the spin sus-ceptibility χ decreases rapidly with temperature, indi-cating the transition (or rather a crossover due to thefinite system size) into the superfluid phase. Of course,the suppressed susceptibility by itself does not yet allowto make a statement on the phase transition to the super-fluid phase since even a rapid suppression may be causedby, e.g., the formation of a pseudogap. However, a peakin the compressibility of the UFG occurs at the tem-peratures associated with the lower end of the plateaus(for βh (cid:38) . ) [50] which can be directly related to thetransition temperature. Finally, we observe that, com-pared to the balanced case, the rapid decrease of thesusceptibility sets in “delayed” when the temperature islowered for increasing βh . Together with the behavior ofthe compressibility, this indicates a decrease of the criti-cal temperature with increasing imbalance (see also ourdiscussion in Sec. V below).We note that the occurrence of a plateau in χ above T c is not observed in studies of the attractive 2D Hub-bard model where an “anomalous suppression” has beenfound [41, 42]. However, it should be pointed out that insuch 2D models the effective interaction is much strongerthan in the UFG, in the sense that in 2D a deep two-bodybound state forms even in the absence of a backgrounddensity. Similarly, Ref. [19] studied the 3D case for cou-plings stronger than the UFG (where a bound state ispresent at the two-body level) and found a clear increasein pairing correlations above the critical temperature T c .From our analysis, we conclude that the UFG doesnot show an anomalously suppressed spin susceptibilityfor imbalances βh (cid:38) . in the sense of the attractive 2DHubbard model. For βh (cid:46) . , the UFG may feature apseudogap region but its precise limits and characteriza-tion can not be conclusively studied with thermodynamicprobes (e.g., the spin susceptibility) used in our presentwork. In the following, we will relate these findings, asfar as possible, to the phase diagram of the spin-polarizedUFG. V. IMPLICATIONS FOR THE PHASEDIAGRAM
Despite intense theoretical and experimental investi-gations during the past two decades, our understand-ing of the full phase diagram of the polarized UFG isstill incomplete. With respect to the overall structure ofthe phase diagram, we summarize main results obtainedfrom non-perturbative studies beyond the mean-field ap-proximation. A more complete discussion of the precisenature of the superfluid phase, including potentially oc-curring exotic superfluids, is beyond this work (see, e.g.,Refs. [75–77] for reviews).
A. Overview on the phase structure
In Fig. 2 we consider the phase diagram in two differ-ent forms: The left panel depicts the phase diagram asspanned by the temperature T measured in units of thechemical potential µ , and the so-called Zeeman field h also given in units of µ . The color coding reflects thephase diagram as obtained from a functional renormal-ization group (fRG) study [13], the black solid line depictsthe corresponding second-order phase transition line be-tween the normal and the superfluid phase which ter-minates in a critical point (black dot) where it meets afirst-order transition line (see also Refs. [78, 79] for earlystudies of non-relativistic Fermi gases with this approachand Refs. [80–82] for corresponding reviews). In the bal-anced limit ( h/µ = 0 ), the fRG result for the criticaltemperature result agrees well with two experimental de-terminations of ( βµ ) c = 0 . [22, 26]. We add that a morerecent study based on LW theory also finds this value for ( βµ ) c in the balanced limit [83].While T c in the balanced case is well understood [84],much less is known about the critical field strength ( h/µ ) c at which the system undergoes a transition from thesuperfluid phase to the normal phase in the zero-temperature limit. This transition is expected to be offirst order [13, 75–77, 83], see also Ref. [85] for a recentdiscussion of quantum Lifshitz points and fluctuation-induced first-order phase transitions in imbalanced Fermimixtures. In this regime of the phase diagram, the resultsfrom fRG [13], experiment [22] and FN-DMC [86] are inaccordance (although not in full quantitative agreement)but disagree with those from LW theory [83] and the (cid:15) -expansion [87] which find a significantly higher criticalvalue ( h/µ ) c .In the right panel of Fig. 2, an experimental mea-surement of the phase diagram (color coding as in theleft panel) is shown in the plane spanned by the tem-perature T in units of the Fermi temperature T F , ↑ andthe polarization p = ( N ↑ − N ↓ ) /N ↑ + N ↓ ) . The solidblack line represents an estimate for the phase transi-tion line as extracted from an analysis of experimentaldata for a trapped gas based a local density approxima-tion, see Ref. [21] for details. In addition, we show thecritical line from LW theory [83] for comparison (blackdash-dotted line) which is identical to a more recent fullyself-consistent T-matrix calculation [88] and also agreeswell with experimental data at | p | = 0 . Although dif-ferent from the experimental determination at p > ,we add that the flatness of the phase transition line atsmall polarization observed in the LW studies is in ac-cordance with Monte Carlo studies based on the Wormalgorithm [89] which cover polarizations | p | (cid:46) . . Thelatter study is the only stochastic determination of T c be-yond p = 0 to date, to the best of our knowledge. Such aweak dependence of the phase transition line on the po-larization is also suggested by a recent CL study of thecompressibility [50]. Note that there are further stud-ies of the phase structure of the UFG based on a range of methods. For example, determinations of the second-order transition line based on NSR [11] and ETMA [90](red dashed line). Both predict a T c for the balanced gaswell above T /T F , ↑ = 0 . . Moreover, an early RG studyreported quantitative agreement with the experimentalvalue for the critical temperature in the balanced case aswell as for the location of the critical point [77], see alsoRef. [47] for a discussion.At zero temperature, the experimental determinationof the critical polarization p c by the ENS group exhibitsa large uncertainty [22] and does therefore not allow todiscriminate between state-of-the-art predictions for thisquantity from theory. However, the latter are all wellbelow the mean-field estimate which is p c ≈ . [77]. B. The pseudogap regime
In addition to the occurrence of (potentially inhomoge-neous) superfluid phases at low temperatures, pre-formedpairs in the normal phase could lead to non-Fermi liquidbehavior in some regions of the phase diagram, which werefer to as “pseudogap regime” in the following. Here, werelate our results for the spin susceptibility χ to variousregions in the phase diagram by extracting an estimatefor the spin-gap temperature T s . The latter is defined tobe the maximum of χ/χ P , see Fig. 1 for our results for χ/χ P as a function of the temperature for various val-ues of the Zeeman field h . As mentioned above, χ/χ P exhibits a relatively broad maximum, which complicatesa precise determination of its position. Nevertheless, asearch for such a maximum allows us to identify the re-gion where χ/χ P starts to display an anomalous suppres-sion, potentially related to the onset of pairing effects.We begin by analyzing our CL data for | βh | (cid:46) . .The position of the maximum is shown as filled squaresin both panels of Fig. 2. As a crude estimate for theuncertainty, we choose half the distance to the neighbor-ing data points which is shown in the plots as a thicksolid line accompanying the corresponding squares. Notethat this estimate does not account for statistical andsystematic errors in our CL study. In the right panel ofthe figure, the original determination of T s for the bal-anced gas from an ETMA calculation [10] as well as froma recent AFQMC study [19] are shown for comparison.While the spin-gap temperature from the former study isfound to be higher than the one presented here (althoughwithin the above described error), it is found to be lowerin the AFQMC calculations.As mentioned above, for . (cid:46) βh (cid:46) . , the CL datafor the normalized spin susceptibility does not show adistinct maximum but is rather characterized by a broadplateau. Towards lower temperatures, we then observe arapid decrease which indicates the onset of the superfluidphase transition. Given the accuracy of our present data,we refrain from indicating the positions of the maximaof χ/χ P by filled squares as we do for | βh | (cid:46) . butonly represent the plateau as solid lines in both panels . . . ./...... / normalsuperfluid . . . . ..... / (ETMA)(AFQMC) normalsuperfluid PS FIG. 2. Phase diagrams of the spin-polarized UFG. (Left) Phase diagram spanned by the dimensionless temperature ( βµ ) − and the dimensionless Zeeman field h/µ . The black solid line depicts the second-order phase transition line from a fRG study [13]where the black dot represents the location of the critical point. The thick line reflects the spin-gap temperature T s as obtainedfrom the PRVE along with an uncertainty estimate (shaded area, see main text). (Right) Phase diagram spanned by T /T F , ↑ and the polarization as measured in experiment [21] (gray symbols reflect experimental measurements, lines are the inferredphase boundaries) and compared to a recent determination via LW theory [83] (dot-dashed lines) as well as ETMA [10] (reddashed line). For the balanced limit T s is shown as determined via AFQMC [19] and ETMA [10]. In both panels, squares reflectCL data (see text) and critical values correspond to experimental values from the MIT group [26] (red triangles) and ENSgroup [22, 25] (green triangles), LW results [83] (orange triangle), FN-DMC calculations [86] (dark blue triangles), (cid:15) -expansion(purple triangle) and Worm-MC data [89] (gray-shaded area). of Fig. 2. The lower ends of these lines correspond tothe last data point before the aforementioned rapid de-crease at lower temperatures sets in. In the phase dia-gram spanned by the temperature and polarization (rightpanel of Fig. 2), we observe that the lower ends of theplateaus for βh = 1 . , . , and . overlap with the su-perfluid phase boundary obtained from LW theory [83].Taking at face value the position of the phase bound-ary obtained from LW theory, our CL results suggestthat there is no room for a regime with an anomalouslysuppressed spin susceptibility in this region of the phasediagram. We emphasize that the lines associated withthe plateaus for βh = 1 . , . , and . should not beconfused with the thick solid lines for | βh | (cid:46) . whichindicate an estimate for the uncertainty of the spin-gaptemperature T s .Similarly, in the phase diagram spanned by the tem-perature and the Zeeman field (left panel of Fig. 2), thelines associated with the plateaus of the spin susceptibil-ity for . ≤ βh ≤ . extend down to the phase bound-ary obtained from a fRG study [13]. In the latter study aprecondensation regime has been identified (blue hatchedband). In this regime, the gap vanishes in the long-rangelimit but “local ordering” is still present on intermedi-ate length scales, leading to a strong suppression of thedensity of states above the superfluid transition. As dis-cussed in Ref. [13], this precondensation regime may beused to estimate the emergence of a pseudogap region inthe phase diagram. However, the definition of this regimecannot be straightforwardly related to the definition ofthe pseudogap regime based on the spin susceptibilityunderlying our present analysis. In addition to the CL results above, we have performedthe same analysis for the PRVE. The position of themaximum of χ/χ P as a function of T /T F is indicatedby the thick line (continuous and dashed, see below) inFig. 2 (left panel). The corresponding uncertainty esti-mate (dark and light shaded area, see below) reflects thedistance between the points where χ/χ P is . of themaximum value. The solid thick line and correspond-ing dark shaded area reflect the parameter region wherewe expect the PRVE to be quantitatively trustworthy;in particular, part of the dark shaded area overlaps (andagrees with) the CL results described above. The dashedthick line and corresponding light shaded area displaythe parameter region where resummation techniques be-gin to fail due to strong interaction effects, in particularat low polarizations.Judging from our analysis of the spin susceptibility inthe light of other phase-diagram studies, we presently ex-pect that a notable pseudogap regime may only developin the regime βh (cid:46) . , if present at all. In particular,we do not observe a pseudogap regime of the form re-ported for the attractive Hubbard model [41, 42]). Ourdata is compatible with the picture of a narrow “pseudo-gap band” above the superfluid phase which continuouslyshrinks as βh is increased until it eventually disappears.In this scenario one could interpret the observation ofthe plateau-like feature in the susceptibility for βh (cid:38) . as the absence of a pseudogap regime in this part of thephase diagram. The black shaded areas in both panels ofFig. 2 highlight the most-likely regions in which a pseu-dogap may develop, based on our findings. VI. DISCUSSION & OUTLOOK
To summarize, we have presented a numerical cal-culation of the magnetic susceptibility for the spin-imbalanced UFG based on the CL approach. In the bal-anced case, our results agree with several previous resultsfrom the literature across the entire temperature rangein the normal phase. For the spin-polarized case, we vali-date the accuracy of these CL results with a PRVE up tofifth order, with good agreement ranging from high tem-peratures until well below T F . Furthermore, we use thisquantity as a probe for the influence of pairing in the nor-mal phase. Our data shows a slight suppression of χ/χ P for small imbalances with a spin-gap temperature wellabove T c . A strong suppression, as it is for instance ob-served in 2D Hubbard models superconductors [41, 42],is not observed. Nevertheless, the occurrence of a pseu-dogap regime for smaller imbalances ( βh (cid:46) . ) cannotbe ruled out with our present study.For βh (cid:38) . , we report a plateau-like feature of thespin susceptibility just above the phase transition anda subsequent abrupt drop at low temperatures. Such asuppression is expected to eventually occur in the low-temperature limit due to the formation of a pairing gap.Notably, we find that the onset of the rapid decreaseof the normalized spin susceptibility χ/χ P is shifted to-wards lower temperatures when βh is increased whichsuggests a decrease of the critical temperature for su-perfluidity with increased imbalance, in agreement withprevious studies of the phase diagram. In our case, thisinterpretation is supported by the fact that the positionof the lower end of the plateau in the susceptibility coin-cides with the peak of the compressibility [50]. However,for a rigorous determination of the βh -dependence of thecritical temperature, a finite-size scaling analysis as wellas an extrapolation to the zero-density limit (as, e.g., inRef. [29]) should be performed.The shrinking of a potential pseudogap regime with in-creasing βh deduced from our analysis of the spin suscep-tibility supports the picture that the nature of the normalphase above the critical polarization at low temperaturesis well described by Fermi liquid theory. This has beenpreviously conjectured based on qualitative results froma T-matrix study [91] as well as FN-DMC [86] and alsoagrees with experimental findings for large imbalances atlow temperatures [22, 23, 25]. It must be noted, how-ever, that our present study cannot be conclusive in thisregard as our analysis solely relies on thermodynamic ob-servables.In summary, our findings further corroborate that ther-modynamic quantities are not ideal probes for pseudogapphysics in the UFG. In particular, these results show thatan analysis of the spin susceptibility and the associatedspin-gap temperature is not sufficient by itself to fullyunderstand pairing effects, low energy excitations andthe potential formation of a pseudogap regime above thecritical temperature.As a consequence for experiments, this suggests that a more complete characterization of a potential pseudo-gap regime is likely best achieved by probing the spectralfunction as in Ref. [3, 27, 36], where a back-bending ofthe dispersion relation near k F would indicate pairingeffects [31]. However, these experiments are subject toambiguous interpretation and consensus has yet to bereached.Perhaps a combination with the recent developmentof flat trap geometries in ultracold Fermi gases [92, 93]will lead to a more complete understanding of the UFGabove the superfluid transition. Moreover, modern spec-troscopic techniques, which allowed to discern betweentwo- and many-body pairing in 2D Fermi gases [28], couldshed some more light on this matter. Finally, we notethat a viable alternative to spectroscopic probes could bethe investigation of density-density correlations in mo-mentum space, the so-called shot-noise, which exhibitsdistinct signatures of pairing [51, 94]. This is work inprogress. ACKNOWLEDGMENTS
The authors would like to thank Tilman Enss andYoram Alhassid for useful comments on the manuscript.J.B. acknowledges support by the DFG under grantBR 4005/4-1 (Heisenberg program) and BR 4005/5-1.J.B. and L.R. acknowledge support by HIC for FAIRwithin the LOEWE program of the State of Hesse. Thismaterial is based upon work supported by the NationalScience Foundation under Grants No. PHY1452635and PHY2013078. Numerical calculations have beenperformed on the LOEWE-CSC Frankfurt.
Appendix A: Comparison of integrated susceptibility
As mentioned in the main text, a recent experimen-tal study [30] presented trap-averaged data for the spin-susceptibility of the spin-balanced UFG. For the bulk sys-tem, as studied here via periodic boundary conditions,also the integrated spin-susceptibility ¯ χ = (cid:90) µ −∞ d µ (cid:48) χ ( µ (cid:48) ) λ T (A1)was measured. We show a comparison of our data inthe normal phase (i.e., βµ (cid:46) . ) to this measurement inFig. 3. We observe overall good agreement. The error-bars of the CL values reflect linearly propagated statisti-cal errors from the bare data. Appendix B: Literature data for critical values
In Tab. I, we summarize values for the critical polar-ization p c and the critical magnetic field h c (in units of VE, 3rd orderexperiment (non SF)CL, interpolated integrationexperiment (SF)
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