Effective theory and universal relations for Fermi gases near a d -wave interaction resonance
EEffective theory and universal relations for Fermi gases near a d -wave interactionresonance Pengfei Zhang, Shizhong Zhang, ∗ and Zhenhua Yu † Institute for Advanced Study, Tsinghua University, Beijing, 100084, China Department of Physics and Center of Theoretical and Computational Physics,The University of Hong Kong, Hong Kong, China (Dated: October 8, 2018)In this work, we present an effective field theory to describe a two-component Fermi gas neara d -wave interaction resonance. The effective field theory is renormalizable by matching with thelow energy d -wave scattering phase shift. Based on the effective field theory, we derive universalproperties of the Fermi gas by the operator product expansion method. We find that beyond thecontacts defined by adiabatic theorems, the asymptotic expressions of the momentum distributionand the Raman spectroscopy involve two extra contacts which provide additional information ofcorrelations of the system. Our formalism sets the stage for further explorations of many-bodyeffects in a d -wave resonant Fermi gas. Finally we generalize our effective field theory for interactionresonances of arbitrary higher partial waves. Introduction.
Correlations of d -wave symmetry are offundamental interest in modern physics. One outstand-ing example is the d -wave Cooper pairing in high- T c superconductors which provides a paradigmatic case ofstrongly correlated electron systems [1]. In cold atomsystems, strong d -wave correlations can also be gener-ated close to a d -wave Feshbach resonance, as has beendemonstrated experimentally in Cr [2, 3]. While it isgenerally believed that, compared with s -wave resonance,atomic gases close to higher partial wave resonances suf-fer more rapid atom loss, recent spectroscopic measure-ments around a p -wave Feshbach resonance indicate thatquasi-equilibrium states of such systems exist and theiruniversal properties can be investigated [4]. Theoreti-cally, however, many-body physics with resonant d -waveinteractions has been rarely studied and, in particular,an appropriate minimal model is still lacking.In this work, we consider a two-component Fermi gasnear a d -wave interaction resonance. We construct aneffective low-energy field theory, the bare coupling con-stants of which are renormalized by matching with the d -wave scattering phase shift cot δ ( k ) = − / ( Dk ) − / ( vk ) − / ( Rk ). The super volume D , the effective vol-ume v and the effective range R are the minimal set of pa-rameters that is needed to parametrize the inter-fermioninteractions. Furthermore, we use the effective theory,combined with the operator product expansion (OPE)method, to derive universal properties of the Fermi gaswhen the average inter-particle distance is much largerthan the range r associated with the inter-fermion in-teraction. We find that the universal behaviour of thesystem is governed by five quantities, three of which arerelated to the variation of the system energy with re-spect to the three d -wave scattering parameters, analo-gous to the contacts defined in the case of s - and p -wavecase [5–15]. However, we find that the sub-leading termsof the tails of momentum distribution and Raman spec-troscopy involve two new contacts, which further charac- terise the correlations of the system at short distances.Our effective field theory provides a minimal model forstudying other many-body physics of Fermi gases near a d -wave resonance. We show that the d -wave contacts re-veal much richer correlation structures than the s -wavecase. Finally we generalize our formalism for resonantinteractions to arbitrary higher partial waves. Effective field theory.
To describe the low energy de-grees of freedom close to a d -wave interaction resonance,we adopt a Lagrangian field theory and requires that theLagrangian density to obey the following symmetry re-quirements: (1) Rotation symmetry. (2) Galilean invari-ance such that the scattering of two fermions in vacuumdoes not depend on their center of mass momentum. Inaddition, we aim to establish a local effective field theory,which should be renormalizable in the low energy limit interms of the minimal set of scattering parameters D, v, R ,describing the d -wave scattering phase shift.The Lagrangian density of the effective field theorythat we construct for the system up to a momentum cut-off Λ is given by L = (cid:88) i =1 ψ † i (cid:18) i∂ t + ∇ M (cid:19) ψ i + (cid:96) (cid:88) m = − (cid:96) ¯ g ( d † (cid:96)m Y m + h.c. )+ η (cid:96) (cid:88) m = − (cid:96) d † (cid:96)m (cid:34) i∂ t + ∇ M + ¯ z (cid:18) i∂ t + ∇ M (cid:19) − ¯ ν (cid:35) d (cid:96)m (1)where (cid:96) = 2 and the operator Y m is given by Y m = 14 (cid:88) a,b = x,y,z C mab [( ∂ a ψ )( ∂ b ψ ) − ( ∂ a ∂ b ψ ) ψ + ( ∂ b ψ )( ∂ a ψ ) − ψ ( ∂ a ∂ b ψ )] . (2)The field operator ψ i is the annihilation operator forfermions in state | i (cid:105) . M is the mass of the fermions. Wetake (cid:126) = 1 throughout. The dimer fields d (cid:96)m of azimuthal a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov quantum number m mediate the d -wave interaction be-tween the two fermions, which we assume to be isotropic. C mab are the Clebsch-Gordon coefficients when transform-ing k i k j /k to the spherical harmonics √ πY m (ˆ k ). Interms of a i, k and b (cid:96)m, k , the Fourier transformations ofthe operators ψ i and d (cid:96)m , the fermion-dimer coupling inthe Lagrangian L = (cid:82) d r L [the second term in Eq. (1)]takes the form L fd = ¯ g (cid:114) πV (cid:96) (cid:88) m = − (cid:96) (cid:88) p , k [ k (cid:96) Y (cid:96)m (ˆ k ) b † (cid:96)m, p a , p + k a , p − k + h.c. ] , (3)where V is the volume of the system. Since we focus onthe effects of the d -wave resonance, we neglect possiblebackground scatterings of either s - or p -wave symmetry,and those due to direct couplings between the fermions.The term proportional to η = ± ν being its detuning. Unlike thecase for p -wave scattering, an extra term proportionalto the bare coupling constant ¯ z is constructed in orderto renormalize the effective range R [see Eq. (9)], whilestill respecting the Galilean invariance. As will be shownlater, it is necessary to take η = − s -wave and p -wave resonance models, and it isworthwhile to point out the differences. In the s -wavecase, Kaplan was the first to use an s -wave dimer field b , k to describe the non-relativistic scattering betweennucleons with a large s -wave ( (cid:96) = 0) scattering length a s [16]. In this case, the zero-range limit Λ → ∞ is welldefined with the choice η = 1 and ¯ z = 0 by matching thescattering matrix with the s -wave phase shift expansion k cot δ s ( k ) = − /a s . The same resonance model wasconstructed independently by Kokkelmans et. al. foratoms close to an s -wave Feshbach resonance [17], forwhich the dimer field b , k naturally represents the closedchannel molecules.Different from the s -wave case, low-energy scatteringin the p -wave channel is described by two parameters, k cot δ p ( k ) = − /v p − k /R p [18]. Here v p is the p -wavescattering volume and R p is the p -wave effective range.In this case, however, to obtain a renormalizable theorywith finite v p and R p in the low energy limit, one has totake η = −
1. This means that the free dimer field b m, k becomes ghost field with negative norm [18]. However,such negative norm is only relevant at a much higherenergy, of order of Λ , which is irrelevant for the low-energy physics described by the scattering phase shift δ p ( k ).In the d -wave interaction resonance, it is first impor-tant to note that the low-energy scattering phase shiftmust be retained up to order k , namely k cot δ d ( k ) = − /D − k /v − k /R ; the three interaction parameters D , v and R are the minimal set. This is because across the resonance, while the magnitude of D can be tunedto be much larger than the interaction range r , v/r and R/r are typically of order unity. Taking the zerolimit v → R → δ ( k ) →
0, which cannot describe theoriginal interacting system. In contrast, it is safe to takethe zero limit of the expansion coefficients of order higherthan k in k cot δ ( k ). Now, we note that in Eq. (1), theterm d † (cid:96)m ( i∂ t ) d (cid:96)m corresponds to the total energy of twoscattering fermions, and the term d † (cid:96)m ( −∇ / M ) d (cid:96)m cor-responds to the center of mass energy. The combination d † (cid:96)m ( i∂ t + ∇ / M ) d (cid:96)m thus corresponds to the relativescattering energy. As a result, we explicitly construct theextra term ¯ zd † (cid:96)m ( i∂ t + ∇ / M ) d (cid:96)m in Eq. (1) to matchthe k -dependence of k cot δ d ( k ) for d -wave resonances.Note that by construction, the Lagrangian Eq. (1) main-tains explicitly the Galilean invariance.The renormalizability of Eq. (1) is manifested by cal-culating the T -matrix, T ( P , k , k (cid:48) , Ω), of scattering be-tween two fermions with relative incoming (outgoing)momentum 2 k (2 k (cid:48) ) and total momentum P . Due tothe Galilean invariance of Eq. (1), one only needs to cal-culate in the center of mass frame, and the T -matrix isgiven by T m ( , k , k (cid:48) , Ω) = − π ¯ g k Y m (ˆ k ) Y ∗ m (ˆ k (cid:48) ) D ( , Ω) , (4)where | k | = | k (cid:48) | due to energy conservation and ˆ k = k / | k | and ˆ k (cid:48) = k (cid:48) / | k (cid:48) | . D ( P , Ω) is the full dimer propa-gator, given in Fig. 1(a) D − ( P , Ω)= ¯ D − ( P , Ω) − ¯ g π (cid:90) Λ0 dq q Ω − P / M − q /M , (5)where ¯ D ( P , Ω) is the bare dimer propagator given by¯ D ( P, Ω) = E p, + − E p, − η ¯ z (cid:18) − E p, + − − E p, − (cid:19) , (6)with E p, ± = P / M − (1 ∓ √ ν ¯ z ) / z the dimers’normal mode energies. In the case 1 + 4¯ ν ¯ z >
0, therealways exits one branch of ¯ D ( P, Ω) with negative weightcorresponding to the presence of ghost fields [27], irre-spective of the sign of η . The appearance of ghost fieldsis inevitable due to the requirement to renormalize notonly v but also R for d -wave interactions [see Eqs. (8) and(9)] [18]. In the case 1 + 4¯ ν ¯ z <
0, the poles of ¯ D ( P, Ω)move away from the real axis into the complex plane andby itself seems problematic. However, the low energyobservables predicted by the full coupled effective fieldtheory remains valid (see below). In Table I, we sum-marize the main differences between our d -wave effectivefield theory with the s - and p -wave cases.Matching T m ( , k ˆ k , k ˆ k (cid:48) , k /M + i
0) with cot δ d ( k ) = − /Dk − /vk − /Rk in the limit k →
0, we find the (cid:96) minimalparameters η ¯ z ghost field s -wave 0 a s p -wave 1 v p , R p − d -wave 2 D, v, R − (cid:54) = 0 YesTABLE I. Differences between our d -wave effective field the-ory with the s - and p -wave cases. Each cases are renormalizedto the minimal interaction parameters listed. renormalization conditions:1 D = − η π ¯ ν ¯ g M + 2Λ π , (7)1 v = η π ¯ g M + 2Λ π , (8)1 R = η π ¯ z ¯ g M + 2Λ π . (9)To keep values of D , v and R finite while taking the limitΛ → ∞ , we require η = −
1. Otherwise if η = 1, fromEq. (8), | v | < π/ and approaches zero. In fact, itturns out not possible to construct a purely fermionicmodel with contact inter-fermion interactions which re-produces the correct d -wave low energy scattering ampli-tude with finite parameters v and R in the limit Λ → ∞ .Thus it is crucial to introduce the dimer field with theconcomitant appearance of the ghost field which, how-ever, does not alter the low energy physics.The applicable regime of our effective field theory canbe analysed from the pole structure of T m in terms of therenormalized parameters T m ( , k ˆk , k ˆk (cid:48) , Ω)= − π k Y m ( ˆk ) Y ∗ m ( ˆk (cid:48) ) /M /D + M Ω /v + ( M Ω) /R + i ( M Ω) / . (10)For simplicity, let us consider the limit 1 /D → + . Thereal pole of T m at Ω → − with positive weight ∼ v corresponds to a physical two-fermion bound state ap-proaching threshold. However, since typically v ∼ r and R ∼ r , there are other complex poles at energies | Ω | ∼ /M r , which apparently violate the unitary con-dition on the S -matrix. The origin of these unphysicalpoles is the truncation of cot δ d ( k ). However, as long aswe are only interested in energy scales much smaller than1 /M r as we shall do in the following, our effective fieldtheory should give physically valid results. D -wave contacts. Effective field theory has served asan ideal formalism to elucidate the universal aspects ofquantum gases [19, 20]; in particular, the derivation ofuniversal relations involving the so-called contacts usingthe operator product expansion (OPE) [6, 9, 20–28]. Thisis an operator relation for the product of two operators (a) (b)(c) (d) FIG. 1. Feynman diagrams for: (a) the T -matrix for twofermions; (b) the matrix element of ψ † i ( R + r / ψ i ( R − r / at small separation [21, 29] O i (cid:16) R + r (cid:17) O j (cid:16) R − r (cid:17) = (cid:88) l f ijl ( r ) O l ( R ) (11)where O i are the local operators and f ijl ( r ) are the expan-sion functions. A similar expansion can also be carriedout in the time domain. OPE is an ideal tool to exploreshort-range physics, r (cid:28) r (cid:28) n − / in a field theorycontext. Here n is the average density.In the case of d -wave interactions, we first define threecontact densities (operators) as the derivatives of the La-grangian density L with respect to D − , v − and R − ,by using Eqs. (7) to (9)ˆ C D M ≡ δ L δ ( D − ) = M ¯ g π (cid:88) m d † (cid:96)m d (cid:96)m , (12)ˆ C v M ≡ δ L δ ( v − ) = M ¯ g π (cid:88) m d † (cid:96)m (cid:18) i∂ t + ∇ M (cid:19) d (cid:96)m , (13)ˆ C R M ≡ δ L δ ( R − ) = M ¯ g π (cid:88) m d † (cid:96)m (cid:18) i∂ t + ∇ M (cid:19) d (cid:96)m . (14)Note that we have used the equation of motion satis-fied by d (cid:96)m to obtain the concise expression of Eq. (13).While ˆ C D is proportional to the total dimer density, ˆ C v and ˆ C R can be considered as proportional to the onesweighted by the powers of the internal energy of thedimers. A similar structure has been found for p -wavecontacts [13]. In addition, as we will see from the tailsof the momentum distribution and the Raman spec-troscopy, it is also useful to introduce two extra d -wavecontact densities asˆ C D,P M ≡ M ¯ g π (cid:88) m d † (cid:96)m (cid:18) − ∇ M (cid:19) d (cid:96)m , (15)ˆ C v,P M ≡ M ¯ g π (cid:88) m d † (cid:96)m (cid:18) i∂ t + ∇ M (cid:19) (cid:18) − ∇ M (cid:19) d (cid:96)m , (16)which, compared with Eqs. (12) and (13), are furtherweighted by the kinetic energy of the dimers, and encap-sulate additional information of correlations at short dis-tances. The spatial integration of the expectation valuesof the contact densities are defined as the d -wave con-tacts: C D = (cid:82) d r (cid:104) ˆ C D (cid:105) , C v = (cid:82) d r (cid:104) ˆ C v (cid:105) , C R = (cid:82) d r (cid:104) ˆ C R (cid:105) , C D,P = (cid:82) d r (cid:104) ˆ C D,P (cid:105) , and C v,P = (cid:82) d r (cid:104) ˆ C v,P (cid:105) . FromEqs. (12-14), one can write down the adiabatic theorems, ∂F∂α − = − C α M ; α = D, v, R, (17)where F is the free energy of the system. To illustratethe use of the effective field theory, we now derive someuniversal relations between the introduced contacts andvarious physical observables. Short distance expansion.
The tails of the momentumdistribution can be extracted from the one-body densitymatrix ρ i ( R , r ) = (cid:104) ψ † i ( R + r / ψ i ( R − r / (cid:105) and canbe measured experimentally by the time-of-flight tech-nique [30, 31]. To relate ρ i ( R , r ) with the d -wave con-tacts, we calculate the OPE by matching the matrix el-ements of operators from an incoming state | I (cid:105) with twofermions of different species having momentum P / k ˆ k and P / − k ˆ k to an outgoing state | F (cid:105) with two fermionshaving momentum P / k ˆ k (cid:48) and P / − k ˆ k (cid:48) . The totalenergy of the fermion pair is E = P / M + k /M . Sincewe are interested in the rotationally invariant case, wewill average over the direction of the total momentum P . The case without rotational invariance can be calcu-lated similarly. The matrix element of ρ i is given by thediagram shown in Fig. 1(b) and the result is (cid:104) F | ρ i ( R , r ) | I (cid:105) = 4 πM ¯ g k (cid:88) m Y m (ˆ k ) Y ∗ m (ˆ k (cid:48) ) D ( P, E ) × (cid:20) δ ( r ) + k πr − r ( k + P k / π (cid:21) + const . + o ( r ) . (18)Likewise, we calculate the matrix elements of the contactdensities according to the diagrams shown in Fig. 1(a, b). We find (cid:104) F | ˆ C D | I (cid:105) = M ¯ g k (cid:88) m Y m (ˆ k ) Y ∗ m (ˆ k (cid:48) ) D ( P, E ) , (19) (cid:104) F | ˆ C v | I (cid:105) = k (cid:104) F | ˆ C D | I (cid:105) , (20) (cid:104) F | ˆ C R | I (cid:105) = k (cid:104) F | ˆ C D | I (cid:105) , (21) (cid:104) F | ˆ C D,P | I (cid:105) = P (cid:104) F | ˆ C D | I (cid:105) / , (22) (cid:104) F | ˆ C v,P | I (cid:105) = P k (cid:104) F | ˆ C D | I (cid:105) / . (23)After Fourier transforming Eq. (18) and matching withEqs. (19) to (23), we find that the momentum distribu-tion n i ( q ) of the i th species has a tail in the large q -limit( n / (cid:28) q (cid:28) /r ) n i ( q ) = 1 V (cid:20) C D π + C v π q + 9 C R + 2 C v,P π q (cid:21) , (24)whose magnitude depends on the d -wave contact densi-ties. The presence of the additional quantity C v,P , whichcan not be derived from the adiabatic theorems (17), inthe momentum tail can be understood in the followingway. Let us consider a single pair of interacting fermions.In the center of mass frame of the pair where C v,P iszero according to Eqs. (16) and (24), the momentumtail n com ( q ) involves only C α for α = D, v, R . How-ever, when we switch to a reference frame moving with arelative velocity u , the momentum tail of the pair in thisnew frame should be n ( q ) = n com ( q − m u ). Expansionof n com ( q − m u ) to order 1 /q leads to an extra term ∼ u C v in n ( q ), which is exactly the generally nonzero C v,P term in Eq. (24) in this case. Note that the Galileaninvariance garrauntees C D and C v having the same val-ues in different reference frames [cf. Eq. (17)]. Quantitiessimilar to C v,P have been introduced for p -wave interac-tions in three dimensions [13, 32, 33].The tails of the momentum distribution n i ( q ) seemsto yield a divergent number of fermions. Actually, bythe U (1) gauge invariance of Eq. (1), the conserved totalparticle number is given byˆ N = (cid:90) d r (cid:110) (cid:88) i =1 , ψ † i ψ i − (cid:88) m ( d † m (cid:2) z (cid:0) i∂ t + ∇ / M (cid:1)(cid:3) d m + h.c. ) (cid:111) . (25)Using the renormalization relations (7), (8) and (9), onecan verify that the divergent part of n i ( q ) at large q is cancelled by the dimer terms; the dimer terms can beconsidered as counterterms to the fermion densities. Notethat the factor ¯ z (cid:0) i∂ t + ∇ / M (cid:1) is due to the expansionof the bare dimer fields in terms of their normal modes. Short distance and time expansion.
Single-particlespectral function, which reveals fundamental propertiesof an interacting many-body system, such as pairing andpseudo-gap phenomena, can be measured using Ramanspectroscopy in atomic gases [34, 35]. When two Ramanlasers of frequency ω and ω and wave-vector k and k are applied, atoms can be excited from the initial inter-nal state | (cid:105) to the final internal state | (cid:105) by absorbingenergy ω = | ω − ω | and momentum q = k − k . Theresultant number of atoms transferred to state | (cid:105) is, bythe Fermi golden rule, proportional to the rate I Ra ( q , ω ) = − π ImΠ Ra ( q , ω ) , (26)Π Ra ( q , ω ) = − iV (cid:90) dtd r e iωt − i q · r (cid:104) T Q ( r , t ) Q † ( , (cid:105) , (27)with Q ( r , t ) ≡ ψ † ( r , t ) ψ ( r , t ).By calculating the OPE of Q ( r , t ) Q † ( ,
0) in boththe time and space domain, we find for ω > (cid:15) q ≡ q / M : πM I Ra ( q , ω ) = (cid:18) M ω − q (cid:19) / C D − q C D,P
M ω − q ) / + (cid:34) q (cid:112) M ω − q + 4 sinh − (cid:32) q (cid:112) M ω − q (cid:33)(cid:35) C v q + 2 q (7 q − q M ω + 60 M ω )3(2 M ω − q ) (4 M ω − q ) / C v,P + q − q M ω + 60 M ω (2 M ω − q ) (4 M ω − q ) / C R . (28)For (cid:15) q > ω > (cid:15) q / I Ra ( q , ω ) is given by Eq. (28)with the factor sinh − [ q/ (cid:112) M ω − q ] replaced bycosh − [ q/ (cid:112) − M ω + 2 q ]. I Ra ( q , ω ) = 0 when ω < (cid:15) q / q → I Ra ( , ω ) gives the radio-frequency re-sponse and involves only C v , C D and C R . The presenceof C D,P and C v,P in Eq. (28) can also be understoodfrom a Galilean covariance argument similar to the onegiven below Eq. (24). Discussion . The construction of the effective field the-ory Eq. (1) for d -wave resonance suggests a general pro-cedure for resonances of arbitrary higher partial waves.Consider a two-component Fermi gas with short-rangeinteractions, the phase shift in the (cid:96) -th scattering chan-nel can be written as k (cid:96) +1 cot δ (cid:96) ( k ) = − (cid:80) (cid:96)α =0 k α /a (cid:96)α + O ( k (cid:96) +2 ) in the low energy limit. To reproduce the phaseshift, we need only to generalize the dimer field term inEq.(1) to L d = (cid:96) (cid:88) m = − (cid:96) (cid:96) (cid:88) α =0 d † (cid:96)m ¯ z (cid:96)α (cid:18) i∂ t + ∇ M (cid:19) α d (cid:96)m , (29)and assume L fd to be the form of Eq. (3) with the fac-tor ¯ g (cid:112) π/V replaced by 4 π/ √ M V , which amounts toa rescaling of the dimer field d (cid:96)m . The relation betweenparameters { ¯ z (cid:96)α } to the physical scattering parameters { a (cid:96)α } can be established similarly by matching the scat-tering T -matrix to that of k (cid:96) +1 cot δ (cid:96) ( k ). One finds1 a (cid:96)α = ¯ z (cid:96)α M α + 2 π Λ (cid:96) − α )+1 (cid:96) − α ) + 1 , (30)for 0 ≤ α ≤ (cid:96) . For fixed { a (cid:96)α } , the zero range limitΛ → ∞ is attainable only if ¯ z (cid:96)α are all negative. Ourformalism sets the stage for the exploration of universalaspects of both few-body and many-body physics closeto a higher partial wave resonance. Further importantquestions remain to be investigated, including the effectsof long-range and multi-body interactions. Acknowledgment.
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