# Theoretical Approach to the Gauge Invariant Linear Response Theories for Ultracold Fermi Gases with Pseudogap

Authors: Hao Guo; Yan He
Journal: Advances in Condensed Matter Physics
 (2015)

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## Abstract

Recent experimental progress allows for exploring some important physical quantities of ultracold Fermi gases, such as the compressibility, spin susceptibility, viscosity, optical conductivity, and spin diffusivity. Theoretically, these quantities can be evaluated from suitable linear response theories. For BCS superfluid, it has been found that the gauge invariant linear response theories can be fully consistent with some stringent consistency constraints. When the theory is generalized to stronger than BCS regime, one may meet serious difficulties to satisfy the gauge invariance conditions. In this paper, we try to construct density and spin linear response theories which are formally gauge invariant for a Fermi gas undergoing BCS-Bose-Einstein Condensation (BEC) crossover, especially below the superfluid transition temperatureTc. We adapt a particular t-matrix approach which is close to the G0G formalism to incorporate noncondensed pairing in the normal state. We explicitly show that the fundamental constraints imposed by the Ward identities and Q-limit Ward identity are indeed satisfied.


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## Body

## 1. Introduction

Recently there is a broad literature on the subjects of response functions in superconductors and atomic Fermi gas superfluids [1–11], where interparticle interaction is strong enough such that the classical BCS theory is not adequate. Related experiments include the studies of the thermodynamic response functions and dynamical response [12–17]. Theoretically, linear response theories have been an important tool for studying the transport and dynamic properties of Fermi gases. Hence, it is important to assess the self-consistency of the linear response theories as well as comparing with the experimental results. There must be some general rules that the theory must follow. In [18, 19], several fundamental constraints associated with the conservation laws/Ward identities and the sum rules were addressed. Since the conservation laws are generically related to certain (gauge) symmetry of the theory, then in the broken-symmetry phase or ordered phase it is particularly difficult for many-body theories to satisfy all these constraints. It was also pointed out that the strict weakly interacting BCS mean field theory does pass all these types of testing both below and above Tc even when the pairing population is unbalanced [18, 19]. In other words, the linear response theories of BCS superfluids can be formulated into a fully gauge invariant theory. In the normal state, the simplest Nozieres-Schmitt-Rink (NSR) is also compatible with these gauge invariance conditions [20].In this paper, we try to build an ideal linear response theory for strongly correlated superfluids undergoing BCS-BEC crossover by a diagrammatic approach such that the fundamental constraints mentioned above can be satisfied. Our selected diagrams bear on those associated with the Goldstone modes due to the symmetry-breaking via the consistent-fluctuation-of-the-order-parameter (CFOP) approach and the conventional contributions, namely, the Maki-Thompson (MT) and Aslamazov-Larkin (AL) diagrams. As a price, we have to adapt a slightly modifiedG0G formalism to incorporate the pairing fluctuation effect. We emphasize that this approach is purely a theoretical attempt until now. However, it might be a necessary step to fully understand the transport properties of strongly correlated Fermi gases.The linear response theories must be consistent with several fundamental constraints [18, 19, 21] imposed by the Ward identities and Q-limit Ward identity [22]. It is well known that the Ward identities guarantee the gauge invariance of the theory. However, the Q-limit Ward identity is quite strange to most researchers in the condensed matter community. It in fact leads to the sum rules of compressibility and spin susceptibilities which further build the consistent connection between the single-particle thermodynamics and two-particle correlation functions. For spin response theory the Q-limit Ward identity is only meaningful to polarized Fermi superfluids [19]. In this paper, we focus on the unpolarized Fermi superfluids. The central difficulty of formulating the consistent linear response theory is to maintain the gauge invariance when the pseudogap self-energy is introduced by the pairing fluctuation effect. This is obviously beyond the CFOP approach since the total energy gap is now different from the order parameter.In the following sections, we first briefly review the CFOP linear response theories in both the density and the spin channels for BCS mean field theory; then we introduce the pairing fluctuation effects via a particulart-matrix formalism. We further carry on extra diagrammatic corrections in the two channels and verify that the new theories do maintain the gauge symmetry, respectively. Throughout this paper, we follow the convention c=ħ=kB=1.


## 2. BCS Mean Field Theory Approach

By using theσ to denote the spin or pseudospin ↑, ↓, the Hamiltonian for a two-component Fermi gas interacting via the attractive contact interaction g is(1)H=∫d3xψσ†xp^22m-μψσx-g∫d3xψ↑†xψ↓†xψ↓xψ↑x,where ψ and ψ† are the annihilation and creation operators of fermions, μ is the chemical potential, and m is the fermion mass. There is an implicit summation over the pseudospin indices σ. The Hamiltonian has a U(1)×U(1) symmetry [23](2)ψσ⟶e-iαψσ,ψσ†⟶eiαψσ†;ψσ⟶e-iSσαψσ,ψσ†⟶eiSσαψσ†,where S↑,↓=±1 and α is the phase parameter of those transformations. The first U(1) symmetry is well known for relating to the electromagnetism (EM). If the particle is charged, this symmetry naturally becomes a gauge symmetry. For a charge neutral system, the symmetry is still associated with the mass current conservation. The second symmetry is the spin rotational symmetry which is associated with the spin current conservation. Our linear response theories in the density and spin channels must respect these two symmetries, respectively. The central idea is to “gauge” the U(1) symmetries by introducing two types of weak external fields. In the density channel, it is the weak EM field Aμ=(ϕ,A), while in the spin channel it is Aμ≡(Bz,m), where Bz is the z component of the magnetic field (assuming z is the axis of spin rotation) and m is the magnetization.After taking BCS mean field approximation, the order parameter or superconducting gap function is introducedΔsc(x)=-g〈ψ↑(x)ψ↓(x)〉. The first U(1) symmetry is spontaneously broken while the second is not. As can be seen in [18], this brings significant difference between the linear response theories in the two channels. For a homogeneous system, the BCS Hamiltonian can be expressed as(3)HBCS=∑kσψpσ†ξpψpσ+∑pΔscψ-p↑ψp↓+∑pΔscψp↓†ψ-p↑†,where ξp=p2/2m-μ. As a familiar result, the BCS Green and anomalous Green functions in the momentum space are(4)Gsciωn,p=uscp2iωn-Escp+vscp2iωn+Escp,Fsciωn,p=-uscpvscpiωn-Escp+uscpvscpiωn+Escp,where iωn is the Fermion Matsubara frequency and Escp=ξp2+Δsc2 is the quasiparticle energy dispersion. Hereinafter we use the subscript “sc” to emphasize that these discussions are only under the BCS mean field approximation. Define P≡(iωn,p); the number and gap equations are determined by n=2∑P‍Gsc(P) and Δsc=-g∑P‍Fsc(P). These identities give(5)n=∑p1-ξscpEscp1-2fEscp,1g=∑p1-2fEscp2Escp.The bare Green function is G0(P)=(iωn-ξp)-1. The Dyson equation gives Gsc-1(P)=G0-1(P)-Σsc(P) where Σsc(P)=-Δsc2G0(-P) is the BCS self-energy.


### 2.1. Density Channel

In the density channel, the system is perturbed by an effective external EM fieldAμ, and the Hamiltonian becomes H=HBCS+HI with(6)HI=∑pqσψp+qσ†γμp+q,pAμqψpσ,where γμ(p+q,p)≡γμ(P+Q,P)=(1,(p+q/2)/m) is the bare EM interaction vertex. Here Q≡qμ=(iΩl,q) is the external four momentum, where Ωl is the boson Matsubara frequency. The bare vertex satisfies the “bare” Ward identity(7)qμγμP+Q,P=G0-1P+Q-G0-1P.In a gauge invariant EM linear response theory, a full EM interaction vertex Γμ (the bare and full EM vertices are shown in Figure 1) which satisfies the full Ward identity(8)qμΓscμP+Q,P=Gsc-1P+Q-Gsc-1Pmust be found so that the perturbed current can be expressed as δJμ(Q)=Kscμν(Q)Aν(Q) with Kscμν(Q) determined by the Kubo formalism(9)KscμνQ=nmhμν+2∑PΓscμP+Q,PGscP+QγνP,P+QGscP,where hμν=-ημν(1-ην0) with ημν=diag(1,-1,-1,-1) being the metric tensor. By using the Ward identity (8), it is easy to show that qμKscμν(Q)=0, which further leads to the conservation of perturbed current qμδJμ(Q)=0. Hence the linear response theory is indeed gauge invariant. Under the framework of BCS mean field theory, such full vertex can be obtained either by Nambu’s integral-equation approach [23, 24] or by the CFOP approach. However, the Q-limit Ward identity provides an independent consistency check of the theory. It is not known if Nambu’s approach can pass this check, while the vertex given by the latter approach is proved to satisfy this condition(10)limq→0⁡Γsc0P+Q,Pω=0=∂Gsc-1P∂μ=1-∂ΣscP∂μ.Details can be found in [18]. This identity not only builds a consistent connection between the one-particle thermodynamics and two-particle response functions but also acts as the sufficient and necessary condition for the compressibility sum rule(11)∂n∂μ=-Ksc000,q⟶0.Figure 1
EM interaction vertex. The left one is the bare vertex, and the right one is the full vertex. The solid line denotes the fermion line and the dashed line denotes the photon line.The expression of this gauge invariant interaction vertex given by the CFOP approach is given in our past work [18](12)ΓscμP+Q,P=γμP+Q,P+CollscμP+Q,P+MTscμP+Q,P,where the second term,(13)CollscμP+Q,P=ΔscΠμQG0-P-Q+ΔscΠ¯μQG0-P,corresponds to the excitations of Nambu-Goldstone modes due to the breaking of the U(1) symmetry and the third term,(14)MTscμP+Q,P=-Δsc2G0-Pγμ-P,-P-QG0-P-Q,is the famous MT diagram which is shown in Figure 2. The expressions of Πμ and Π¯μ are given in Appendix A. These two terms originates from summing up the diagrams with photon-fermion interaction lines inserted at any possible position. By using the equalities [18] qμΠμ(Q)=2Δsc and qμΠ¯μ(Q)=-2Δsc, it can be shown that(15)qμCollscμP+Q,P=2ΣscP-2ΣscP+Q.Combining with the equality(16)qμMTscμP+Q,P=ΣscP+Q-ΣscP,the Ward identity (8) in the BCS mean field level can be proved.Figure 2
Maki-Thompson diagram. The wavy line represents the pair propagator.


### 2.2. Spin Channel

In the spin channel, theU(1) spin rotational symmetry is also “gauged” by introducing an effective external field. However, this symmetry is not broken by the order parameter. Therefore, we expect that the structure of the spin linear response theory is simpler than that of its density counterpart. The “bare” spin interaction vertex is dependent on the pseudospin: γSσμ(p+q,p)≡γSσμ(P+Q,P)=Sσ(1,(p+q/2)/m) with S↑=1, S↓=-1 and the subscript “S” referring to the “spin.” The quantity Sσ indicates that the vertex has different signs for different pseudospin indices; see Figure 3. It also respects the “bare” Ward identity in the spin channel(17)qμγSσμP+Q,P=SσG0-1P+Q-G0-1P.The spin interaction Hamiltonian is given by(18)HIS=∑pqσψp+qσ†γSσμp+q,pAμqψpσ.Similarly, the perturbed spin current is also evaluated via the Kubo formalism δJSμ(Q)=KSscμν(Q)Aν(Q) where(19)KSscμνQ=nmhμν+∑PσΓSscσμP+Q,PGscP+QγSσνP,P+QGscP.The full spin interaction vertex ΓSscσμ is given by(20)ΓSscσμP+Q,P=γSσμP+Q,P+MTSscσμP+Q,P,where the MT term is expressed as(21)MTSscσμP+Q,P=-Δsc2G0-PγSσμ-P,-P-QG0-P-Q.Since the U(1) spin rotational symmetry is not broken below Tc, then the full spin interaction vertex does not contain contributions associated with the Nambu-Goldstone modes. Moreover, the Ward identity is indeed satisfied in the mean field theory level(22)qμΓSscσμP+Q,P=SσGsc-1P+Q-Gsc-1P.Therefore, the perturbed spin current is conserved qμδJSμ(Q)=0. In the spin channel, there is no well-defined Q-limit Ward identity for unpolarized Fermi superfluids although such identity does exist for polarized Fermi superfluids [19]. This is because the equal-population case can not be approached from the population imbalanced case by simply letting the particle number difference approach zero.Figure 3
Spin interaction vertex. It has different signs for different pseudospin indices.The above discussions show that the linear response theories in the density and spin channels are fully consistent with the BCS mean field approximation for Fermi superfluids. However, when generalized to the whole BCS-BEC crossover regime, the mean field approximation overestimates the critical temperature in the unitarity and BEC side since the fluctuations of the noncondensed pairs are ignored. We next show a formally theoretical scheme in which the fundamental constraints are still satisfied when the pairing fluctuation effects are included.


## 2.1. Density Channel

In the density channel, the system is perturbed by an effective external EM fieldAμ, and the Hamiltonian becomes H=HBCS+HI with(6)HI=∑pqσψp+qσ†γμp+q,pAμqψpσ,where γμ(p+q,p)≡γμ(P+Q,P)=(1,(p+q/2)/m) is the bare EM interaction vertex. Here Q≡qμ=(iΩl,q) is the external four momentum, where Ωl is the boson Matsubara frequency. The bare vertex satisfies the “bare” Ward identity(7)qμγμP+Q,P=G0-1P+Q-G0-1P.In a gauge invariant EM linear response theory, a full EM interaction vertex Γμ (the bare and full EM vertices are shown in Figure 1) which satisfies the full Ward identity(8)qμΓscμP+Q,P=Gsc-1P+Q-Gsc-1Pmust be found so that the perturbed current can be expressed as δJμ(Q)=Kscμν(Q)Aν(Q) with Kscμν(Q) determined by the Kubo formalism(9)KscμνQ=nmhμν+2∑PΓscμP+Q,PGscP+QγνP,P+QGscP,where hμν=-ημν(1-ην0) with ημν=diag(1,-1,-1,-1) being the metric tensor. By using the Ward identity (8), it is easy to show that qμKscμν(Q)=0, which further leads to the conservation of perturbed current qμδJμ(Q)=0. Hence the linear response theory is indeed gauge invariant. Under the framework of BCS mean field theory, such full vertex can be obtained either by Nambu’s integral-equation approach [23, 24] or by the CFOP approach. However, the Q-limit Ward identity provides an independent consistency check of the theory. It is not known if Nambu’s approach can pass this check, while the vertex given by the latter approach is proved to satisfy this condition(10)limq→0⁡Γsc0P+Q,Pω=0=∂Gsc-1P∂μ=1-∂ΣscP∂μ.Details can be found in [18]. This identity not only builds a consistent connection between the one-particle thermodynamics and two-particle response functions but also acts as the sufficient and necessary condition for the compressibility sum rule(11)∂n∂μ=-Ksc000,q⟶0.Figure 1
EM interaction vertex. The left one is the bare vertex, and the right one is the full vertex. The solid line denotes the fermion line and the dashed line denotes the photon line.The expression of this gauge invariant interaction vertex given by the CFOP approach is given in our past work [18](12)ΓscμP+Q,P=γμP+Q,P+CollscμP+Q,P+MTscμP+Q,P,where the second term,(13)CollscμP+Q,P=ΔscΠμQG0-P-Q+ΔscΠ¯μQG0-P,corresponds to the excitations of Nambu-Goldstone modes due to the breaking of the U(1) symmetry and the third term,(14)MTscμP+Q,P=-Δsc2G0-Pγμ-P,-P-QG0-P-Q,is the famous MT diagram which is shown in Figure 2. The expressions of Πμ and Π¯μ are given in Appendix A. These two terms originates from summing up the diagrams with photon-fermion interaction lines inserted at any possible position. By using the equalities [18] qμΠμ(Q)=2Δsc and qμΠ¯μ(Q)=-2Δsc, it can be shown that(15)qμCollscμP+Q,P=2ΣscP-2ΣscP+Q.Combining with the equality(16)qμMTscμP+Q,P=ΣscP+Q-ΣscP,the Ward identity (8) in the BCS mean field level can be proved.Figure 2
Maki-Thompson diagram. The wavy line represents the pair propagator.


## 2.2. Spin Channel

In the spin channel, theU(1) spin rotational symmetry is also “gauged” by introducing an effective external field. However, this symmetry is not broken by the order parameter. Therefore, we expect that the structure of the spin linear response theory is simpler than that of its density counterpart. The “bare” spin interaction vertex is dependent on the pseudospin: γSσμ(p+q,p)≡γSσμ(P+Q,P)=Sσ(1,(p+q/2)/m) with S↑=1, S↓=-1 and the subscript “S” referring to the “spin.” The quantity Sσ indicates that the vertex has different signs for different pseudospin indices; see Figure 3. It also respects the “bare” Ward identity in the spin channel(17)qμγSσμP+Q,P=SσG0-1P+Q-G0-1P.The spin interaction Hamiltonian is given by(18)HIS=∑pqσψp+qσ†γSσμp+q,pAμqψpσ.Similarly, the perturbed spin current is also evaluated via the Kubo formalism δJSμ(Q)=KSscμν(Q)Aν(Q) where(19)KSscμνQ=nmhμν+∑PσΓSscσμP+Q,PGscP+QγSσνP,P+QGscP.The full spin interaction vertex ΓSscσμ is given by(20)ΓSscσμP+Q,P=γSσμP+Q,P+MTSscσμP+Q,P,where the MT term is expressed as(21)MTSscσμP+Q,P=-Δsc2G0-PγSσμ-P,-P-QG0-P-Q.Since the U(1) spin rotational symmetry is not broken below Tc, then the full spin interaction vertex does not contain contributions associated with the Nambu-Goldstone modes. Moreover, the Ward identity is indeed satisfied in the mean field theory level(22)qμΓSscσμP+Q,P=SσGsc-1P+Q-Gsc-1P.Therefore, the perturbed spin current is conserved qμδJSμ(Q)=0. In the spin channel, there is no well-defined Q-limit Ward identity for unpolarized Fermi superfluids although such identity does exist for polarized Fermi superfluids [19]. This is because the equal-population case can not be approached from the population imbalanced case by simply letting the particle number difference approach zero.Figure 3
Spin interaction vertex. It has different signs for different pseudospin indices.The above discussions show that the linear response theories in the density and spin channels are fully consistent with the BCS mean field approximation for Fermi superfluids. However, when generalized to the whole BCS-BEC crossover regime, the mean field approximation overestimates the critical temperature in the unitarity and BEC side since the fluctuations of the noncondensed pairs are ignored. We next show a formally theoretical scheme in which the fundamental constraints are still satisfied when the pairing fluctuation effects are included.


## 3. Gauge Invariant Linear Response Theories in theG0G Formalism

When we consider the situation that the interaction between fermions is stronger than the BCS attraction, the self-energy obtains corrections from the noncondensed pairs. Hence the interaction vertex must be corrected correspondingly to ensure an exact validity of the Ward identity andQ-limit Ward identity. In this paper, we adapt the G0G formalism [25], which is compatible with the BCS-Leggett ground state, to discuss the pseudogap effect. The self-energy due to the noncondensed pair is given by(23)ΣpgP=∑QtpgQG0Q-P,where tpg(Q)=-g′/(1-g′χ(Q)) is the t-matrix due to noncondensed pairs. The pair susceptibility is constructed in the G0G formalism χ(Q)=∑K‍G(K)G0(Q-K), where G is the full Green function with pairing fluctuation effect included. Here we assume that the coupling constant between fermions in noncondensed pairs, g′, is not necessarily equal to g, the coupling constant between fermions in condensed pairs. To determine the pairing onset temperature T∗, we still use the Thouless criterion; that is, tpg(0) is divergent at T∗ or(24)tpg-10=1+g′χ0=0.One possible reason that g′ may not be equal to g is that the Thouless criterion can not reduce to the BCS gap equation even when g′ is equal to g. Similarly, the t-matrix due to the condensed pair is tsc(Q)=-(Δsc2/T)δ(Q), and the BCS self-energy is also expressed as Σsc(P)=∑Q‍tsc(Q)G0(Q-P). The order parameter is still determined by Δsc=-g∑p‍〈ψp↑ψ-p↓〉, which is nonzero below Tc. Now the full inverse Green function is given by(25)G-1P=G0-1P-ΣP=G0-1P-ΣscP-ΣpgP,where Σ(P) is the total self-energy. We emphasize that no further approximation is introduced now. The full theory is quite complicated due to the inclusion of the undetermined pseudogap self-energy Σpg and new coupling g′; this brings difficulties to the numerical work in the future. If we adapt the approximation that the pair propagator is highly peaked at zero momentum as discussed in [25], the numerical calculation will be greatly simplified. However, this is not the aim of this paper. We hope to find a theory which can pass all stringent consistent types of testing in the first step.


### 3.1. Density Channel

In the density channel, to get a new gauge invariant interaction vertex, we must find a vertex correction which is consistent with the new self-energyΣpg. Such vertex does exist if we adapt the upper modified G0G formalism(26)ΓμP+Q,P=γμP+Q,P+CollscμP+Q,P+MTscμP+Q,P+MTpgμP+Q,P+AL⁡1μP+Q,P+AL⁡2μP+Q,P.Here Collscμ and MTscμ are still given by (13) and (14), respectively, and MTscμ can be further expressed in a more general style by including the t-matrix tsc:(27)MTscμP+Q,P=∑KtscKG0K-PγμK-P,K-P-QG0K-P-Q.Hence the MT diagram for noncondensed pairs is given by(28)MTpgμP+Q,P=∑KtpgKG0K-PγμK-P,K-P-QG0K-P-Q.The fifth and sixth terms are two different types of AL diagrams (shown in Figure 4), (29)AL⁡1μP+Q,P=-∑K,LtpgKtpgK+QG0K-PGK-LG0L+QγμL+Q,LG0L,AL⁡2μP+Q,P=-∑K,LtpgKtpgK+QG0K-PG0K-LGL+QΓμL+Q,LGL.We see that AL2μ contains a full vertex; hence expression (26) is in fact a series. In this scheme, the pseudogap effect does not enter into the terms related to the collective modes; then the Collscμ term vanishes above Tc; hence there are no Nambu-Goldstone modes excitations, which is consistent with the fact that the U(1) EM symmetry is unbroken.Figure 4
The left one is the AL1 diagram, and the right one is the AL2 diagram. The thin and thick lines denote the bare and full Green’s functions, respectively.The full vertex (26) satisfies the Ward identity(30)qμΓμP+Q,P=G-1P+Q-G-1P,and the gauge invariant response functions now can be expressed as(31)KμνQ=nmhμν+2∑PΓμP+Q,PGP+QγνP,P+QGP.To prove the Ward identity we need a lemma(32)qμ12AL⁡1μP+Q,P+12AL⁡2μP+Q,P+MTpgμP+Q,P=0.This proof of this lemma is outlined in Appendix B. Moreover, by applying the bare Ward identity (7), we can show that(33)qμMTpgμP+Q,P=∑KtpgKGK-P-Q-GK-P=ΣpgP+Q-ΣpgP.Hence(34)qμAL⁡1μP+Q,P+AL⁡2μP+Q,P+MTpgμP+Q,P=ΣpgP-ΣpgP+Q.Finally, (7), (15), (16), and (34) lead to the Ward identity (30) self-consistently by assuming that AL2μ(P+Q,P) has a full gauge invariant interaction vertex.Moreover, the full vertex should also respect theQ-limit Ward identity(35)limq→0⁡Γ0P+Q,Pω=0=∂G-1P∂μ=1-∂ΣP∂μ.A brief derivation shows that this identity ensures the compressibility sum rule(36)∂n∂μ=-2∑PG2P∂G-1P∂μ=-limq→0⁡∑PΓ0P+Q,PGP+Qγ0P,P+QGPω=0=-limq→0⁡K00Qω=0,where n=2∑P‍G(P) now. Note that(37)ΓμP+Q,P=ΓscμP+Q,P+MTpgμP+Q,P+AL⁡1μP+Q,P+AL⁡2μP+Q,Pand that Γscμ(P+Q,P) already satisfies (10); we only need to show that(38)limq→0⁡MTpg0P+Q,P+AL⁡10P+Q,P+AL⁡20P+Q,Pω=0=-∂ΣpgP∂μ.It can be proved as follows:(39)-∂ΣpgP∂μ=-∑KtpgK∂G0K-P∂μ-∑K∂tpgK∂μG0K-P=∑KtpgKG02K-P∂G0-1K-P∂μ+∑Ktpg2K∂tpg-1K∂μG0K-P=∑KtpgKG02K-P+∑Ktpg2K∂χpgK∂μG0K-P=∑KtpgKG02K-P+∑K,Ltpg2K∂G0K-L∂μGLG0K-P+∑K,Ltpg2KG0K-L∂GL∂μG0K-P=∑KtpgKG02K-P-∑K,Ltpg2KG02K-LGLG0K-P-∑K,Ltpg2KG0K-LG2L∂G-1L∂μG0K-P.Since the 0-component of the bare vertex γμ is always 1, then right-hand side of (39) is indeed(40)MTpg0P,P+AL⁡10P,P+AL⁡20P,Pif we compare with expressions (28) and (29). Therefore, the Q-limit Ward identity is also satisfied.


### 3.2. Spin Channel

In the spin channel, the linear response theory is formulated in a similar but simpler way. The central idea is to find the gauge invariant spin interaction vertex. According to the expression of the spin interaction Hamiltonian (18), it is convenient to restore the pseudospin dependence of the Green function. Hence G0↑(P)=G0↓(P)=G0(P), G↑(P)=G↓(P)=G(P). Generically, the vertex also contains the MT and AL diagrams when going beyond BCS theory:(41)ΓSσμP+Q,P=γSσμP+Q,P+MTSscσμP+Q,P+MTSpgσμP+Q,P+AL⁡S1σμP+Q,P+AL⁡S2σμP+Q,P.The MT diagram associated with the contributions from the order parameter and pseudogap are, respectively, given by(42)MTSscσμP+Q,P=∑KtscKG0σ¯K-PγSσ¯μK-P,K-P-QG0σ¯K-P-Q,MTSpgσμP+Q,P=∑KtpgKG0σ¯K-PγSσ¯μK-P,K-P-QG0σ¯K-P-Q.Here we emphasize again that the spin interaction vertices have different signs for different pseudospin indices, which leads to an important result that the contributions from the AL diagrams automatically cancel out. This can be shown by a straightforward verification. Figure 5 shows the vertex correction from the two spin-up AL1 diagrams (with two sets of different pseudospin attributions). We have(43)AL⁡S1↑μP+Q,P=-∑K,K′G0↓K-PtpgK+QtpgK′G↑K-K′G0↓K′+QγS↓μK′+Q,K′G0↓K′+G↓K-K′G0↑K′+QγS↑μK′+Q,K′G0↑K′=0,where the fact that G0↑=G0↓, G↑=G↓, and γS↑μ=-γS↓μ has been applied. Similar calculation indicates that the vertex corrections from the two spin-down AL1 diagrams also vanish. The vertex corrections from spin-up AL2 are(44)AL⁡S2↑μP+Q,P=-∑K,K′G0↓K-PtpgK+QtpgK′G0↑K-K′G↓K′+QΓS↓μK′+Q,K′G↓K′+G0↓K-K′G↑K′+QΓS↑μK′+Q,K′G↑K′=0.Hence the contributions from two spin-down AL2 vanish too. From the equalities(45)qμMTSscσμP+Q,P=SσΣscP+Q-ΣscP,qμMTSpgσμP+Q,P=SσΣpgP+Q-ΣpgPone can show that the Ward identity for the full spin interaction vertex is satisfied:(46)qμΓSσμP+Q,P=SσG-1P+Q-G-1P.The spin linear response theory is gauge invariant too.Figure 5
Vertex corrections from spin-upAL1 diagrams.Therefore, all consistency constraints are satisfied within this scheme in both the density and the spin channels when pairing fluctuation effects are considered. However, the approach in the density channel is not a useful form for numerical application. If any approximation is applied, it may most possibly violate some of the constraints. In a certain situation, some constraint may survive; hence these consistency conditions can play an indicator to “measure” how good the approximation is.


### 3.3. Applications

After formulating the theoretical framework, it is possible to evaluate some physical quantities under some conditions. As we have pointed out previously, if certain approximations are introduced, the fundamental constraints may not be satisfied. Our ideal future goal is to control the badness that those constraints are broken, which seems terribly difficult till now. For an instructive discussion, here we show how to derive physical quantities by applying certain approximation. We assume that the pair propagator is highly peaked at zero momentum [25]; then the pseudogap self-energy can be approximated as(47)Σpg-P≃∑QtpgQG0-P=-Δpg2G0-P,where Δpg2=-∑Q‍tpg(Q) is the pseudogap, which is still hard to determine by numerics. If we define the total energy gap Δ2=Δsc2+Δpg2, then full Green’s function given, G(P), formally has a similar expression as BCS Green’s function, Gsc(P), only with Δsc replaced by Δ. The MT diagram for noncondensed pairs can also be simplified as(48)MTpgμP+Q,P=-Δpg2G0-Pγμ-P,-P-QG0-P-Q.Using these results and applying Lemma (32) we can get the expression for the response kernel Kμν(Q), from which we further derive an approximated expression of the compressibility. Consider(49)∂n∂μ=∑pΔsc2Ep31-2fEp-21-Δpg2Ep2ξp2Ep2∂fEp∂Ep+∑pξp/Escp21-2fEscp/Escp+2∂fEscp/∂Escp2∑p1/Escp1-2fEscp/Escp+2∂fEscp/∂Escp,where Ep=ξp2+Δ2. The second term comes from the breaking of the U(1) symmetry in the superfluid phase; it vanishes above Tc since the symmetry is unbroken in the normal phase. This approximation obviously breaks the Ward identity, and we hope to find a way to control the approximation in the future.


## 3.1. Density Channel

In the density channel, to get a new gauge invariant interaction vertex, we must find a vertex correction which is consistent with the new self-energyΣpg. Such vertex does exist if we adapt the upper modified G0G formalism(26)ΓμP+Q,P=γμP+Q,P+CollscμP+Q,P+MTscμP+Q,P+MTpgμP+Q,P+AL⁡1μP+Q,P+AL⁡2μP+Q,P.Here Collscμ and MTscμ are still given by (13) and (14), respectively, and MTscμ can be further expressed in a more general style by including the t-matrix tsc:(27)MTscμP+Q,P=∑KtscKG0K-PγμK-P,K-P-QG0K-P-Q.Hence the MT diagram for noncondensed pairs is given by(28)MTpgμP+Q,P=∑KtpgKG0K-PγμK-P,K-P-QG0K-P-Q.The fifth and sixth terms are two different types of AL diagrams (shown in Figure 4), (29)AL⁡1μP+Q,P=-∑K,LtpgKtpgK+QG0K-PGK-LG0L+QγμL+Q,LG0L,AL⁡2μP+Q,P=-∑K,LtpgKtpgK+QG0K-PG0K-LGL+QΓμL+Q,LGL.We see that AL2μ contains a full vertex; hence expression (26) is in fact a series. In this scheme, the pseudogap effect does not enter into the terms related to the collective modes; then the Collscμ term vanishes above Tc; hence there are no Nambu-Goldstone modes excitations, which is consistent with the fact that the U(1) EM symmetry is unbroken.Figure 4
The left one is the AL1 diagram, and the right one is the AL2 diagram. The thin and thick lines denote the bare and full Green’s functions, respectively.The full vertex (26) satisfies the Ward identity(30)qμΓμP+Q,P=G-1P+Q-G-1P,and the gauge invariant response functions now can be expressed as(31)KμνQ=nmhμν+2∑PΓμP+Q,PGP+QγνP,P+QGP.To prove the Ward identity we need a lemma(32)qμ12AL⁡1μP+Q,P+12AL⁡2μP+Q,P+MTpgμP+Q,P=0.This proof of this lemma is outlined in Appendix B. Moreover, by applying the bare Ward identity (7), we can show that(33)qμMTpgμP+Q,P=∑KtpgKGK-P-Q-GK-P=ΣpgP+Q-ΣpgP.Hence(34)qμAL⁡1μP+Q,P+AL⁡2μP+Q,P+MTpgμP+Q,P=ΣpgP-ΣpgP+Q.Finally, (7), (15), (16), and (34) lead to the Ward identity (30) self-consistently by assuming that AL2μ(P+Q,P) has a full gauge invariant interaction vertex.Moreover, the full vertex should also respect theQ-limit Ward identity(35)limq→0⁡Γ0P+Q,Pω=0=∂G-1P∂μ=1-∂ΣP∂μ.A brief derivation shows that this identity ensures the compressibility sum rule(36)∂n∂μ=-2∑PG2P∂G-1P∂μ=-limq→0⁡∑PΓ0P+Q,PGP+Qγ0P,P+QGPω=0=-limq→0⁡K00Qω=0,where n=2∑P‍G(P) now. Note that(37)ΓμP+Q,P=ΓscμP+Q,P+MTpgμP+Q,P+AL⁡1μP+Q,P+AL⁡2μP+Q,Pand that Γscμ(P+Q,P) already satisfies (10); we only need to show that(38)limq→0⁡MTpg0P+Q,P+AL⁡10P+Q,P+AL⁡20P+Q,Pω=0=-∂ΣpgP∂μ.It can be proved as follows:(39)-∂ΣpgP∂μ=-∑KtpgK∂G0K-P∂μ-∑K∂tpgK∂μG0K-P=∑KtpgKG02K-P∂G0-1K-P∂μ+∑Ktpg2K∂tpg-1K∂μG0K-P=∑KtpgKG02K-P+∑Ktpg2K∂χpgK∂μG0K-P=∑KtpgKG02K-P+∑K,Ltpg2K∂G0K-L∂μGLG0K-P+∑K,Ltpg2KG0K-L∂GL∂μG0K-P=∑KtpgKG02K-P-∑K,Ltpg2KG02K-LGLG0K-P-∑K,Ltpg2KG0K-LG2L∂G-1L∂μG0K-P.Since the 0-component of the bare vertex γμ is always 1, then right-hand side of (39) is indeed(40)MTpg0P,P+AL⁡10P,P+AL⁡20P,Pif we compare with expressions (28) and (29). Therefore, the Q-limit Ward identity is also satisfied.


## 3.2. Spin Channel

In the spin channel, the linear response theory is formulated in a similar but simpler way. The central idea is to find the gauge invariant spin interaction vertex. According to the expression of the spin interaction Hamiltonian (18), it is convenient to restore the pseudospin dependence of the Green function. Hence G0↑(P)=G0↓(P)=G0(P), G↑(P)=G↓(P)=G(P). Generically, the vertex also contains the MT and AL diagrams when going beyond BCS theory:(41)ΓSσμP+Q,P=γSσμP+Q,P+MTSscσμP+Q,P+MTSpgσμP+Q,P+AL⁡S1σμP+Q,P+AL⁡S2σμP+Q,P.The MT diagram associated with the contributions from the order parameter and pseudogap are, respectively, given by(42)MTSscσμP+Q,P=∑KtscKG0σ¯K-PγSσ¯μK-P,K-P-QG0σ¯K-P-Q,MTSpgσμP+Q,P=∑KtpgKG0σ¯K-PγSσ¯μK-P,K-P-QG0σ¯K-P-Q.Here we emphasize again that the spin interaction vertices have different signs for different pseudospin indices, which leads to an important result that the contributions from the AL diagrams automatically cancel out. This can be shown by a straightforward verification. Figure 5 shows the vertex correction from the two spin-up AL1 diagrams (with two sets of different pseudospin attributions). We have(43)AL⁡S1↑μP+Q,P=-∑K,K′G0↓K-PtpgK+QtpgK′G↑K-K′G0↓K′+QγS↓μK′+Q,K′G0↓K′+G↓K-K′G0↑K′+QγS↑μK′+Q,K′G0↑K′=0,where the fact that G0↑=G0↓, G↑=G↓, and γS↑μ=-γS↓μ has been applied. Similar calculation indicates that the vertex corrections from the two spin-down AL1 diagrams also vanish. The vertex corrections from spin-up AL2 are(44)AL⁡S2↑μP+Q,P=-∑K,K′G0↓K-PtpgK+QtpgK′G0↑K-K′G↓K′+QΓS↓μK′+Q,K′G↓K′+G0↓K-K′G↑K′+QΓS↑μK′+Q,K′G↑K′=0.Hence the contributions from two spin-down AL2 vanish too. From the equalities(45)qμMTSscσμP+Q,P=SσΣscP+Q-ΣscP,qμMTSpgσμP+Q,P=SσΣpgP+Q-ΣpgPone can show that the Ward identity for the full spin interaction vertex is satisfied:(46)qμΓSσμP+Q,P=SσG-1P+Q-G-1P.The spin linear response theory is gauge invariant too.Figure 5
Vertex corrections from spin-upAL1 diagrams.Therefore, all consistency constraints are satisfied within this scheme in both the density and the spin channels when pairing fluctuation effects are considered. However, the approach in the density channel is not a useful form for numerical application. If any approximation is applied, it may most possibly violate some of the constraints. In a certain situation, some constraint may survive; hence these consistency conditions can play an indicator to “measure” how good the approximation is.


## 3.3. Applications

After formulating the theoretical framework, it is possible to evaluate some physical quantities under some conditions. As we have pointed out previously, if certain approximations are introduced, the fundamental constraints may not be satisfied. Our ideal future goal is to control the badness that those constraints are broken, which seems terribly difficult till now. For an instructive discussion, here we show how to derive physical quantities by applying certain approximation. We assume that the pair propagator is highly peaked at zero momentum [25]; then the pseudogap self-energy can be approximated as(47)Σpg-P≃∑QtpgQG0-P=-Δpg2G0-P,where Δpg2=-∑Q‍tpg(Q) is the pseudogap, which is still hard to determine by numerics. If we define the total energy gap Δ2=Δsc2+Δpg2, then full Green’s function given, G(P), formally has a similar expression as BCS Green’s function, Gsc(P), only with Δsc replaced by Δ. The MT diagram for noncondensed pairs can also be simplified as(48)MTpgμP+Q,P=-Δpg2G0-Pγμ-P,-P-QG0-P-Q.Using these results and applying Lemma (32) we can get the expression for the response kernel Kμν(Q), from which we further derive an approximated expression of the compressibility. Consider(49)∂n∂μ=∑pΔsc2Ep31-2fEp-21-Δpg2Ep2ξp2Ep2∂fEp∂Ep+∑pξp/Escp21-2fEscp/Escp+2∂fEscp/∂Escp2∑p1/Escp1-2fEscp/Escp+2∂fEscp/∂Escp,where Ep=ξp2+Δ2. The second term comes from the breaking of the U(1) symmetry in the superfluid phase; it vanishes above Tc since the symmetry is unbroken in the normal phase. This approximation obviously breaks the Ward identity, and we hope to find a way to control the approximation in the future.


## 4. Conclusion

We have constructed gauge invariant density and spin linear response theories for a Fermi gas undergoing BCS-BEC crossover by including adequate diagrams in the interaction vertices using thet-matrix formalism based on a slightly corrected G0G scheme. We verified that the Ward identities and the important Q-limit Ward identity are both satisfied when the contributions due to the order parameter (condensed pairs) and pseudogap (noncondensed pairs) are both included. This justifies Nambu’s assertion that the modification of the vertex must be consistent with the way that the self-energy is included in the quasiparticle. Those constraints guarantee the self-consistency of the theories. Until now our approach is a purely theoretical formalism without including any approximation, yet we believe it will shed light on the reliable theoretical predictions of the transport properties of strongly correlated Fermi gases and help us to understand more about the many particle theory. Future improvements include trustworthy numerical calculations by taking suitable approximations.



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*Source: 582734-2015-10-01.xml*