Squeezed-field path-integral description of BCS superconductors
SSqueezed-field path-integral description of BCS superconductors
Kazuma Nagao,
1, 2, ∗ Dapeng Li,
1, 2 and Ludwig Mathey
1, 2 Zentrum f¨ur Optische Quantentechnologien and Institut f¨ur Laserphysik, Universit¨at Hamburg, 22761 Hamburg, Germany The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: February 25, 2021)We develop a squeezed-field path-integral representation for BCS superconductors utilizing ageneralized completeness relation of squeezed-fermionic coherent states. We derive a Grassmannpath integral of fermionic quasiparticles that explicitly includes the collective degrees of freedomof the order-parameter dynamics governed by the classical Anderson pseudospin model. Based onthis method, we analyze the spectral function of the single-particle excitations, and show that thesqueezed-field path integral for the BCS Hamiltonian describes the dispersion relation and the massgap of the Higgs amplitude mode of BCS superconductors, as well as the quasiparticle and quasiholeexcitation branches described by the BCS mean-field approximation.
I. INTRODUCTION
Superconductors are macroscopic quantum states ofmatter, described by order parameters of spontaneously-broken symmetries [1–3]. The phenomenological mean-field model for superconductors has been first developedby Ginzburg and Landau [2], giving a universal descrip-tion of the thermodynamic phases. In 1957, Bardeen,Cooper, and Schrieffer (BCS) have provided a micro-scopic description of conventional s -wave superconduc-tors [2, 4, 5], establishing key notions, such as Cooperpairing of electrons, the phonon-mediated pairing mech-anism of Cooper pairs due to lattice vibrations, and theinstability of the Fermi surface due to attractive interac-tions.Fermionic quasiparticle excitations and bosonic collec-tive excitations are key features of quantum liquids of su-perfluid fermions exemplified by superconductors [2, 4],and by liquid He [2, 6]. Recent pump-probe experimentshave explored light-induced dynamics of conventionalBCS and high- T c cuprate superconductors near and far-from equilibrium [7–19]. A striking direction of researchis the exploration of the Higgs amplitude mode in anNbN superconductor with terahertz light [7, 12]. Collec-tive and single-particle properties of superfluid fermionshave also been studied in ultracold gases [20–31].For a complete understanding of the experimental ob-servations of fermionic superfluid systems, a microscopicdescription for the co-existence and interaction of thesingle-particle and collective degrees of freedom has tobe established. For example, quasiparticle pair-breakingprocesses of Cooper pairs have a significant influence onthe collective excitation branch of superconductors andsuperfluid Fermi gases [13, 32]. Despite considerable the-oretical advances, a unifying framework of treating bothquasiparticle and collective degrees of freedom has notbeen established so far.In this paper, we develop a generalized Grassmannpath-integral approach to describe the fermionic quasi- ∗ [email protected] particles and the bosonic collective excitations on equalfooting. Our formulation presented in this work is basedon the squeezed-field path-integral description, whichhas been developed in the context of Bose–Einstein con-densates of ultracold gases [33], and more recently, ex-tended to one-dimensional gapless systems described byTomonaga–Luttinger liquid theory [34]. The squeezed-field path integrals are a field theoretical representationof quantum systems, in which one inserts the complete-ness relation of the squeezed coherent states, not the co-herent states, into the time evolutions of the path inte-gral. The constructed path integrals have an extendedphase space of the squeezing parameters for quasipar-ticles, which include quantum and thermal fluctuationsthat are hard to access in the standard coherent-staterepresentation. See Sec. IV for a detailed discussion.In this work, we generalize the formalism of squeezed-field path integrals to fermionic superfluid systems,specifically BCS superconductors. Our formalism canbe readily applied to more complex orders as well, suchas unconventional p -wave superconductors and superflu-ids [6, 35], and strongly-correlated Hubbard-type sys-tems [36]. We demonstrate that the quantized motionof the squeezing parameters, referred to as the squeezingfields [33, 34], is described by the classical Anderson pseu-dospin model [12, 37, 38], whose Hamiltonian is given byan energy functional of electrons with respect to the BCSvariational state. We analyze experimentally-relevantquantities, in particular, the single-particle spectral func-tion, based on the method developed here, and demon-strate that the squeezed-field path integral describes keyfeatures of the Higgs modes of BCS superconductors inthe weakly-interacting BCS regime.This paper is organized as follows: In Sec. II, we in-troduce the BCS Hamiltonian for electrons interactingvia attractive interactions and explain the assumptionsfor the analysis of the following sections. In Sec. III, wepresent a brief overview of the mean-field properties ofthe BCS Hamiltonian. In Sec. IV, we derive a general-ized Grassmann-field path-integral representation of theBCS Hamiltonian by utilizing the completeness relationof squeezed-fermionic coherent states, and give a physi- a r X i v : . [ c ond - m a t . s up r- c on ] F e b cal interpretation of the motion of the squeezing field. InSec. V, we analyze the energy spectrum of the linearizedapproximation of the Anderson pseudospin model. InSec. VI, we apply the formalism to the single-particlespectral function. Finally, in Sec. VII, we conclude. II. MODEL
We consider a system of electrons with an attractiveinteraction, described by the BCS Hamiltonian [5, 39]ˆ H = (cid:88) k ,σ = ↑ , ↓ ξ k ˆ c † k ,σ ˆ c k ,σ − gV (cid:88) k , k (cid:48) , q ˆ c † k + q , ↑ ˆ c †− k , ↓ ˆ c − k (cid:48) + q , ↓ ˆ c k (cid:48) , ↑ , (1)where { ˆ c k ,σ , ˆ c † k (cid:48) ,σ (cid:48) } = ˆ c k ,σ ˆ c † k (cid:48) ,σ (cid:48) + ˆ c † k (cid:48) ,σ (cid:48) ˆ c k ,σ = δ k , k (cid:48) δ σ,σ (cid:48) , µ is the chemical potential, g > V is the total volume. The free-particle dispersion ξ k = (cid:15) k − µ is measured relative to thechemical potential. Throughout this work we assume aspherical band structure, i.e. (cid:15) k = (cid:126) k m , where m is theelectron mass. Furthermore, we assume that the interac-tion range is restricted inside an energy region around thefree Fermi energy E F ≡ (cid:126) k m with a width 2 (cid:126) ω D [39, 40].Here ω D is the Debye frequency of solids. We assume thatthe system is in homogeneous three-dimensional space.In real solids, electrons couple to the electromagneticfield, which leads to gapped plasmon modes [39, 41]. Thiscoupling is ignored in the BCS Hamiltonian (1). As aconsequence, the BCS Hamiltonian possesses a Nambu–Goldstone (NG) mode [39, 42, 43] as a gapless excita-tion branch with spontaneously-broken U(1) symmetry.However, this paper is aimed at demonstrating our the-oretical formalism. Hence we ignore the coupling to thegauge fields in the following discussion, to be includedelsewhere. III. MEAN-FIELD PROPERTIES OF THE BCSHAMILTONIAN
The mean-field properties of the ground state of theBCS Hamiltonian (1) are derived from the variationalBCS state [5, 37, 39, 44]: | Ω BCS (cid:105) = (cid:89) k (cid:104) u k + v k ˆ c † k ↑ ˆ c †− k ↓ (cid:105) | (cid:105) , (2)where k runs over momentum space and | (cid:105) denotes thevacuum of fermions. The coefficients u k and v k de-note the occupation probability of the states | k − k (cid:105) and | ↑ k ↓ − k (cid:105) . The normalization of the BCS state implies aconstraint for a spinor ( u k , v k ) T , i.e. | u k | + | v k | = 1.The expectation value of the Hamiltonian, i.e. E BCS = (cid:104) Ω BCS | ˆ H| Ω BCS (cid:105) , gives a variational functional with re-spect to the independent coefficient v k . From the varia- tional condition δE BCS /δv k = δE BCS /δv ∗ k = 0, the equi-librium values of u k and v k are determined as [39, 40] u k = 12 (cid:18) ξ k − gn E k (cid:19) ,v k = 12 (cid:18) − ξ k − gn E k (cid:19) . (3)The function ∆ = gV (cid:80) k (cid:104) ˆ c − k , ↓ ˆ c k , ↑ (cid:105) = gV (cid:80) k u k v k is thegap function of the superconducting phase, and E k = (cid:112) ∆ + ( ξ k − gn ) . In addition, gn ≡ gV (cid:80) k v k de-notes the Hartree shift to the bare chemical potential µ [45]. The value of the gap function is evaluated by solv-ing the self-consistent gap equation [40]. If the interac-tion strength g is weak, i.e., gN F (cid:28)
1, the gap function isapproximately given by ∆ ≈ (cid:126) ω D exp ( − / ( gN F )) ≡ ∆ (cid:48) with ˜ µ ≡ µ + gn ≈ E F [40, 45]. Here N F denotes the den-sity of states per Cooper pair and per the unit volume onthe Fermi surface. For a three-dimensional system, N F is given by N F = (2 m ) / π (cid:126) √ E F [40].The Bogoliubov mean-field approximation [46] reducesthe many-body Hamiltonian (1) to a quadratic form, i.e.ˆ H ≈ V | ∆ | g + (cid:88) k ξ k (ˆ c † k , ↑ ˆ c k , ↑ + ˆ c †− k , ↓ ˆ c − k , ↓ ) − (cid:88) k (cid:16) ∆ ∗ ˆ c − k , ↓ ˆ c k , ↑ + ∆ˆ c † k , ↑ ˆ c †− k , ↓ (cid:17) . (4)This reduced Hamiltonian is readily diagonalized by us-ing the Bogoliubov transformation [40, 44, 46], definedby ˆ c k , ↑ = u k ˆ γ k , ↑ + v k ˆ γ †− k , ↓ , ˆ c − k , ↓ = u k ˆ γ − k , ↓ − v k ˆ γ † k , ↑ . (5)The transformed Hamiltonian takes the form ˆ H → (cid:80) k E k (ˆ γ † k , ↑ ˆ γ k , ↑ +ˆ γ †− k , ↓ ˆ γ − k , ↓ ), and the BCS state is foundto be the vacuum state of the fermionic Bogoliubov quasi-particles ˆ γ k , ↑ and ˆ γ − k , ↓ , i.e. bogolons. Hence, the energy E k represents the dispersion relation of the single quasi-particle and quasihole excitations of the BCS state. IV. SQUEEZED FIELD PATH INTEGRAL FORTHE BCS HAMILTONIAN
The key ingredient to build a squeezed-field path inte-gral for fermions is the two-mode squeezing operator tocreate the BCS state:ˆ S ( η ) = (cid:89) k exp (cid:16) η k ˆ c † k , ↑ ˆ c †− k , ↓ − η ∗ k ˆ c − k , ↓ ˆ c k , ↑ (cid:17) . (6)The parameter η k = θ k e iϕ k denotes the rotation angle ofthe Bogoliubov transformation. Indeed, u k = cos( θ k / v k = sin( θ k / e iϕ k . We choose a specific gaugeof θ k and ϕ k as 0 ≤ θ k ≤ π and 0 ≤ ϕ k < π .Due to the unitarity of the transformation, the identityˆ S † ( η ) ˆ S ( η ) = ˆ S ( η ) ˆ S † ( η ) = 1 is satisfied for arbitrary η k . In terms of this unitary operator, the BCS stateis written as a squeezed vacuum state of electrons, i.e. | Ω BCS (cid:105) = ˆ S ( η ) | (cid:105) . See also, e.g., Refs. [37, 44, 47].In the standard path-integral method for fermions, thefermionic coherent states of Grassmann numbers are usedto span the phase space for classical trajectories [39, 44].These are defined as a ket vector | c (cid:105) = e ˆ c † · c − c · ˆ c | (cid:105) ,and its conjugated bra is (cid:104) c | = (cid:104) | e − ˆ c † · c + c · ˆ c . The vec-tors, c = ( · · · , c k ,σ , · · · ) and c = ( · · · , c k ,σ , · · · ), are theGrassmann-number fields, and (cid:104) c | c (cid:105) = 1. Similar to thebosonic coherent state, the fermionic coherent state sat-isfies a completeness relation [39, 44], given by (cid:90) d c d c | c (cid:105)(cid:104) c | = 1 . (7)The integral measure is d c d c = (cid:81) k dc k dc k . This com-pleteness relation is used to derive a classical action ofthe Grassmann fields for the quantum-mechanical Hamil-tonian (1). The Grassmann-field representation hasbeen widely utilized to formulate perturbative and non-perturbative frameworks for many-body problems of in-teracting fermions [39, 44, 48, 49]. For example, applica-tions of the Grassmann path integrals to renormalizationgroup analyses of interacting fermions have been compre-hensively reviewed in [49].To obtain a squeezed-field path integral for ˆ H , we firstsqueeze the completeness relation (7) with ˆ S ( η ), andthen integrate over it with respect to the variationalparameters η k . The η -integration is normalized by us-ing the Haar invariant measure of the SU(2) group, i.e. d Ω = (cid:81) k dθ k dϕ k π sin θ k [44, 48, 50]. As a result, we arriveat the relation (cid:90) d Ω d c d c ˆ S ( η ) | c (cid:105)(cid:104) c | ˆ S † ( η ) = 1 . (8)This equation (8) provides an extended completeness re-lation for the squeezed-fermionic coherent state | η , c (cid:105) ≡ ˆ S ( η ) | c (cid:105) . In the path integrals built with the squeezed-fermionic coherent states, the classical trajectories movethrough the extended phase space ( c , c , Ω ), which in-cludes the phase space ( c , c ) of Eq. (7) as its subspace.The additional motion along the direction of Ω includesadditional fluctuations around the BCS state, which arenot easily accessible in the standard phase-space choiceof the Grassmann path integrals, i.e., Eq. (7).Consider the thermodynamic partition function forthe BCS Hamiltonian Z ( β, µ ) = Tr e − β ˆ H , where β =( k B T ) − is the inverse temperature. We insert the ex-tended completeness relation of the squeezed-fermioniccoherent state and take the continuum limit to obtain Z = (cid:90) ≤| v k | ≤ D ( c , c , v ∗ , v ) e − (cid:126) S ( c , c , v ∗ , v ) . (9)We impose the constraint condition, 0 ≤ | v k | ≤ k on the path integral to be consistent with the integration area of the Haar measure. The Euclideanaction S = (cid:82) (cid:126) β dτ L ( c , c , v ∗ , v ) is the time integral of thesqueezed-field Lagrangian L ( c , c , v ∗ , v ) = (cid:88) k ,σ c k ,σ (cid:126) ∂ τ c k ,σ + (cid:88) k v ∗ k (cid:126) ∂ τ v k + H ( c , c , v ∗ , v ) + L NLD . (10)The classical Hamiltonian H ( c , c , v ∗ , v ) = (cid:104) c , η | ˆ H| η , c (cid:105) is the expectation value of the BCS Hamiltonian ˆ H with respect to the squeezed fermionic coherent state | η , c (cid:105) . The dynamical term, L dyn ≡ (cid:80) k ,σ c k ,σ (cid:126) ∂ τ c k ,σ + (cid:80) k v ∗ k (cid:126) ∂ τ v k + L NLD , is the Berry-phase term of theimaginary-time path integral, which stems from thecontinuum limit of the direct product of the adjacentoverlaps (cid:81) ∞ j =1 (cid:104) c j , η j | η j − , c j − (cid:105) [48]. See also Ap-pendix A for the derivation. The first term L dyn , ≡ (cid:80) k ,σ c k ,σ (cid:126) ∂ τ c k ,σ is the dynamical term of the standardGrassmann path integrals [39]. The second dynamicalterm, L dyn , ≡ (cid:80) k v ∗ k (cid:126) ∂ τ v k , describes the dynamics ofthe squeezing fields v k and v ∗ k . Furthermore, as an ad-ditional feature of squeezed-field path integrals, L dyn in-cludes a nonlinear dynamical term [33, 34], given by L NLD = (cid:88) k ˙ θ k (cid:0) e iϕ k c k , ↑ c − k , ↓ − e − iϕ k c − k , ↓ c k , ↑ (cid:1) − (cid:88) k i ˙ ϕ k | v k | ( c k , ↑ c k , ↑ + c − k , ↓ c − k , ↓ ) (11)+ (cid:88) k i ˙ ϕ k ( u k v k c k , ↑ c − k , ↓ + u k v ∗ k c − k , ↓ c k , ↑ ) . A similar set of dynamical terms also appears insqueezed-field path integrals for bosonic systems, seeRefs. [33, 34].The classical Hamiltonian H can be expressed as H [ c , c , v ∗ , v ] = E BCS [ v ∗ , v ]+ (cid:104) c | : ˆ S † ( η ) ˆ H ˆ S ( η ) : | c (cid:105) . Thedouble colons represent the normal ordering operation forthe Bogoliubov-transformed fermions. The expression ofthe energy functional E BCS is [37] E BCS [ v ∗ , v ] = (cid:88) k ξ k | v k | − gV (cid:88) k , k u k v k u k v ∗ k − gV (cid:88) k , k | v k | | v k | . (12)To advance our analysis of this classical model, we intro-duce Anderson’s pseudospins for the BCS state [12, 37,38] s x k = 12 sin θ k cos ϕ k = 12 u k ( v k + v ∗ k ) ,s y k = 12 sin θ k sin ϕ k = 12 i u k ( v k − v ∗ k ) , (13) s z k = 12 cos θ k = 12 ( u k − | v k | ) . In this spin representation, the energy functional (12)becomes a long-range interacting classical-spin model de-fined on the k -lattice [37]. Indeed, the first term of (12)translates to a linear term of s z k coupled to an exter-nal inhomogeneous magnetic field , i.e., ξ k . The secondterm describes the long-range XY couplings between s x,y k and s x,y k (cid:48) with the strength g/V , and the last term de-notes long-range Ising interactions between s z k and s z k (cid:48) .The interaction range of the pseudospins is over the re-stricted momentum region around the Fermi surface. Inthe squeezed-field path integral for BCS superconduc-tors, the squeezing parameters v k are dynamical fields ofthe path integral, expressing quantum and thermal fluc-tuations of the order-parameter field. In terms of thepseudospins, the dynamics can be visualized as precess-ing vectors ( s x k , s y k , s z k ) for each momentum k [12, 38]. Wenote that the Anderson pseudospin model has been usedto analyze the pump-probe response of the Higgs modeof s -wave superconductors [7, 12].To simplify the squeezed-field Lagrangian, we considersmall fluctuations around the mean-field ordered state.We split the field variable v k into a mean value v k andits fluctuations b k , i.e. v k = v k + b k . The fluctuations b k are assumed to be small, relative to u k . We expandthe field u k = (cid:112) − v ∗ k v k = u k − v k u k ( b k + b ∗ k ) + · · · toobtain L ≈ const. + L (2)F + L (2)B + L int ( c , c , b ∗ , b , ˙ θ , ˙ ϕ ), inwhich L (2)F = (cid:88) k ,σ c k ,σ ( (cid:126) ∂ τ + (cid:126) ω k ) c k ,σ , (14) L (2)B = (cid:88) k b ∗ k (cid:126) ∂ τ b k + (cid:88) k , k A k , k b ∗ k b k + (cid:88) k , k B k , k (cid:0) b k b k + b ∗ k b ∗ k (cid:1) . (15)The matrices A k , k and B k , k are given in Appendix B.The fermionic part of the quadratic Lagrangian, L (2)F ,describes the dispersion relation of the fermionic Bogoli-ubov modes (cid:126) ω k = E k . The bosonic part, L (2)B , describescollective modes of the order parameter expressed as aprecessing motion of the pseudospins. Note that thelinear terms of the bosons b k vanish due to the varia-tional condition to determine v k . Since the Hamiltonianof L (2)B is not diagonal with respect to b k and b ∗ k , thebosons with different momenta have a finite correlationwith each other. In Sec. V, we discuss the energy eigen-values of this classical Hamiltonian in detail. Moreover,the nonlinear interaction term L int ( c , c , b ∗ , b , ˙ θ , ˙ ϕ ) de-scribes the couplings between the fermionic and bosonicdegrees of freedom as well as the self-interaction of eachcomponent.Before proceeding, we make several remarks on thegeneralized path integral and the fluctuation expansion.First, the pseudospin representation is reminiscent ofthe Holstein–Primakoff (HP) representation of the SU(2) spin operators, see e.g. [48]. It is defined viaˆ S x = ˆ a † ˆ b + ˆ b † ˆ a , ˆ S y = ˆ a † ˆ b − ˆ b † ˆ a i , ˆ S z = ˆ a † ˆ a − ˆ b † ˆ b , (16)in which ˆ a = ˆ a † = (cid:112) S − ˆ b † ˆ b is assumed, and S is thestrength of the SU(2) spins. The operators ˆ b and ˆ b † sat-isfy [ˆ b, ˆ b † ] = 1 and [ˆ b, ˆ b ] = [ˆ b † , ˆ b † ] = 0. We note thatthere is a formal correspondence as ˆ a ↔ u k and ˆ b ↔ v k .In this analogy, the fluctuation expansion that we havepresented above can be seen as the HP spin-wave ex-pansion around a mean-field ground state [48, 51]. Wenote that, unlike the usual HP expansion for a ferromag-netic ground state [48], ˆ b has the mean value v k for theBCS state. In the squeezed-field path-integral formula-tion, the field variables b k are quantized, in analogy tothe quantized spin waves of a magnetic state.Having this observation in mind, the nonlinear term L int ( c , c , b ∗ , b , ˙ θ , ˙ ϕ ) can be regarded as a sum ofinfinitely-many nonlinear processes involving three, four,and infinite spin waves in k -space. Furthermore, in ad-dition to the contributions to L int from the Hamiltonian,the nonlinear contributions also derive from L NLD of theBerry phase, see Eq. (11). We note that the time deriva-tive terms are essential in determining the equations ofmotion in the Lagrange formalism [52, 53]. However,in the Hamilton formalism, the same equations are de-rived by introducing Poisson brackets for the canonicalphase-space variables, which are independently given ofthe Hamiltonian [52, 53]. Therefore, the presence of thenonlinear time-derivative terms implies that the Poissonbrackets for bosons and fermions are geometrically de-formed by such perturbations [54]. We will discuss thisconsideration more concretely elsewhere. We note that aperturbative analysis of the nonlinear interactions of thesqueezing fields has been discussed in Ref. [34].In spin-wave theory, corrections due to nonlinear pro-cesses are small, if the fluctuations around the referencestate of the expansion are sufficiently small. Otherwise,the truncation of the expansion is no longer valid, seealso Ref. [48]. In Sec. VI, we discuss a parameter regime,in which the corrections due to the fluctuations of thesqueezing fields are small compared to the BCS mean-field results. In the following sections, we focus on prop-erties of the quadratic Lagrangian L (2)F + L (2)B , and vi-sualize the consequences of this quadratic Lagrangian toobservable quantities relevant to real systems.Finally, we point out a similarity of the generalizedGrassmann path integral to specific features of gauge fieldtheories [3]. The path-integral expressions of quantumsystems are equivalent representations of the operator-based formalisms. For the generalized path integral, ifwe go back to the operator language, we observe an ad-ditional state space for the constrained bosons as well asfor the fermionic quasiparticles. Therefore, an additionalbosonic Hilbert space has emerged from the original de-scription of the BCS Hamiltonian. We note that theemergence of the intrinsic dynamics of the squeezing field,i.e., the quantization of v k , can be attributed to the mix-ing property of ˆ S ( η ) for particles and holes, which allowsto yield multiple operator products, which are not nor-mal ordered in the time evolution of the path integral, seealso Appendix A. Gauge-field descriptions for strongly-correlated systems, such as the t - J model and the Hub-bard model, have been reviewed in Refs. [3, 48, 55]. Sincethe ratio v k /u k or the product u k v k quantify the amountof quantum entanglement between ( k , ↑ ) and ( − k , ↓ ) ofthe BCS state [56, 57], in this sense, the bosonic field v k has similarities to a gauge field, which classically medi-ates quantum correlations between the fermionic single-particle states. V. ENERGY SPECTRUM OF THELINEARIZED PSEUDOSPIN MODEL
We analyze the energy spectrum of the linearized pseu-dospin Hamiltonian. As a preparational step, we intro-duce an energy-shell representation of the momentumsum, i.e. (cid:80) k → V N F (cid:80) i ∆ ξ . We assume that eachshell has a spherical shape of area V N F and width ∆ ξ .We note that this replacement is valid only for the s -wave systems, which have a spherical Fermi surface, anda spherical pairing symmetry of the Cooper pairs [2]. Forthis representation, the linearized Hamiltonian of L (2)B isformally expressed as H (2)B = b · M · b . (17)The matrix M is the Bogoliubov matrix of 2 N di-mensions, where N is the number of the energyshells. The notation b = ( b , · · · , b N , b ∗ , · · · , b ∗ N ) T isa composite vector of complex-valued fields, and b =( b ∗ , · · · , b ∗ N , b , · · · , b N ) is its Hermitian conjugation.To obtain the energy spectrum of H (2)B , we performthe bosonic Bogoliubov transformation [51, 58] for theclassical variables, i.e. b = W · β . For later use, especiallyin Sec. VI, we parametrize the 2 N × N matrix W as W = (cid:18) U VV ∗ U ∗ (cid:19) . (18)The bosonic Bogoliubov transformation has to preservethe Poisson bracket of the bosons, so that the condition W Σ N W H = Σ N , (19)is required [51, 58]. W H represents the Hermitian conju-gation of W . The matrix Σ N = diag( N , − N ) is a met-ric tensor of a 2 N -dimensional Minkowski space. Hence,the matrix W should be normalized as it is pseudo-unitary. See references [51, 58] for details and furtherapplications.The eigenvalue problem that we solve is (Σ N M ) · W = W · Λ. The eigenvalue matrix is Λ = diag( λ , · · · , λ N ). (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:2)(cid:2)(cid:3)(cid:2)(cid:4) (cid:2)(cid:3)(cid:5) (cid:2)(cid:3)(cid:6) (cid:2)(cid:3)(cid:7) (cid:4)
0, and also the approximatevalues estimated by ∆ (cid:48) = 2 (cid:126) ω D e − / ( gN F ) for reference.In this regime, we find that the gap function is smallerthan the Debye frequency (cid:126) ω D = 0 . E F , and the chem-ical potential µ (cid:48) is sufficiently close to 1. In this work,we only focus on this weakly-interacting regime, i.e., theBCS regime [59, 60], in three dimensions. (cid:1)(cid:2)(cid:3)(cid:4) (cid:3) (cid:5) (cid:6) (cid:1) (cid:3)(cid:7) (cid:8)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:3)(cid:4)(cid:1)(cid:6)(cid:3)(cid:4)(cid:1)(cid:7)(cid:3)(cid:4)(cid:7) (cid:6) (cid:5) (cid:8) (cid:9) (cid:6)(cid:10) (cid:2)(cid:5)
0. The horizontal axis represents N , rang-ing from N = 100 to N = 3200. The x -axis is displayed as logscale with basis 2. We put gN F = 0 .
8. (Right): The size de-pendence of the second eigenvalue (cid:126) ω sq2 /E F > gN F , implying its convergence. As seen in Fig. 1, the values of (cid:126) ω sq s from s = 2 to s = N have positive values for all gN F . The energy eigen-values are continuously distributed over an energy range,for each gN F . As discussed later, this finite-size rangeof the eigenvalues implies the band width of the Higgsamplitude mode of the BCS superconductors [7, 8]. InSec. VI, we will demonstrate that these allowed statesform a band dispersion of the Higgs mode in physically-relevant dynamical quantities with a mass gap at theFermi surface. We also find that the range of the distri-bution decreases as gN F increases, implying the reduc-tion of the band width of the Higgs mode with increasinginteraction. We note that the minimum and maximumvalues of (cid:126) ω sq s> , i.e., (cid:126) ω sq2 and (cid:126) ω sq N , show good conver-gence for N ≥ (cid:126) ω sq2 is shown in Fig. 2. Moreover, as seen in Fig. 1, theenergy difference between (cid:126) ω sq2 and (cid:126) ω sq1 increases with gN F , see also Fig. 2.Figure 1 indicates that the eigenvalue for s = 1, (cid:126) ω sq1 ,exhibits small, but finite negative values for all gN F . Thismode can be regarded as an onset of the NG mode withzero momentum, i.e., a zero-energy excited state. Seealso Sec. VI. As shown in Fig. 2, the finite value of thelowest mode approaches zero as the matrix size N in-creases. Therefore, the gap opening of the lowest modeseems to be a finite-size effect, which is similar to theAnderson tower of states of quantum spin chains [61, 62].Given this numerical insight, we expect that the energy TABLE I. Numerically computed values of the gap function.∆ denotes the numerical solution of the gap equation, and∆ (cid:48) = 2 (cid:126) ω D e − / ( gN F ) . The Debye frequency is (cid:126) ω D = 0 . E F . gN F ∆ (cid:48) ( E F ) ∆ ( E F )0.7 4 . × − . × − . × − . × − . × − . × − . × − . × − gap of the lowest mode asymptotically approaches zerofrom the negative side for larger values of N , constitut-ing the NG mode with zero momentum in the thermody-namic limit.The negativity of the lowest eigenvalue means thatthe fluctuation expansion of v k up to the quadratic or-der fails to reproduce the physical properties of the NGmodes. For the negative mode, the Bose distributionfunction, f B ( ω sq1 ) ≡ e β (cid:126) ω sq1 − , can acquire negative oc-cupations, meaning unphysical contributions to the pathintegral. Indeed, the definition of the squeezed-field pathintegral (9) does not allow the fields b k to occupy redun-dant states with 1 < | v k | . Thus, in order to describethe NG-mode branch, we need to take higher-order ver-tices of the HP expansion into account beyond the linearapproximation.However, we note that the NG modes are gapped outto high-energy plasmon modes in real solids because ofthe coupling to the electromagnetic gauge field, i.e., theAnderson–Higgs mechanism [39, 63]. As we will see be-low, the linearized Lagrangian leads to the dispersion re-lation of the Higgs mode in the spectral functions. Thebranch of the Higgs modes exists as low-energy stateswith or without the coupling to the gauge field [39, 63].Therefore, we expect that our approach can be appliedto analyzing dynamical properties of the Higgs mode inreal BCS superconductors. VI. OBSERVABLES
In this section, we apply the squeezed-field path-integral formalism to experimentally relevant quanti-ties. Specifically, we discuss the properties of the single-particle spectral function [21].To obtain the single-particle spectral function, we con-sider the single-particle Green’s function in the imaginarytime axis [46] G ( k , τ − τ (cid:48) ) = − (cid:68) T τ (cid:104) ˆ c k , ↑ ( τ )ˆ c + k , ↑ ( τ (cid:48) ) (cid:105)(cid:69) . (21)where T τ {· · · } is the time-ordering operation for theHeisenberg field operators ˆ c k , ↑ ( τ ) ≡ e τ ˆ H/ (cid:126) ˆ c k , ↑ e − τ ˆ H/ (cid:126) and ˆ c + k , ↑ ( τ ) ≡ e τ ˆ H/ (cid:126) ˆ c † k , ↑ e − τ ˆ H/ (cid:126) . The Matsubara fre-quency expansion of finite-temperature quantum-fieldtheory [46] decomposes G ( k , τ − τ (cid:48) ) into the Fourier com-ponents ˜ G ( k , iω n ) with ω n = π (2 n + 1) k B T for n ∈ Z .The analytical continuation of ˜ G ( k , iω n ) from the imag-inary axis to the real axis of the complex plane leads tothe spectral function of single-particle excitations, see,e.g., Ref. [21]: A ( k , ω ) = − π Im (cid:104) ˜ G ( k , iω n → ω + iδ ) (cid:105) , (22)where δ > A ( k , ω ) reduces to a sum of two δ -functions at ω = ω k HiggsBCS
2, ˜ ξ k = 0, and ˜ ξ k = (cid:126) ω D /
2, respectively.(e-g): The peak weights for gN F = 0 . N = 1000, and at (e) ˜ ξ k = − (cid:126) ω D /
2, (f) ˜ ξ k = 0, and (g) ˜ ξ k = (cid:126) ω D / and at ω = − ω k , corresponding to the quasi-particleand quasi-hole excitations of the superconductor, respec-tively [46]: A ( k , ω ) ≈ u k δ ( ω − ω k ) + v k δ ( ω + ω k ) (23) ≡ A MF ( k , ω ) . The spectral weight functions u k and v k are given by thevalues of the mean-field ground state from the minimiza-tion of the BCS energy functional. We note that pertur-bative evaluations of this spectral function for attractiveBCS-type models beyond the BCS approximation havebeen reported in the literature [21, 24].We discuss a higher-order correction to the mean-field result of the single-particle spectral function, whicharises due to the order-parameter fluctuations of thesqueezing fields v k . We assume once again that thefluctuations of the fields v k are small, to approxi-mate the squeezed-field Lagrangian at quadratic or-der, via L (2)B , see Eq. (15). We assume that the sys-tem is at zero temperature. Within the linear approx-imation, the quasiparticles and the squeezing modesare not explicitly correlated to each other. There-fore, we can utilize the Wick theorem of Gaussianfunctional integrations [39]. The Wick theorem statesthat G ( k , τ ) ≈ −(cid:104) u k ( τ ) u k (0) (cid:105) sq (cid:104) c k , ↑ ( τ ) c k , ↑ (0) (cid:105) sq −(cid:104) v k ( τ ) v ∗ k (0) (cid:105) sq (cid:104) c − k , ↓ ( τ ) c k , ↓ (0) (cid:105) sq . Note that (cid:104)· · · (cid:105) sq im- plies the functional average with the squeezed-field pathintegral of the quadratic-order Lagrangian. Utilizing theresults of Appendix C, we obtain the spectral functionwith the fluctuation corrections A ( k , ω ) ≈ u k δ ( ω − ω k ) + v k δ ( ω + ω k )+ v k u k N (cid:88) s =2 |U k ,s + V k ,s | δ ( ω − ω k − ω sq s )+ N (cid:88) s =2 |V k ,s | δ ( ω + ω k + ω sq s ) . (24)The first two δ -functions are the mean-field contribu-tions of A MF ( k , ω ). We find that the correction termsto A MF ( k , ω ) give sideband peaks to the total spectrumat ω = ω k + ω sq s and at ω = − ω k − ω sq s , respectively,reflecting the fluctuations of the order parameter. Thespectral weights of the sideband peaks are determined bythe transformation matrices of the bosonic Bogoliubovtransformation U and V , which have been introduced inSec. V.We note that in the expression (24) we have not in-cluded the zero-mode contributions of H (2)B . If we eval-uate these within the linear approximation, we find thatthe zero modes change the spectral weight of the quasi-particle at ω = ω k − ω sq1 = ω k + 0 + and that of the quasi-hole at ω = − ω k + ω sq1 = − ω k − + , see also Appendix C.However, as discussed in Ref. [64], the zero-momentumand zero-energy Bogoliubov mode describes the momen-tum of the condensed state, rather than an intrinsic exci-tation. Furthermore, as mentioned above, in a real solidthis NG mode is not gapless, but has a non-zero energywhich is the plasmon energy. For this reason we have notincluded this contribution in Eq. (24). In the following,we focus on the finite-energy contributions that give theproperties of the Higgs mode. Ignoring the zero-energycontributions does not affect the following consequenceson the Higgs mode.In Fig. 3(a), we plot Eq. (24) as a function of ω and˜ ξ k = (cid:15) k − µ (cid:48) = (cid:15) k − E F . The peak contrast of A ( k , ω )indicates that there are two peaks above and below thepeaks of the quasiparticle and quasihole excitations of theBCS approximation. These high-energy peaks are theHiggs amplitude mode, which are intuitively visualizedas the fluctuation modes of the amplitude of the order-parameter field ∆ = (cid:80) k u k v k [8]. In Figs. 3(c) and (f),we display the spectral peak weights of A ( k , ω ) near theFermi surface, i.e., around ˜ ξ k = 0. There, the weightsof the Higgs mode are sufficiently small compared to theweight of the quasiparticle excitation, which is given by u k . This supports that the HP expansion of u k is justifiednear the Fermi surface in describing the corrections tothe mean-field result. In addition, the particle regime for ξ (cid:48) k (cid:38) (cid:126) ω D / ξ (cid:48) k (cid:46) − (cid:126) ω D / v k is close to unity, andthe sideband peaks have larger values than u k . There-fore, the dispersion relation of the Higgs mode of thisregime is only qualitatively correct within the linear ap-proximation of the squeezed-field Lagrangian. Further-more, the large values of the squeezing-field correctionsalso imply that the Higgs dispersion in this regime maybe significantly modified by higher-order nonlinear cor-rections of the vertices. We note that modifications of thesideband curvature of the squeezing mode due to higher-order nonlinear couplings have been discussed in previousworks [33, 34].The information of the dispersion relation of the Higgsmode is embedded in the Bogoliubov matrices U = ( U k ,s )and V = ( V k ,s ). In Fig. 4(a), we display |U k ,s + V k ,s | asa function of ˜ ξ k and (cid:126) ω s for gN F = 1 . (cid:126) ω D = 0 . E F ,and N = 400. The maximum value of |U k ,s + V k ,s | for each k forms a clear band dispersion with an en-ergy gap (cid:126) ω H at the Fermi surface. We find that theHiggs dispersion has a larger energy gap than that of thequasiparticle dispersion predicted by the BCS approxi-mation, for these parameters. Moreover, the dispersionrelation of the Higgs mode has a larger curvature thanthat of the quasiparticle dispersion. We note that theband width of the Higgs-mode branch stems from thewidth of the continuous distribution of λ s , as shown inFig. 1. We also note that, if we plot |U k ,s | or |V k ,s | instead of |U k ,s + V k ,s | , the resulting band dispersion is (cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:3)(cid:1)(cid:2)(cid:1)(cid:4)(cid:1)(cid:2)(cid:1)(cid:4)(cid:3) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:1)(cid:2)(cid:7) (cid:4) (cid:1) (cid:2)(cid:3) (cid:4) (cid:5) (cid:6) (cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:10)(cid:15)(cid:16)(cid:9) BCSHiggs (a)(b)
0. The colorbar indicates the magnitude of |U k ,s + V k ,s | . The horizontalaxis is ˜ ξ k / ( (cid:126) ω D ). The vertical axis is measured in units of E F .(b): Comparison between the Higgs gap (cid:126) ω H (black square)and the gap function ∆ (blue circle). The horizontal axis is gN F . The energies are measured in units of E F . The matrixsize is N = 1000. the same as that of |U k ,s + V k ,s | .In Fig. 4(b), we display the extracted Higgs gap (cid:126) ω H as a function of gN F . For all gN F , we find that theHiggs gap (cid:126) ω H is larger than the mean-field gap func-tion ∆, but smaller than 2∆, implying an inequality∆ < (cid:126) ω H < (cid:126) ω MFAH = 2∆ [8, 38]. Hence, our nu-merical result displays a reduction of the Higgs gap fromthe mean-field Higgs gap. This reduction is due to a com-bined effect of the quantum fluctuations of the squeezingfields, and the nonzero correlations between the differentmomentum modes within the linear approximation.
VII. CONCLUSIONS
We have developed a squeezed-field path integral de-scription of BCS superconductors. Utilizing a complete-ness relation of squeezed-fermionic coherent states, wehave constructed a generalized Grassmann path integral,that describes collective excitations of the order parame-ter and quasi-particle excitations in a single framework.The collective excitations of the order parameter are de-scribed by the dynamics of the coupled squeezing param-eters of the system, which are naturally expressed as adynamical Anderson pseudospin model.In particular, we have demonstrated that the eigen-modes of the squeezing sector of the path integral de-scribe the Higgs mode of the superconductor. We deter-mine the Higgs energy gap and the Higgs dispersion vianumerical evaluation of the eigenmodes of the squeezingsector. The resulting Higgs spectrum is reflected in thesingle-particle spectral function, which displays sidepeaksof the quasi-particle peaks.In this discussion we have ignored the Nambu–Goldstone mode that is predicted by the BCS model.The BCS model predicts this gapless excitation mode, incontrast to the gapped plasmon mode observed in solids,and predicted by the Anderson-Higgs mechanism. Wewill include our description to capture this feature of su-perconductors elsewhere.In conclusion, we emphasize that the formalism thatwe have put forth in this paper advances the theoreticaldescription of superconductors in a fundamental way, andgives a new perspective on collective excitations, such asthe Higgs mode, and their coupling to quasi-particle ex-citations of the superconductor. Furthermore we notethat the formalism that we have developed here, lays outa framework that can be applied not only to a wide rangeof superconducting states, such as of higher orbital sym-metry or of topological nature, but also to any orderedstate that emerges in a fermionic system and can be de-scribed by an order parameter composed of two fermions.Therefore our formalism enables new insight into a broadrange of physical systems that are central to many-bodytheory.
ACKNOWLEDGMENTS
This work was supported by the Deutsche Forschungs-gemeinschaft (DFG) through the SFB 925 and the Clus-ter of Excellence ‘Advanced Imaging of Matter’ of theDFG EXC 2056 - project ID 390715994. D.L. acknowl-edges financial support under a Katholischer Akademis-cher Auslandsdienst (KAAD) stipend. We thank IliasM. H. Seifie and Caroline Nowoczyn for interesting andfruitful discussions.
Appendix A: Evaluation of the path-integraloverlaps
We write | Ψ j (cid:105) ≡ ˆ S ( η j ) | c j (cid:105) as the squeezed-fermioniccoherent state at time τ = j ∆ τ . The overlaps of thesqueezed-field Grassmann path integrals are given by (cid:104) Ψ j | Ψ j − (cid:105) = (cid:104) c j | ˆ S † ( η j ) ˆ S ( η j − ) | c j − (cid:105) . (A1)We perform an expansion ˆ S † ( η j ) ˆ S ( η j − ) = ˆ1 +∆ τ δ ˆ S ( η j ) + · · · to obtain δ ˆ S ( η j ) = − (cid:88) k ˙ θ k ,j ˆ S † ( η j ) ∂ θ k ,j ˆ S ( η j ) − (cid:88) k ˙ ϕ k ,j ˆ S † ( η j ) ∂ ϕ k ,j ˆ S ( η j ) . (A2)One can prove thatˆ S † ( η j ) ∂ θ k ,j ˆ S ( η j ) = e iϕ k ,j c † k , ↑ ˆ c †− k , ↓ − e − iϕ k ,j c − k , ↓ ˆ c k , ↑ , (A3)ˆ S † ( η j ) ∂ ϕ k ,j ˆ S ( η j ) = i S † ( η j )(ˆ n k , ↑ + ˆ n − k , ↓ ) ˆ S ( η j ) − i n k , ↑ + ˆ n − k , ↓ ) . (A4)See also Ref. [34]. These relations lead to (cid:104) Ψ j | Ψ j − (cid:105) = (cid:104) c j | c j − (cid:105) e − ∆ τ (cid:80) k ˙ θ k ,j ( e iϕ k ,j c k , ↑ c − k , ↓ − e − iϕ k ,j c − k , ↓ c k , ↑ ) × e − ∆ τ (cid:80) k i ˙ ϕ k ,j ( | v k ,j | −| v k ,j | c k , ↑ c k , ↑ −| v k ,j | c − k , ↓ c − k , ↓ + u k ,j v k ,j c k , ↑ c − k , ↓ + u k ,j v ∗ k ,j c − k , ↓ c k , ↑ ) + O (∆ τ ) , (A5) (cid:104) c j | c j − (cid:105) = exp ∆ τ (cid:88) k ,σ c k ,σ ( τ j ) − c k ,σ ( τ j − )∆ τ c k ,σ ( τ j − ) − ∆ τ (cid:88) k ,σ c k ,σ ( τ j ) c k ,σ ( τ j ) − c k ,σ ( τ j − )∆ τ . (A6) Appendix B: N -dimensional Bogoliubov matrix The bosonic Hamiltonian H (2)B can be expressed as H (2)B = N (cid:88) i =1 (cid:0) b ∗ i b i (cid:1) (cid:18) A ii B ii B ii A ii (cid:19) (cid:18) b i b ∗ i (cid:19) + (cid:88) i (cid:54) = i (cid:0) b ∗ i b i (cid:1) (cid:18) A i i B i i B i i A i i (cid:19) (cid:18) b i b ∗ i (cid:19) . (B1) Using these matrices A and B , the Bogoliubov matrix M is written as M = A A · · · B B · · · A A · · · B B · · · ... ... ... ... B B · · · A A · · · B B · · · A A · · · ... ... ... ... . (B2)0Each matrix element is defined as follows: A ii = (cid:20) ξ i + E F − µ (cid:48) + v i u i ∆2 + ∆4 v i u i (cid:21) − gN F ∆ ξ v i u i − gN F ∆ ξ ,B ii = ∆4 v i u i − gN F ∆ ξ v i − gN F ∆ ξ v i u i ,A ij = − gN F ∆ ξv i v j − gN F ∆ ξ v i u i v j u j − gN F ∆ ξ u i u j + gN F ∆ ξ (cid:34) v i u i u j + v j u j u i (cid:35) , for i (cid:54) = j,B ij = − gN F ∆ ξv i v j − gN F ∆ ξ v i u i v j u j + gN F ∆ ξ (cid:34) v i u i u j + v j u j u i (cid:35) , for i (cid:54) = j, where ξ i ∈ [ − (cid:126) ω D , (cid:126) ω D ]. Appendix C: The imaginary-time Green’s functionwithin the linear approximation
For the linearized path-integral action, the imaginary-time Green’s function reduces to G ( k , τ ) ≈ − (cid:104) u k ( τ ) u k (0) (cid:105) sq (cid:104) c k , ↑ ( τ ) c k , ↑ (0) (cid:105) sq − (cid:104) v k ( τ ) v ∗ k (0) (cid:105) sq (cid:104) c − k , ↓ ( τ ) c − k , ↓ (0) (cid:105) sq . (C1)We expand u k ( τ ) in b k ( τ ) = v k ( τ ) − v k to obtain theleading order correction to the BCS mean-field result,i.e. G ( k , τ ) ≈ − u k [1 − f F ( ω k )] e − ω k τ − v k f F ( ω k ) e ω k τ + G ( k , τ ) + G ( k , τ ) + G ( k , τ ) + G ( k , τ ) . (C2)For this equation, we have defined G ( k , τ ) = − v k u k [1 − f F ( ω k )] N (cid:88) s =1 |U k ,s + V k ,s | × [1 + f B ( ω sq s )] e − τ ( ω sq s + ω k ) , (C3) G ( k , τ ) = − v k u k [1 − f F ( ω k )] N (cid:88) s =1 |U k ,s + V k ,s | × f B ( ω sq s ) e − τ ( ω k − ω sq s ) , (C4) G ( k , τ ) = − f F ( ω k ) N (cid:88) s =1 |U k ,s | [1 + f B ( ω sq s )] e τ ( ω k − ω sq s ) , (C5) G ( k , τ ) = − f F ( ω k ) N (cid:88) s =1 |V k ,s | f B ( ω sq s ) e τ ( ω k + ω sq s ) . (C6)Notice that U and V can be assumed to be real for thematrix Σ N M , because it has no imaginary term. Thefree propagators of each field at finite temperatures can be calculated [39] as (cid:104) c k , ↑ ( τ ) c k , ↑ (0) (cid:105) sq = [1 − f F ( ω k )] e − τω k , (C7) (cid:104) c − k , ↓ ( τ ) c − k , ↓ (0) (cid:105) sq = f F ( ω k ) e τω k , (C8) (cid:104) β s ( τ ) β ∗ s (0) (cid:105) sq = [1 + f B ( ω sq s )] e − τω sq s , (C9) (cid:104) β ∗ s ( τ ) β s (0) (cid:105) sq = f B ( ω sq s ) e τω sq s . (C10) f B / F ( ω ) = e β (cid:126) ω ∓ is the Bose or Fermi distribution func-tion.The Matsubara-Fourier transforms of G , G , G , and G with ω n = π (2 n + 1) / ( (cid:126) β ) read˜ G ( k , iω n ) = v k u k N (cid:88) s =1 |U k ,s + V k ,s | Θ ( ω k , ω sq s ) iω n − ( ω k + ω sq s ) , (C11)˜ G ( k , iω n ) = v k u k N (cid:88) s =1 |U k ,s + V k ,s | Θ ( ω k , ω sq s ) iω n − ( ω k − ω sq s ) , (C12)˜ G ( k , iω n ) = N (cid:88) s =1 |U k ,s | Θ ( ω k , ω sq s ) iω n + ( ω k − ω sq s ) , (C13)˜ G ( k , iω n ) = N (cid:88) s =1 |V k ,s | Θ ( ω k , ω sq s ) iω n + ( ω k + ω sq s ) . (C14)Notice that e iβ (cid:126) ω n = e π (2 n +1) = −
1. The functionsΘ i ( ω k , ω sq s ) for each Green’s function are defined byΘ ( ω k , ω sq s ) = [1 − f F ( ω k )][1 + f B ( ω sq s )][1 + e − β (cid:126) ( ω k + ω sq s ) ] , Θ ( ω k , ω sq s ) = [1 − f F ( ω k )] f B ( ω sq s )[1 + e − (cid:126) β ( ω k − ω sq s ) ] , Θ ( ω k , ω sq s ) = f F ( ω k )[1 + f B ( ω sq s )][1 + e (cid:126) β ( ω k − ω sq s ) ] , Θ ( ω k , ω sq s ) = f F ( ω k ) f B ( ω sq s )[1 + e β (cid:126) ( ω k + ω sq s ) ] . The zero-temperature limits of Θ i ( ω k , ω sq s ) depend onthe sign of (cid:126) ω sq s . For example, Θ ( ω k , ω sq s ) becomes unityfor (cid:126) ω sq s > ( ω k , ω sq s ) = 1 + e β (cid:126) ( ω k + ω sq s ) ( e β (cid:126) ω k + 1)( e β (cid:126) ω sq s − e − β (cid:126) ( ω k + ω sq s ) + 11 + e − β (cid:126) ω k − e − β (cid:126) ω sq s − e − β (cid:126) ( ω k + ω sq s ) → . However, for (cid:126) ω sq s <
0, it becomes zero. Indeed,Θ ( ω k , ω sq s ) = e − β (cid:126) ω k + e β (cid:126) ω sq s e β (cid:126) ω sq s + e β (cid:126) ( ω sq s − ω k ) − − e − β (cid:126) ω k → . The dependence of the signs of lim T → Θ i ( ω k , ω sq s ) issummarized as follows:lim T → Θ ( ω k , ω sq s ) = lim T → Θ ( ω k , ω sq s ) = 1 − δ s, , (C15)lim T → Θ ( ω k , ω sq s ) = lim T → Θ ( ω k , ω sq s ) = − δ s, . (C16)Therefore, we obtain Eq. (24) in the text.1 [1] P. W. Anderson, Basic notions of condensed matterphysics (CRC Press, 2018).[2] A. J. Leggett et al. , Quantum liquids: Bose condensationand Cooper pairing in condensed-matter systems (Oxforduniversity press, 2006).[3] X.-G. Wen,
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