Leggett modes in iron-based superconductors as a probe of Time Reversal Symmetry Breaking
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Leggett modes in iron-based superconductors as a probe ofTime Reversal Symmetry Breaking
M. Marciani, L. Fanfarillo,
2, 1
C. Castellani, and L. Benfatto Institute for Complex Systems (ISC), CNR, U.O.S. Sapienza andDepartment of Physics, Sapienza University of Rome, P.le A. Moro 2, 00185 Rome, Italy Instituto de Ciencia de Materiales de Madrid, ICMM-CSIC,Cantoblanco, E-28049 Madrid, Spain (Dated: August 14, 2018)Since their discovery, it has been suggested that pairing in pnictides can be mediated by spinfluctuations between hole and electron bands. In this view, multiband superconductivity wouldsubstantially differ from other systems like MgB , where pairing is predominantly intraband. Indeed,interband-dominated pairing leads to the coexistence of bonding and antibonding superconductingchannels. Here we show that this has profound consequences on the nature of the low-energysuperconducting collective modes. In particular, the so-called Leggett mode for phase fluctuationsis absent in the usual two-band description of pnictides. On the other hand, when also the repulsionbetween the hole bands is taken into account, a more general three-band description should be used,and a Leggett mode is then allowed. Such a model, that has been proposed for strongly hole-doped122 compounds, can also admit a low-temperature s + is phase which breaks the time reversalsymmetry. We show that the (quantum and thermal) transition from the ordinary superconductorto the s + is state is accompanied by the vanishing of the mass of Leggett-like phase fluctuations,regardless the specific values of the interaction parameters. This general result can be obtained bymeans of a generalized construction of the effective action for the collective degrees of freedom thatallows us also to deal with the non-trivial case of dominant interband pairing. I. INTRODUCTION
At microscopic level the appearance of superconduc-tivity requires the pairing of electrons into Cooper pairs,which can then form a macroscopic coherent state re-sponsible for the superfluid behavior. Within BCS the-ory, which successfully explained the superconducting(SC) phenomenon in the so-called conventional super-conductors, electrons can overcome their mutual repul-sion thanks to the presence of phonons, which overscreenthe Coulomb repulsion leading to a residual attractionresponsible for the pairing. However, such a mechanismposes an upper limit to the attainable transition tempera-ture, that has been widely exceeded in the so-called high-temperature superconductors, as cuprate or iron-basedsystems.
In all these cases superconductivity emergesand/or competes with strong electron-electron repulsion,that can be accommodated by Cooper pairs by meansof an unconventional form of the wave function, as it isthe case in cuprates, where the d -wave symmetry of pair-ing allows the pairs to overcome the on-site Hubbard-like repulsion. In the case of pnictides the mechanismis somehow similar, once that the multiband nature ofthe Fermi surface is taken into account. Indeed, it hasbeen suggested by several microscopic approaches thatat low energy the intraband Coulomb repulsion is over-come by the interband repulsion, which allows the pairsto be formed in different bands with a gap having oppo-site sign, the so-called s ± symmetry. Roughly speaking,such a sign change converts a repulsion in attraction,making the pairs formation possible. Notice that sucha pairing mechanism is fundamentally different from theone observed in other multiband superconductors, as e.g. MgB . Here indeed the largest pairing channel is the in-traband phononic one, and the interband interaction isonly responsible for a relatively small Josephson-like cou-pling of pairs in different bands. In this respect, pnictidesuperconductors represent a completely different class ofSC systems with respect to MgB .A fundamental question associated to the unconven-tional nature of pairing is how it can affect the behaviorof the SC collective modes, which in turn can influencethe observable physical quantities, giving indirect infor-mation on the nature of the underlying SC state. Suchan issue has been widely discussed in the past withinthe context of cuprate superconductors, and it hasbeen the subject of intense investigation in the recent lit-erature on pnictide superconductors. Here the issueis made even more involved by the presence of severalbands, that would suggest the presence of multiple col-lective modes associated to the fluctuations of the ampli-tude and phase of the condensates in the various bands.For example, it has been discussed the possibility toobserve the so-called Leggett mode, that correspondsto the relative density (phase) fluctuations of the con-densate in the various bands. As it has been shownlong ago in a seminal paper by Leggett, such a mas-sive mode could eventually lie below the threshold forparticle-hole excitations, avoiding then its overdamping.Such a situation is partly realized in MgB , whereindeed experimental signatures of the Leggett mode havebeen identified in Raman spectroscopy. In the case ofpnictides it has been also suggested the intriguing possi-bility that the Leggett mode becomes massless atthe quantum transition between an ordinary s ± state anda time-reversal-symmetry broken (TRSB) state. Sucha TRSB state can emerge for example in a three-bandcase when interband repulsion is equally large between allthe bands: in this situation the sign change betweenone band and the remaining two is frustrated, leading toan intrinsically complex order parameter (∆ ∗ , ∆ ∗ , ∆ ∗ ) =(∆ , ∆ , ∆ ) . Since the emergence of a massless col-lective mode could bear several observable consequencesin physical observable, as e.g. Raman response or in-tervortex interactions, it could be used as a smokinggun to test the appearance or not of a TRSB state inpnictides.Quite interestingly, the theoretical investigation of theproperties of collective modes in pnictides suffered un-til now of a fundamental limitation. Indeed, as wediscussed at the beginning, pairing in pnictides arisemainly from interband interactions. However, very of-ten a modelization has been used in the literature basedon multiband models with predominant intraband pairinginteractions. While this makes it possible to derivethe collective modes using standard procedures based onthe construction of the effective action for the collectivedegrees of freedom, it makes these results unsuitablefor the specific case of pnictides. On the other hand,an alternative derivation based on the direct diagram-matic derivation of the collective response functions, asthe one used in Refs. , does not allow for a simplegeneral understanding of the number and nature of thecollective modes. As we discuss in the present paper, thedifference between the two cases is not only quantita-tive but qualitative. Indeed, when interband interactiondominates, as it is the case physically relevant for pnic-tides, the number itself of available low-energy collectivemodes is smaller than the number of bands involved inthe problem. In this case the correct understanding ofthe SC collective modes should be based on the numberof
SC bonding channels , that is usually smaller than thenumber of bands involved. This leads to several profounddifferences between pnictides and ordinary (intraband-dominated) multiband superconductors, like e.g. MgB .A powerful root to enlighten these differences is the ex-plicit construction of the action for the collective modesstarting from a microscopic model for pnictides that as-sumes predominant interband pairing. In the ordinarycase of intraband-dominated pairing such a procedurerelies on the use of the so-called Hubbard-Stratonovich(HS) decoupling of the SC interaction by means of abosonic fields associated to the pairing operators. Thisapproach has been successfully applied to two or three-band models with predominant intraband pairing.However, when interband coupling dominates, as it is thecase for pnictides, the HS decoupling must be properlymodified to account for the presence of antibonding SCchannels, an issue that has been often overlooked in therecent literature in the derivation of effective functionalsboth above and below T c . Here we follow insteadthe strategy outlined recently in Ref. [29], where the cor-rect implementation of the HS procedure has been usedto describe the fluctuations above T c . We then intro- duce a transformation of the pairing fields in the variousbands that allows us to show that below T c the fluc-tuations associated to the antibonding SC channels donot give rise to observable collective modes. This re-sult follows immediately from a general correspondencebetween the low-energy collective phase fluctuations andthe multiband mean-field equations. When applied to thetwo- or three-band case with dominant interband pairing,relevant for pnictides, this correspondence allows one toshow that: (i) in the two-band case the Leggett modeis absent, in contrast to intraband-dominated supercon-ductors as MgB ; (ii) in the three-band case a Leggettmode is present, it becomes massless at the TRSB transi-tion and it acquires again a small mass inside the TRSBphase due to the mixing to amplitude fluctuations. A sec-ond low-energy mode appears in the TRSB state, eventhough it is usually found very near to the threshold forsingle-particle excitations. In contrast to the previousliterature, which focused on the softening of the Leggettmode at T = 0 as a function of the SC coupling lead-ing the system through a quantum phase transition to aTRSB state, we discuss its occurrence as a functionof temperature. Indeed, the thermal phase transition be-tween a TRS and TRSB phase is possibly realized ina much wider range of parameters for realistic systems,and then it has definitively more chances to be observedexperimentally.The structure of the paper is the following. In Sec.II-A we outline the main steps that lead to the effectiveaction for the collective degrees of freedom starting froma microscopic two-band model with interband-dominatedpairing. The character of the amplitude and phase modesis discussed in Sec. II-B, where we also show the absenceof the ordinary Leggett mode, found instead in two-bandsuperconductors with intraband-dominated pairing. Thethree-band case is discussed in Sec. III. Sec. III-A is de-voted to a brief review of the possible relevance for pnic-tides of three-band models which admit a TRSB state.The general structure of the collective modes is discussedin Sec. III-B, where it is established the correspondencebetween the TRSB transition and the vanishing of themass of a Leggett-like mode. In Sec. III-C we consider aspecific set of SC couplings to show explicitly the temper-ature (and quantum) evolution of the low-energy modesacross the TRSB transition. The results of Fig. 4 sum-marize the main physical messages relevant for the readerwho is not interested in the theoretical aspects of theirderivation, and Sec. III-D contains a general discussionon the experimental probes that can be used to test thebehavior of the phase collective modes near the TRSBstate. Sec. IV contains our final remarks and the sum-mary of the main results of the paper. Additional techni-cal details, which are useful to make a direct comparisonwith previous work in the literature, are reported in theappendices. Appendix A shows the equivalence betweenthe derivation of the Gaussian action for SC fluctuationsdone in polar or cartesian coordinates. Appendix B dis-cusses the two-band case with dominant intraband pair-ing by means of the formalism of the present manuscript.Finally, in Appendix C we discuss the general connectionbetween the TRSB transition and the effective action forthe three-band model. II. THE EFFECTIVE ACTION FOR ATWO-BAND MODELA. Construction of the effective action
To show explicitly the peculiar role of interband inter-actions in determining the nature of the collective modeswe first describe the two-band case. Having in mind pnic-tide systems, such an effective modelization is usually ap-propriate for systems not too far away from half-filling.Indeed, in this case one can assume that the most relevantinteractions are between the two hole pockets centered at Γ and the two electron ones centered at M, with no in-teraction between the hole bands (see also discussion inSec. IIIA below). Assuming also that the electron bandsare degenerate, this four-band model can be mapped into an effective BCS-like two-band one as H = H + H int ,H = X k ,l,σ ξ l k c l † k σ c l k σ H int = − X q ,ij ˆ g lm φ † l, q φ m, q ( l, m = 1 , (1)where φ l, q = X k c l k + q ↓ c l − k ↑ (2)is the pairing operator in each band and the matrix ˆ g lm ˆ g = (cid:18) α γγ β (cid:19) , det ˆ g < (3)describes predominant interband pairing. The bare elec-tronic dispersion in Eq. (1) will be approximated with aparabolic one, ξ l k = ε l ± k / m l − µ , with the plus or mi-nus sign for electrons or holes, respectively, and the chem-ical potential µ will be taken equal to zero. We noticethat while to account quantitatively for the correct spec-tral and thermodynamic properties of pnictides a morerefined Eliashberg-like multiband approach is needed, the Hamiltonian (1) can be considered an appropriatestarting point to discuss the general structure of collec-tive modes in most pnictides.As customary, the microscopic effective model for thecollective modes can be derived by considering the actioncorresponding to the Hamiltonian (1), within the finite-temperature Matsubara formalism, S = Z β dτ X l, k σ c l † k σ ( τ )[ ∂ τ + ξ k ] c l k σ ( τ ) dτ + H I ( τ ) , (4) where τ is the imaginary time and β = 1 /T . To obtainthe effective action in terms of the order-parameter col-lective degrees of freedom, the interaction H I is usuallydecoupled in the particle-particle channel by means ofthe Hubbard-Stratonovich field h HS : e ± Λ φ † φ = Z D h HS e −| h HS | / Λ+ √± φ † h HS + h.c. ) . (5)In the above equation the imaginary unit √− ≡ i sig-nals the presence of a repulsive particle-particle inter-action. In the usual single-band case one deals withan interaction attractive in the particle-particle channel,so no imaginary unit appears. However in the presentmultiband case with predominant interband coupling thediagonalization of the ˆ g matrix with a proper rotation R will lead in general also to a negative eigenvalue, corre-sponding to repulsion in the particle-particle channel: ˆ g = R − ˆΛ R = R − (cid:18) Λ − Λ (cid:19) R, Λ , > . (6)As we shall see below, the saddle-point values of the HSfields h HS are connected to the SC gaps in the variousbands. However, the imaginary unit in the transfor-mation (5) would force us to shift the integration con-tour of Reh by a finite imaginary quantity, so that Reh ∈ R + iA , see discussion below Eq. (18). Topreserve an ordinary integration contour we will enforce A = 0 by taking advantage of the fact that the interactionHamiltonian H I can be put in the diagonal form under amore general transformation T = ˆ H ϕ R (with det T = 1 ),where the matrix ˆ H ϕ ˆ H ϕ = (cid:18) / √ Λ
00 1 / √ Λ (cid:19) (cid:18) cosh ϕ sinh ϕ sinh ϕ cosh ϕ (cid:19) (cid:18) √ Λ √ Λ (cid:19) (7)leaves ˆΛ invariant: ˆ H Tϕ ˆΛ ˆ H ϕ = ˆΛ . (8)As one can see, the ˆ H ϕ matrix is essentially proportionalto the matrix of hyperbolic rotations, which commuteswith the diag (1 , − matrix which arises when the twoeigenvalues of ˆ g have opposite sign. The relation (8) holdsregardless the value of the parameter ϕ , which will bechosen to decouple the two SC channels, see Eq. (22)below. Indeed, thanks to Eq. (8) ˆ g can be diagonalizedby T as well: T = ˆ H ϕ R ⇒ ˆ g = T T ˆΛ T, ˆΛ − = T ˆ g − T T . (9)Thus, if we introduce the new combinations of fermionicfields: (cid:18) ψ ψ (cid:19) = T (cid:18) φ φ (cid:19) , (10) H int can be rewritten as: H int = − X q ,lm ˆ g lm φ † l, q φ m, q == − X q (cid:16) Λ ψ † , q ψ , q − Λ ψ † , q ψ , q (cid:17) . (11)Once defined the new combinations of fermionic fields ψ i we can use the HS decoupling (5) to write the followingpartition function: Z = Z D c lσ D c l † σ D h i D h † i e − S ,S = S + X q | h ,q | Λ + | h ,q | Λ − X q (cid:16) h ∗ ,q ψ ,q + h.c. (cid:17) − i (cid:16) h ∗ ,q ψ ,q + h.c. (cid:17) , (12)where q ≡ ( i Ω m , q ) . The action (12) is now quadratic inthe fermionic fields, that can be integrated out exactly.By introducing the Nambu operators N † l,k = ( c l † k, ↑ , c l − k, ↓ ) we can indeed rewrite the action as: S = X lk,k ′ N † l,k h − ¯ G − k,l δ k,k ′ + Σ lk,k ′ i N l,k ′ ++ X q | h ,q | Λ + | h ,q | Λ , (13)where: ¯ G − k,l = (cid:18) iω n − ξ k,l T l ¯ h + iT l ¯ h T l ¯ h ∗ + iT l ¯ h ∗ iω n + ξ k,l (cid:19) , (14) Σ lq = k − k ′ = r TV (cid:18) T l h ,q + iT l h ,q T l h ∗ ,q + iT l h ∗ ,q (cid:19) . (15)In Eq. (14) we put ¯ h i = p T /V h i, . By integrating outthe fermions one gets as usual a contribution to the ac-tion equal to − ln det ( ¯ G − − Σ) = − T r ln( ¯ G − − Σ) = − T r ln ¯ G − − T r ln(1 − ¯ G Σ) , where the trace acts bothon momentum and Nambu space. One can then separatethe mean-field action from the fluctuating part as: S = S MF + S F L (16) S MF = ¯ h Λ + ¯ h Λ − X l T r ln ¯ G − l (17) S F L = X q | h ,q | Λ + | h ,q | Λ + X l X n T r ( ¯ G l Σ) n n (18)From Eq. (14) one can see that the HS fields play the roleof the SC gaps in each band, provided that one assumesa saddle-point value of the antibonding field such that ¯ h = iA, ¯ h ∗ = iA, (19)to guarantee the Hermitian form of the saddle-point ac-tion. Here we will use instead the generalized transfor-mation (10) to impose ¯ h = ¯ h ∗ = 0 at the saddle point.This can be understood by minimizing the mean-field ac-tion (17), which gives the set of equations: (cid:18) − P l T l Π l − P l T l Π l T l − P l T l Π l T l − − P l T l Π l (cid:19) (cid:18) ¯ h i ¯ h (cid:19) = 0 , (20) where we defined the Cooper bubble Π l as: Π l = TV X k ,n ω n + E k ,l , (21)with the identification E k ,l = ξ k ,l + (cid:0) T l ¯ h ∗ ¯ h − T l ¯ h ∗ ¯ h + 2 iT l T l (¯ h ∗ ¯ h + ¯ h ¯ h ∗ ) (cid:1) . Onceagain this quantity cannot be identified with the energyof the quasiparticles in each band, unless we use Eq. (19).On the other hand, we can choose the ϕ parameter ofthe transformation (9) to decouple the two saddle-pointequations (20): X l T l Π l T l = 0 . (22)In this case one immediately sees that since Π l > theequation for ¯ h can only be satisfied for ¯ h = 0 , so thatthe SC transition is only controlled by the bonding field ¯ h , whose self-consistent equation is " − X l T l Π l ¯ h = 0 , (23)where we also assumed that ¯ h is real. This choice cor-responds to the gauge where both gaps are real, as givenby (see Eq. (14)): ∆ l = T l ¯ h . (24)We stress once more that even if the saddle-point valueof the antibonding HS field h vanishes both gaps are ingeneral different from zero, and their relative strenght ortemperature dependence is controlled by the microscopiccouplings via the elements of the T matrix. The possi-bility to describe the SC state as a function of a singleorder parameter reflects the fact that at T c only one SCchannel becomes active. To make the connection witha more standard notation, we observe that the matrix T in practice diagonalizes the multiband self-consistencyequation, that is usually written as: (ˆ g − − ˆΠ) ~ ∆ = 0 , (25)where ˆΠ ij = δ ij Π i and ~ ∆ is a vector formed by the gaps ∆ l in each band. The above equation admits a non-zerosolution ~ ∆ when the determinant vanishes, i.e. when (atleast) one eigenvalue is zero. By means of the relations(9) above, we see that the set of equations (20) and (22)corresponds to put the matrix ( ˆΠ − ˆ g − ) in diagonal form ˆΛ − − T ˆΠ T T = T (ˆ g − − ˆΠ) T T ≡ Y i δ ij (26)so that the SC state is reached when the element Y ≡ / Λ − P l T l Π l corresponding to the bonding eigenvaluevanishes, leading to Eq. (23) above. It should be noticedthat in the two-band case, regardless the intra-band orinter-band dominated nature of the pairing, the eigen-values of the matrix (25) cannot be both zero, unless theinterband coupling vanishes (see discussion in AppendixC). On the other hand, in the three-band case discussedin Sec. III below the matrix (26) has three eigenvalues:as we shall see, when only one of them vanishes one is inthe usual SC phase, while the vanishing of a second eigen-value signals the emergence of a TRSB phase. Finally, wenotice also that the procedure introduced here to describea multiband superconductor in terms of a single orderingfield can be applied also to the case of spatially inhomo-geneous superconducting condensates, whose Ginzburg-Landau expansion near T c has attracted some interest inthe recent literature. B. Collective modes in the SC state
Within the present formalism the collective modes inthe SC state can be easily obtained by expanding theaction (18) up to second order in the HS fields. In thesingle-band case, where a single HS field is used to de-couple the interaction, one can follow two alternative butequivalent roots. Indeed, as relevant variables one canuse either (i) the amplitude and the phase (polar coordi-nates) or (ii) the real and imaginary part (cartesian coor-dinates) of the HS field. In Appendix A we show how torecover the equivalence between the two approaches. Inour case, where a single HS field condenses at the tran-sition, the second root is the only available one. On theother hand, when the interaction has a dominant intra-band character one does not need to use the transfor-mation (10) to get rid of the antibonding field, and onecan introduce HF fields associated directly to the twogaps in each band. In this case the approach (i) can beagain used, as it has been done for example to study theLeggett mode in MgB in Ref. [20], and more recently toinvestigate Leggett modes across a TRSB transition forintraband-dominating interactions in Refs. [13,15]. How-ever, this is not the case physically relevant for pnictides,as we discussed in the introduction.From Eq. (18) one can see that the coefficients of theGaussian action for the HS field will be given in generalby BCS correlation functions computed with the mean-field Green’s functions (14), with the identification (24) ofthe band gaps. Following the straightforward proceduredescribed in Appendix A one then finds that S F L = X q η T − q ˆ S F L ( q ) η q , (27) η Tq = ( Reh ,q , iReh ,q , Imh ,q , iImh q ) . (28)Notice that once fixed ¯ h as real, see Eq. (24), one canidentify the real and imaginary parts of the h fluctua-tions as the leading orders in the amplitude and phasefluctuations of the field, respectively: Reh ,q = | h ,q | , Imh ,q = ¯ h θ ,q . (29)While the same identification cannot be done for the h field, we can still associates its fluctuations to the real and imaginary parts of the gap fluctuations in each band.Indeed, by following the same root described in AppendixA to derive the relation between the averages of the HSfields and the averages of the physical fermionic operators(10), one can show that: h ψ ,q i = 1Λ h h ,q i , h ψ ,q i = − h ih ,q i , (30) h ψ ∗ ,q ψ , − q i = 1Λ (cid:0) h h ∗ ,q h , − q i − Λ (cid:1) , (31) h ψ ∗ ,q ψ , − q i = 1Λ (cid:0) h ih ∗ ,q ih , − q i + Λ (cid:1) . (32)Since the gap operators in each bands are given by ∆ l = g lm φ m = T Tlm ˆΛ mn ψ n one can also express the averagevalues of the gap fluctuations in terms of fluctuations ofthe h i HS fields as: h ∆ l,q + ∆ ∗ l,q i = T l h Reh ,q i + T l h iReh ,q i , (33) h ∆ l,q − ∆ ∗ l,q i = T l h Imh ,q i + iT l h iImh ,q i , (34)and analogous expressions for the correlations functions.As a consequence, we included the imaginary unit in the h components of the fluctuating vector (28) and we willrefer in what follows to the first two components of η q as“amplitude” fluctuations and to the last two as “phase”fluctuations. Such a decomposition allows one also to eas-ily identify the character of the fermionic bubbles whichappear in the Gaussian action. Indeed, from Eq. (15)one sees that amplitude fluctuations are associated toa σ Pauli matrix in the Nambu notation, while phasefluctuations to σ (see also Eq. (A3) in Appendix A).Moreover, as shown in Appendix A, at long-wavelengththe amplitude and phase sectors decouple, so that theyare described respectively by the following × matrices: ˆ S AF L ( q ) = T ˆΛ T T / − == (cid:18) P l Λ l ( q ) T l + P l Λ l ( q ) T l T l P l Λ l ( q ) T l T l P l Λ l ( q ) T l − (cid:19) (35) ˆ S PF L ( q ) = T ˆΛ T T / − == (cid:18) P l Λ l ( q ) T l + P l Λ l ( q ) T l T l P l Λ l ( q ) T l T l P l Λ l ( q ) T l − (cid:19) (36)where the Λ ijl bubbles are defined in the Appendix A andthe corresponding diagonal matrices are ˆΛ iilm ≡ Λ iil δ lm .In what follows we shall investigate the possibility thatany collective mode is defined in the two sectors, by hav-ing in mind that a mode corresponds to a solution of theequation det ˆ S F L ( ω = m, q = 0) = 0 with m < min ,where ∆ min is the smallest gap. In practice we are in-terested in well-defined resonances below the thresholdof the quasiparticle excitations: thus it is enough to takeinto account the real part of the bubbles Λ l , Λ l afteranalytical continuation i Ω m → ω + iδ to real frequencies,since the imaginary parts vanish at ω < min . Thedifferent behavior of the collective modes will then fol-lows simply from the different frequency and momentumdependence of these two bubbles, whose value at q = 0 is connected to the Cooper bubble (21). Moreover, as itis shown in the Appendix A (Eq. (A29)), at small q andlow T one can write: Λ l ( q ) = − l + A l ∆ l + O ( q ) (37) Λ l ( q ) = − l + 14∆ l (cid:18) Ω m κ l + q ρ s,l m l (cid:19) (38)where κ, ρ s /m represent the compressibility and su-perfluid density of each band, respectively, and A l = P k tanh( βE k / /E k (see Eq. (43) below).By using Eq. (38) one can write down the q = 0 limitof the phase sector (36) as: ˆ S PF L ( q = 0) = (cid:18) − P l Π l T l + − P l Π l T l − (cid:19) (39)where we used the constraint (22) for the T matrix tocancel out the off-diagonal terms at q = 0 . Eq. (39) isone of the first crucial results of the use of the general-ized transformation T : indeed, not only it decouples thesaddle-point equations, but it also decouples the phasefluctuations at long wavelengths, connecting their massesto the eigenvalues of the saddle-point equations them-selves, leading to a straightforward interpretation of theroles of the HS fields. Indeed, since below T c ¯ h = 0 ,the self-consistent equation (23) implies that the quan-tity in square brackets vanishes, so that one immediatelysees that Imh ,q fluctuations describe a massless mode.This is not surprising, since from Eq.s (24), (29) and (34)one sees that a phase fluctuation for the ordering h fieldcorresponds to a simultaneous change of the overall SCphase in all the gaps: ∆ l + iT l Imh = T l ¯ h + iT l ¯ h θ ≃ ∆ l e iθ . (40)As a consequence, Imh is the Goldstone mode of theSC transition, that is expected to be massless in the SCphase. For what concerns instead the fluctuations of theantibonding field h we can first analyze the small fre-quency expansion of Eq. (36) that follows from Eq. (38),i.e.: ˆ S PF L = − ω P l κ l T l l ω P l κ l T l T l l ω P l κ l T l T l l ω P l κ l T l l + (cid:16)P l Π l T l + (cid:17) , (41) where the analytic continuation i Ω m → ω + iδ has beenmade. As ω → one sees that Imh and Imh decouple,and one recovers the massless Imh mode, as discussedabove. On the other hand, the fluctuations of the anti-bonding h field do not give rise to any collective mode.Indeed, the element of the matrix (41) does not admitany real solution for ω , due to the fact that the quan-tity in brackets is strictly positive. This result, which is confirmed by the explicit calculation of Λ l ( ω ) at allfrequencies and temperatures, is a direct consequence ofthe fact that the h field is associated to the antibondingSC channel of the system. Indeed, as we show in detailsin the Appendix B, if h were associated to a bondingSC channel (i.e. a positive eigenvalue in Eq. (6)), the − / Λ term in Eq. (36) would be replaced by +1 / Λ ,leading to a well-defined mode in Eq. (41), that coin-cides with the usual Leggett mode, see Eq. (B12). It isalso worth stressing that the absence of the Leggett modein a two-band modelization of pnictides does not meanthat relative phase fluctuations of the gaps in the twobands are absent, but simply that these fluctuations donot define a coherent collective mode of the system.For what concerns instead the amplitude sector (35),by using again the self-consistent equations (22)-(23) andthe relation (37) one sees that at q = 0 in general ˆ S AF L = (cid:18) A + O ( ω ) B + O ( ω ) B + O ( ω ) − C + O ( ω ) (cid:19) (42)where A = P l A l ∆ l T l , B = P l A l ∆ l T l T l , C = P l C l T l + are positive constants, with C l = P k ( ξ k /E k ) tanh( βE k / . As one could expect, thereis no massless mode in the amplitude sector, since am-plitude fluctuations are always costly in the SC phase.One could then wonder if massive modes are present.In the single-band case one knows that amplitude fluc-tuations at q = 0 correspond to a well-defined modewith frequency m = 2∆ , which get easily damped byinteractions . This result follow from the fact thatthe coefficient of the amplitude fluctuations reduces (seeEq. (A5)) to ( g being the SC coupling) Λ ( ω, q = 0) + 2 g == X k tanh (cid:18) E k T (cid:19) (cid:20) − ξ k E k (cid:18) E k + ω + 12 E k − ω (cid:19) + 1 E k (cid:21) . (43)This function of ω vanishes at ω = 2∆ with a square-rootsingularity, and it is positive everywhere else, see Fig.1a. In the multiband case described by Eq. (36) aboveone is then mixing the Λ l bubbles of the two bands,which have in general zeros for two different values l .For this reason, unless one considers strictly identicalbands, the det ˆ S PF L ( ω, q = 0) never vanishes, as shownin Fig. 1b, so that well-defined amplitude modes are ab-sent. This example shows also that one should be verycareful in computing the collective modes by making alow-energy expansion of the Λ l ≃ − l + A l ∆ − B l ω bubbles. Indeed, one could obtain either spurious re-sults or masses which are quantitatively wrong, especiallyin the TRSB phase where amplitude and phase fluctua-tions get mixed.
We will come back to this pointat the end of the next Section.Finally, we observe that above T c the phase and am-plitude sectors become degenerate, as expected, and one ω/2∆ -1012 / g + Λ ( e V - ) ω (meV) -5-4-3-2-10 d e t S A F L ( ω , q = ) ( e V - ) (a) (b) FIG. 1: (color online) Left panel: frequency dependence of theamplitude fluctuations as given by Eq. (43) in the single-bandcase. The vanishing at ω = 2∆ signals the presence of an am-plitude mode with mass m = 2∆ . Notice that the quadraticlow-frequency expansion, given by the dashed line, would leadto a wrong mass m ≃ √ . Right panel: determinant of theaction (35) at T = 0 in the amplitude sector as a function of ω for coupling values N = 1 eV − , N = 2 . eV − , g = 0 . eV, ω = 15 meV. The determinant never vanishes, so that nowell-defined mode is found in this case. The overall negativesign is due to the presence of the antibonding channel, seealso Eq. (42). recovers the results discussed in Ref. [29]. Indeed, the Λ l and Λ l bubbles coincide, and the leading terms atsmall q go like η q , γ | Ω m | . More specifically, we observethat at q = 0 the action for the Gaussian fluctuationscoincides with the usual quadratic expansion of the freeenergy, and it s given by: S F L ( q = 0) = " − X l T l Π l | h | ++ " + X l T l Π l | h | . (44)As one can see the coefficient of the h field is always pos-itive , showing that it never orders. In contrast, a wrongapplication of the HS transformation (5) lead the au-thors of Refs. [17,30] to the counterintuitive result thatthe coefficient of the antibonding field is always negative ,making it difficult to justify why it should not order.This shows once more that an extra care is needed toextend to interband-dominated interactions the resultsknown for single-band systems, where a single bondingSC channel exists. III. THREE-BAND MODEL FOR THE TRSBTRANSITIONA. Occurrence of a TRSB state in pnictides
Once established the general properties of the collec-tive modes in a superconductor where bonding and an-
FIG. 2: (color online) Schematic of the band structuresin pnictides in the unfolded Brollouin zone. (a) Typicalband structure for optimally-doped 122 compounds (like e.g.Ba . K . Fe As ), formed by two hole pockets around Γ andtwo electron ones at (0 , π ) and ( π, . In this case the largestcoupling is an interband repulsion between the hole and elec-tron Fermi sheets, leading to the s ± symmetry of the orderparameter with a sign change of the gap between hole andelectron bands. (b) Strong hole doping: in this case the elec-tron pockets reduce considerably and a third hole pocket ap-pears at ( π, π ) . In the case of KFe As (c) the electron pock-ets disappear completely. It has been argued that s -wave (b)and d -wave (c) symmetries are nearby in energy at stronghole doping. In the s -wave symmetry the change of sign ofthe gaps occurs between the two hole pockets at Γ , while onthe remaining bands the order parameter is very small. In the d -wave symmetry (c) instead the largest gap is on the thirdhole pocket and nodes are present on all the Fermi surfaces. tibonding SC channels coexist, let us now focus morespecifically on the case of a three-band model for pnic-tides, where an additional repulsion between the twohole bands is considered. This case has attracted con-siderable interest in the recent literature due to the ex-perimental advances in making 122 samples heavilyhole-doped away from half-filling, until the end memberKFe As is reached. Even though a full agreementbetween theoretical predictions and experimental resultshas not been reached yet, we would like to summarizehere some results relevant for the focus of the presentmanuscript. A schematic of the band-structure evolutionfrom Ba − x K x Fe As to KFe As in the unfolded Bril-louin zone (one Fe atom per unit cell) is shown in Fig.2. At intermediate doping (Fig. 2a) the system admitstwo hole pockets at Γ = (0 , and two electron pock-ets at ( π, and (0 , π ) . The largest interactions in thissituation are the spin-fluctuations mediated inter-pocketrepulsions between hole and electron bands, that leadto the s ± symmetry of the order parameter, i.e. constantgaps on all the FS with a change of sign between hole andelectron bands. In this situation, by neglecting nematiceffects making the electron pockets inequivalent, an ef-fective two-band description as the one discussed in theprevious section is possible. As doping increases theelectron pockets shrink and a third hole pocket around ( π, π ) appears (Fig. 2b), until that only hole pockets re-main for KFe As (Fig. 2c). In this compound severaltheoretical calculations have shown that s -waveand d -wave symmetry are almost degenerate in energy.However, the gap hierarchy would be very different in thetwo cases: in the s -wave case the leading interaction isan interband repulsion at small momentum between thehole pockets at Γ , so that the sign change between thegaps is now realized between the two central hole bands(having eventually accidental nodes ), while on the re-maining pockets the gap is vanishing. Instead the d -wavesymmetry is driven by a large intraband repulsion withinthe hole pocket at ( π, π ) , so that the gap is largest hereand nodes are present on all the FS. The experimentalsituation is quite controversial: while ARPES measure-ments show no nodes at large ( x = 0 . ) K doping or accidental nodes for KFe As , thermal probes of thequasiparticle excitations indicate nodal gaps. From the point of view of the general description ofthe collective modes that we will give here, the relevantaspect is that once that two SC channels are almost de-generate in energy one can eventually access a phasewhere both of them coexist, leading to a TRSB state.To slightly simplify the notation and to make contactwith previous work on this topic we will discusshere the case where the order parameter remains in the s -wave symmetry class, so that the most relevant interac-tions are interband repulsion between hole and electronbands, and within the hole pockets at Γ . By assumingagain degenerate electron pockets one can then inves-tigate for example the minimal three-band model pro-posed in Ref.[24] where the two hole bands (bands 1,2)are equal, so that the matrix ˆ g of Eq. (3) becomes forthis three-band case: ˆ g = − V hh V he V hh V he V he V he (45) λ/η T ( K ) η =0.5 c T TRSB (a)(b) (c)
FIG. 3: (color online) Phase diagram of the model (45) ob-tained by numerical solution of the mean-field equations (25).Here we used as bosonic scale ω = 15 meV. Notice that the T TRSB line ends at a finite value of λ/η for λ > η , while for λ < η T
TRSB is in principle always finite but it is exponen-tially suppressed as one moves to small λ values. As it has been noticed in Ref. [24], despite the fact thatthe mean-field equations in this three-band model appearas a straightforward generalization of the two-band casediscussed in the previous Section, the intrinsic frustration hidden in the SC model (45) leads to the appearance ofa qualitatively new effect, i.e. the possible emergence ofa s + is state which breaks time-reversal symmetry. In-deed, in the model (45) each band would like to have agap of opposite sign with respect to the gap in the otherbands: when three gaps compete one then realizes a situ-ation analogous to the antiferromagnet in the triangularlattice, where spins orients themselves at relative π/ angles. In the SC problem the frustration occurs in therange of parameters (i.e. interactions and/or tempera-ture) where two eigenvalues of the matrix ˆΠ − ˆ g − vanish(see Sec. III-B and Appendix C), allowing for an intrin-sically complex SC order parameter. Since in this case ∆ ∗ l = ∆ l time reversal (which corresponds to complexconjugation) is spontanously broken and the system is ina s + is TRSB state. To give a general idea of the rangeof parameters for the TRSB phase we show in Fig. 3 thephase diagram of the specific model (45). Here we as-sumed for simplicity that the DOS N l ≡ N in all bandsare equal, so that we can introduce the dimensionlesscouplings η = V hh N, λ = V he N. (46)As it has been discussed previously , the T T RSB sep-arating the normal superconductor from the TRSB stateends at a finite value λ cr ( η ) > η , while for λ < η the TRSB phase is always present at T = 0 , but the T T RSB is exponentially suppressed. By using η = 0 . and ω = 15 meV for the BCS bosonic scale in the Π l bubbles, as roughly appropriate for pnictides, one ob-tains λ cr ≃ . , leading to a reasonable wide range ofparameters where the TRSB transition can occur. Eventhough these numbers have to be considered only indica-tive for real materials, due to the simplifications of themodel (45) and the overestimation of the critical temper-atures in mean-field like calculations, for a specific sampleone could indeed observe one of the thermal transitionsmarked by vertical lines in Fig. 3.As we discussed above, several other possibilities ex-ist for an intermediate TRSB state in pnictides, depend-ing on the nature of the competing SC channels, thatreflects on the structure of the matrix (45) and on itseigenvalues. For example, to account for a possible d -wave symmetry in KFe As one should include a thirdhole pockets with a large intraband-repulsion term. Onthe other hand, the same matrix structure (45) but witha different identification of the bands can be used todescribe the s + id state that has been proposed for theelectron-doped pnictides. In this case, by addingto the schematic structure of Fig. 2a an interband re-pulsion between the electron pockets one could in-duce a SC state with a sign change of the gap betweenthe (0 , π ) and ( π, pockets, that corresponds to d -wavesymmetry, even if without nodes on the FSs. As we shallsee below, our approach allows us to establish a gen-eral correspondence between the structure of mean-fieldequations (25) and the evolution of the collective modes,providing thus a general scheme to test experimentallywhether or not a TRSB state is realized, regardless thespecific symmetry of the two degenerate SC channels ac-tive in the TRSB phase. For this reason, while previouswork has focused on the T = 0 behavior of the collectivemodes as a function of the tuning parameter (i.e. thedoping) for the quantum TRSB transition, here we focuson the possibility to identify the occurrence of a thermal
TRSB transition. Indeed, while the quantum phase tran-sition between the TRS and TRSB state has in generalonly two end points, the thermal transition occurs in amuch wider range of parameters, making eventually itsidentification a more accessible experimental task.
B. Collective mode across the TRSB transition
To extend the collective-modes derivation of Sec. IIwe will start by considering the most general three-bandmodel which admits a TRSB state and has one antibond-ing SC channel. Thus, ˆ g must have two positive and onenegative eigenvalue, so that after the rotation R the di-agonal matrix ˆΛ is: ˆ g = R − Λ
00 0 − Λ R, (47)where we also set by definition Λ ≥ Λ . For example, inthe simplified model (45) one has that for η > λ is Λ = η , Λ = ( p η + 8 λ − η ) / and Λ = ( p η + 8 λ + η ) / ,while for η < λ the role of Λ and Λ is interchanged.The derivation of the effective action is then a straight-forward generalization of the procedure used in Sec. II.In particular, also in this case one has to introduce a HSfield h associated with the repulsive channel Λ , and onecan take advantage of the generalized transformation T (depending now on three parameters, see Appendix C)to impose ¯ h = 0 at the saddle point. Indeed, the equiv-alent of the saddle-point Eq.s (20) can be made diagonalagain by using the three conditions which generalize Eq.(22), i.e. X l T il Π l T j = i,l = 0 , i, j = 1 , , , (48)so that one is left with − P l T l Π l − P l T l Π l
00 0 − − P l T l Π l ¯ h ¯ h i ¯ h ! == Y ¯ h + Y ¯ h + iY ¯ h = 0 , (49)where the Y i are the eigenvalues of the matrix ˆ g − − ˆΠ which enters the usual mean-field equations (25). Asone can see, in full analogy with the two-band case (20)above, the coefficient Y which multiplies the antibondingHS field ¯ h is always positive, so that one imposes ¯ h = 0 at the saddle point. The remaining two coefficients Y , Y can both in principle vanishing, leading to finite saddle-point values of the corresponding HS fields.Let us discuss the thermal evolution equivalent to oneof the paths (a),(b) in Fig. 3, starting from the non-SCstate. As T decreases and the Cooper bubbles increasethe first coefficient which vanish at T c in Eq. (49) is forexample Y . Then h is the first HS field which orders.Its phase can be chosen real ¯ h = R real, so all the gapsare given by Eq. (24) and are real. As the temperaturedecreases further, according to the range of parametersof the matrix g , it is possible that at T = T T RSB also Y ( T T RSB ) vanishes, Y ( T T RSB ) = X l T l ( T T RSB )Π l ( T T RSB ) − / Λ = 0 . (50)In this case, as we discuss in the Appendix C, one can alsoshow that at lower temperatures the imaginary part of ¯ h acquires a finite saddle-point value. More specifically, onecan always choose a gauge where ¯ h is purely imaginary,i.e. ¯ h = iI . As a consequence, the mean-field gaps at T < T
T RSB are given by ∆ l = T l ¯ h + T l ¯ h = T l R + iT l I = | ∆ l | e i ¯ ϑ l (51)so that they are intrinsically complex and a TRSB state isreached. Moreover, the additional Z symmetry betweenthe two possible time-reversal-symmetry breaking groundstates (51) is encoded in the complex conjugation for the ¯ h field, that leads to a change of sign of all the phases ϑ l without changing the ground-state energy.The emergence of a finite imaginary part of ¯ h belowTRSB has a precursor effect on the behavior of the col-lective phase modes above T T RSB . Indeed, in the TRSphase where all the gaps have trivial phases one canobtain a straightforward extension of Eq.s (35)-(36) forthe amplitude and phase fluctuations of the HS fields.In particular, by using again the constraints (48) forthe T transformation, the equivalent of Eq. (39) for thephase sector η Tq = ( Imh ,q , Imh ,q , iImh ,q ) in the long-wavelength q ≃ limit can be written as: ˆ S PF L ( q = 0) == − P l Π l T l + − P l Π l T l +
00 0 − P l Π l T l − (52)Eq. (52) is one of the central results of our paper. Indeed,it establishes a direct correspondence between the massesof the phase modes and the saddle-point equations (49),showing that as soon as one reaches the TRSB state,defined by Eq. (50), the fluctuations of the Imh HSfield become massless . It must be emphasized that thethis result holds regardless the structure of the couplingmatrix. Indeed, one can prove (Appendix C) that neces-sary and sufficient condition to have gaps with non-trivialphases is that two eigenvalues of the matrix ˆ g − − ˆΠ , i.e.0two Y i coefficients in the diagonal form (49), must van-ish. Since the T transformation decouples also the phasemodes and connects their masses at T ≥ T T RSB to the Y i coefficients, it makes possible to show in full gener-ality that at the boundary between a TRS and TRSBphase one additional phase mode becomes massless. Byconsidering then the phase diagram of Fig. (3), such amassless mode emerges along all the line T T RSB , as wellas for isothermal transitions as a function of thecoupling parameters for the matrix ˆ g , like path (c). Inthis case the TRSB state would be equally determinedby the condition Y = 0 , considering Y a function e.g.of the SC coupling λ : Y ( λ T RSB ) = X l T l ( λ T RSB )Π l ( λ T RSB ) − / Λ = 0 (53)It is worth stressing that our derivation shows also thatin the three-band case only one additional mode (otherthan the Bogoliubov-Anderson Goldstone mode) can bemassless at the TRSB transition. Indeed, for interband-dominated coupling the fluctuations of the antibonding h field in Eq. (52) do not identify a mode, as explainedin Sec. II. On the other hand, if also the third eigen-value Λ of the matrix (47) were positive, the associ-ated h fluctuations would describe a Leggett-like modethat cannot become massless, since at least one eigen-value of the decomposition (49) must be finite (see Ap-pendix C). The possibility to establish these results ongeneral grounds is crucial to identify the total numberof massless modes a-priori. Indeed, an explicit numericalcalculations of the collective modes, done e.g. by usingthe low-frequency expansion of the bubbles, becomesvery delicate when one of the gap vanishes, as we shalldiscuss in more details in the next Section.Below T T RSB the behavior of the collective modes ismore complex, due to the mixing between amplitude andphase fluctuations.
Indeed, when the SC gaps ∆ l in each band are complex numbers the fermionic bubbleswhich appears in Eq.s (35)-(36) acquire an explicit de-pendence on the saddle-point values ¯ ϑ l of the phases ofthe SC order parameters. More specifically one has that Λ l ( q ) = ¯Λ l ( q ) + 2 cos ¯ ϑ l F l ( q ) , (54) Λ l ( q ) = ¯Λ l ( q ) + 2 sin ¯ ϑ l F l ( q ) , (55) Λ l ( q ) = 2 sin ¯ ϑ l cos ¯ ϑ l F ( q ) + O ( q ) , (56)where ¯Λ l is a function of | ∆ l | , so it coincides with theexpression (A7) of the Λ bubble computed assuming areal gap, and F l ( q ) = 2 | ∆ l | TV X k ,n m + ω n ) + E k + q ω n + E k (57)is also a function only of the gap amplitude | ∆ l | . Whenthe gaps have trivial phases ϑ l = 0 , π these definitionscoincide with the ones given in Appendix A and one re-covers the expansion (37)-(38) used above. Below T T RSB the most important difference is that the bubbles Λ l which appear in the coupling between the amplitude andphase sectors (see Eq. (A5)) cannot be neglected, makingthe structure of the Gaussian fluctuations (27) consider-ably more complicated. In this situation the structure ofthe collective modes is not simplified by the use of thetransformation T . Thus, in order to simplify the numer-ical computation, we will take advantage of the fact thatthanks to Eq.s (35)-(36) the overall action for mixed am-plitude and phase fluctuations is a six times six matrixgiven by: ˆ S F L = 12 (cid:18) T ˆΛ T T + 2 ˆΛ − T ˆΛ T T T ˆΛ T T T ˆΛ T T + 2 ˆΛ − (cid:19) == 12 ˆ T (cid:18) Λ + 2ˆ g − ˆΛ ˆΛ ˆΛ + 2ˆ g − (cid:19) ˆ T T ≡ ˆ T ˆ M ˆ T T (58)where we used the property (9) that ˆΛ − = T g − T T andwe defined ˆ T as a 6 × × T on the diagonal. Since det ˆ T = 1 the collec-tive modes will be given by the solutions of the equation det ˆ M = 0 . It is worth noting that the correspondingeigenvectors can be associated to amplitude and phasefluctuations in the various bands: indeed, the relations(33)-(34) between the fermionic operators and the HSfields will read in this case: h Re ∆ l,q i = T l h Reh ,q i + T l h Reh ,q i + T l h iReh ,q ) i (59) h Im ∆ l,q i = T l h Imh ,q i + T l h Imh ,q i + T l h iImh ,q i (60)which correspond in a short notation to e.g. h Re ∆ i = T T h Reh i , with the usual inclusion of the imaginary unitin the fluctuations of the antibonding field h . Thus itis not surprising that the ˆ M matrix coincides with thederivation done in Refs. [16,17] by means of linear re-sponse theory in the band basis. In addition, in the caseof dominant intraband pairing, where no imaginary unitis associated to the HS fields, the relations (59)-(60) canbe used to define new bosonic variables. In this case,when all the gaps are opened so that Im ∆ l,q = ∆ l ϑ l,q ,by means of the identity (A29) one recovers for the phasesector the same structure reported in Ref. [15]. Noticealso that the coupling between fluctuations in differentbands is provided by the inverse matrix ˆ g − of the SCcouplings, while the coupling between the amplitude andphase sector is diagonal in the band index and it is givenby the Λ l bubbles of Eq. (56), which are proprtionalto the sin ¯ ϑ l , so that they differ from zero only in theTRSB state. This result s very general, and indeed itcan be found also within the phenomenological multibandGinzburg-Landau approach of Ref. [14], where the inter-band couplings are provided by Josephson-like terms. C. Temperature and coupling dependence of theLeggett mode
To show explicitly the temperature evolution of theLeggett modes we will refer for simplicity to the set of1 -0.500.5 Re ∆ Re ∆ Re ∆ Im ∆ Im ∆ Im ∆ T (K) -0.6-0.300.30.6 Re ∆ Re ∆ Re ∆ Im ∆ Im ∆ Im ∆ ∆ ( m e V ) , φ / π (r ad ) T (K) ω ( m e V ) L1L2 min | ∆ | | ∆ |φ /πφ /π (a) (b)(c) L1 L2 FIG. 4: (color online) Temperature evolution of the low-energy modes along the path (a) of Fig. 3, correspondingto λ = 1 . η . Here T c = 35 . K and T TRSB = 23
K. (a):Temperature dependence of the low-energy mode along withthe minimum gap threshold, obtained by the temperature de-pendence of the gaps reported in the inset along with thephases of the hole gaps. (b) and (c): components of theeigenvectors corresponding to the modes labeled as L1 andL2 in panel (a). (d): schematic structure of the modes belowand above T TRSB . Here big full arrows denote the equilib-rium gaps while the thin arrows denote the gaps includingthe fluctuations, identified by the big empty arrows. As onecan see, L1 evolves in the ordinary Legget mode for the orderparameters in the two hole bands, while L1 evolves towardsan amplitude mode. coupling constants defined by eq. (45), which gives riseto the phase diagram shown in Fig. 3. As one can see,while Eq. (58) does not allow for a simple identificationof the number and nature of the collective modes, it sim-plifies the numerical evaluation of the modes since onedoes not need to determine also the T matrix. We thensolved self-consistently the gap equations and computedthe matrix ˆ M in Eq. (58), looking for well-defined modesbelow the threshold min provided by the smallest gapin the problem. We assume conventionally that the gapin the electron band ∆ is real and positive, while thegaps in the hole bands are ∆ = ∆ e iφ , ∆ = ∆ e iφ . Ac-cording to the phase diagram of Fig. 3, the three phases correspond respectively to: − SC : ∆ = 0 , φ = φ = π (61) TRSB − SC : ∆ = 0 , φ = φ, φ = − φ (62) − SC : ∆ = 0 , φ = π/ , φ = − π/ (63)Let us start from the path labeled by (a) in Fig. 3, seeFig. 4. Here we identify a mode L1 which softens atthe T T RSB and remains always below the gap threshold.Above T T RSB
L1 is an ordinary Leggett mode associ-ated to the phase fluctuations in the two hole bands.Indeed, in this state the pairing in each hole band isprovided by the interband coupling to the third electronband. Thus, within the hole-bands sector the problemis formally equivalent to a two-gaps superconductor withdominant intraband pairing, and the Leggett mode is welldefined. Below the T T RSB the SC order parameter in thehole bands becomes complex, so that the Leggett-like os-cillation drives also amplitude fluctuations both in thehole and electron bands. Observe that below T T RSB asecond low-energy mode appears, labeled L2 in Fig. 4,which is only slightly below the gap threshold. Indeed,at
T > T
T RSB this mode coincides with pure amplitudefluctuations in the two hole bands, and thus it appearsright at the gap edge , ≡ min . However, as onemoves at higher λ/η values or one makes the two holepockets inequivalent this mode approaches rapidly thegap edge, becoming then overdamped. On the other handat the full symmetric point λ = η L1 is exactly degener-ate with the L2 mode. Indeed, at the λ = η point thethree bands are completely equivalent, and the L1 andL2 describe the same oscillation: the gaps in two bandsapproach each other, inducing a change of modulus ofthe third gap. When one moves in the regime λ < η (path (b) inFig. 3) the role of the two modes in the TRSB statechanges and L2 becomes softer. More interestingly, at
T > T
T RSB the situation is completely different in thiscase, since no soft mode can be found. This result can beeasily understood: at
T > T
T RSB the gap in the electronband closes and the system is formally equivalent to atwo-band superconductor with dominant interband cou-pling. This is the situation discussed in Sec. II, whereno Leggett-like mode is present since only one bondingSC channel exists. By close inspection of the eigenvectorcomponents in Fig. 5b,c,d one sees that as T → T − T RSB the L2 mode tends to the Goldstone mode while the L1mode would coincide to the ordinary Leggett oscillation,which does not identify a mode above T T RSB for thereason explained above. We then recover the same resultdiscussed below Eq. (52) in the language of the h , fields,i.e. that exactly at T = T T RSB there are two solutions at ω = 0 . However, while the Goldstone mode is always welldefined and it remains massless, the other solution canbe connected to a well-defined mode only below T T RSB ,where all the three gaps are opened. In this respect, assoon as one modifies slightly the coupling matrix (45)2
T(K) -0.6-0.4-0.200.20.40.6 Re ∆ Re ∆ Re ∆ Im ∆ Im ∆ Im ∆ -0.4-0.200.20.40.60.8 Re ∆ Re ∆ Re ∆ Im ∆ Im ∆ Im ∆ ∆ ( m e V ) , φ / π (r ad ) T (K) ω ( m e V ) L1L2 min | ∆ |φ /π | ∆ |φ /π (a) L1 (c)(b) L2 FIG. 5: (color online) Temperature evolution of the low-energy modes along the path (b) of Fig. 3, corresponding to λ = 0 . η . Here T c = 18 . K and T TRSB = 23
K. (a): Tem-perature dependence of the low-energy mode along with theminimum gap threshold, obtained by the temperature depen-dence of the gaps reported in the inset along with the phasesof the hole gaps. (b) and (c): component of the eigenvectorscorresponding to the modes labeled as L1 and L2 in panel(a). (d): schematic structure of the modes below and above T TRSB , with the notation of Fig. 4. In contrast to the case λ > η shown in Fig. 4 here the fluctuations above T TRSB donot identify a mode. Nonetheless, exactly at T = T TRSB
L2coincides with the Goldstone mode while L1 appears as anordinary Leggett-like oscillation of the gaps in the two holebands. in order to make the two hole pockets inequivalent, thegap in the electronic band in general survive up to T c .In this case, a soft mode can be found also in the wholetemperature interval T T RSB < T < T c , with a similartemperature dependence as the one shown in Fig. 4.It is worth noting that in evaluating numerically thecollective modes we retained the full frequency depen-dence of the electronic bubbles in Eq. (58). Indeed,the close proximity of one of the soft modes in theTRSB phase to the gap edge makes the low-frequencyexpansion dangerous, as observed also in Ref. [17].This is shown explicitly in Fig. 6, where we report thetemperature dependence of the collective modes obtained ω ( m e V ) L1L2 min ω ( m e V ) L1L22| ∆| min (a) (b) FIG. 6: (color online) Evaluation of the collective modes bymeans of a low-frequency expansion of the fermionic bubblesfor the case (a) λ = 1 . η and (b) λ = 0 . η , whose exactsolutions are reported in Fig. 4 and 5, respectively. Noticethat in both cases the frequencies of the low-energy modes arelargely overestimated, and no soft mode is found for λ = 0 . ,in contrast to the correct result. by using the low-frequency approximation (37)-(38) ofthe fermionic bubbles that appear in Eq. (58). As onecan see, while the massless character of the h fluctua-tions is correctly recovered at T = T T RSB , the absolutevalue of the low-energy modes in the TRSB state is com-pletely wrong in this approximation. In particular, inthe case η < λ (Fig. 6b) no mode is found below thethreshold for the quasiparticle excitations. Even addingthe next-order term in the low-frequency expansion (37),as suggested in Ref. [17], only one mode moves below thegap edge, in contrast to the correct result (Fig. 5a). -0.500.5 L2 Re ∆ Re ∆ Re ∆ Im ∆ Im ∆ Im ∆ λ/η ∆ ( m e V ) , φ / π (r ad ) λ/η ω ( m e V ) L1L22 ∆ min λ/η -0.500.5 L1 Re ∆ Re ∆ Re ∆ Im ∆ Im ∆ Im ∆ | ∆ |φ /π | ∆ |φ /π (a) (b)(c)L1L2 FIG. 7: (color online) (a) Evolution of the low-energy modesalong the path (c) of Fig. 3, i.e. as a function of the ratio λ/η at T = 0 . The corresponding gap and phase values areshown in the inset. (b) and (c): eigenvectors components ofthe two modes. Notice that at λ = η the L1 and L2 modeare degenerate, as already observed before. . On the otherhand as soon as one moves away from the symmetric pointone mode moves rapidly towards the gap edge. we show in Fig. 7 the evolution of the low-energy modeacross the quantum TRSB transition, i.e. the path la-beled with (c) in Fig. 3. Here (see Fig. 7a) the crossingbetween L1 and L2 at λ = η is evident, and it is alsoclear that in the regime λ < η the vanishing of the elec-tronic gap will make it more difficult to resolve experi-mentally the soft mode. Indeed, even if ω L remains wellbelow the gap in the hole bands, it rapidly approaches min ≡ . As we discussed above, this mode is miss-ing in Ref. [17] since the authors used a low-energy ex-pansion of the fermionic bubbles in the numerical evalua-tion of the collective modes. On the other hand, as soonas one makes the hole pockets inequivalent the TRSBstate admits an end point at a finite critical value λ minc also for η < λ , so that all the gaps are finite at T T RSB and only one mode becomes massless at λ minc . D. Experimental signatures of the Leggett mode
Let us discuss now the relevance of the present re-sults to the experimental investigation of a TRSB statein pnictides. A natural probe for the identification oflow-energy phase mode is Raman spectroscopy, in fullanalogy with the case of the intra-band dominated su-perconductor MgB . Even though a full calculationof the Raman response is beyond the scope of the presentmanuscript, by following the results of several previousworks, we outline the basic mechanism whichcan make phase modes visible in Raman. Raman scat-tering allows one to measure the response function for acharge density ˜ ρ ( q ) = P k γ k ρ ( q ) weighted with a struc-ture factor γ k that accounts for the specific geometry ofthe incoming/outgoing light polarization. Since densityand phase fluctuations are conjugate variables, phase fluctuations couple to the Raman response as well,with some caveats on the allowed symmetry for multi-band superconductors. In practice, this means that ontop of the bare Raman response due to quasiparticle ex-citations, that vanishes below | ∆ min | at T ≪ T c , col-lective phase excitations manifest themselves in the Ra-man response as a peak at the typical frequency ω L ofthe corresponding modes. More specifically, when ω L lies below the treshold | ∆ min | for quasiparticle ex-citations the mode at low T is weakly damped by resid-ual impurity-induced scattering processes, so the peakis sharp. In the specific case of pnictides we have shownthat Fig. 4 represents the typical thermal evolution of thephase modes across the transition from the TRS to theTRSB state, see path (a) of Fig. 3 . Since the Leggett-likemode which becomes massless at the T T RSB lies alwaysbelow | ∆ min | it should be visible already when enteringthe normal SC state, with a non-monotonic temperaturedependence: it first softens until T T RSB is reached andthen hardens again, saturating at low T , see ω L ( T ) inFig. 4. Observe that in Fig. 4 the L1 mode involves above T c fluctuations in the two hole bands, which are assumed here for simplicity to have the same DOS, so that theinteraction anisotropy can be tuned by a simple param-eter λ/η . However, in real materials the two hole bandshave different DOS, and then different weighting factors γ l k , which are simply proportional to the band masses.According to the discussion of Ref. [23], this guarantessthat the L1 mode will be visible in Raman, despite it in-volves phase fluctuations between bands having the samecharacter.We also verified that even in the region at λ > λ cr ≃ . of the phase diagram of Fig. 3, where the TRSB tran-sition does not explicitly occurr, the Leggett mode lieswell below the gap treshold min in a wide range oftemperatures and couplings, as shown also by the T = 0 results reported in Fig. 7. Thus, if in realistic materialswith more anisotropic interactions the phase space wherea TRSB state is realized will shrink, making more dif-ficult to realize a sample which displays the s + is state,its proximity can be still evidenced by the emergence of aphase mode at low (but finite) energy. In this situation,one can also investigate the effects of the Leggett modeon other quantities, as for example the superfluid den-sity. Indeed, by expanding the phase-only action (52) atlow frequency one finds that the Goldstone mode and theLeggett one are coupled at finite frequency (see e.g. Eq.(41)). Thus, whenever the Leggett modes is massive itcouples to fluctuations of the overall SC phase, whichin turn can affect the temperature dependence of thesuperfluid density, that has been recently shown tobe highly non-monotonic when 122 systems are stronglyhole doped. This issue, which requires to account prop-erly also for density fluctuations and long-range Coulombinteractions, not included so far, will be the subject ofa future investigation. IV. DISCUSSION AND CONCLUSIONS
In the present manuscript we analyzed the behavior ofcollective phase and amplitude modes in a multiband su-perconductor with predominant interband pairing, whichis the case physically relevant for pnictide superconduc-tors. The interband nature of the pairing mixes bond-ing (attractive) and antibonding (repulsive) SC channels,that must be treated with care while deriving the effectiveaction for the SC fluctuations by means of the standardHubbard-Stratonovich decoupling. Here we implementa generalized transformation T of the multiband pairingoperators which has two crucial consequences:(1) It allows to put the mean-field equations in a diag-onal form: (ˆ g − − ˆΠ) ~ ∆ = 0 ⇒ (ˆΛ − − T ˆΠ T T ) ~ ¯ h ≡ X l Y l ¯ h l = 0 (64)The saddle-point values of the HS fields ~ ¯ h characterizethe SC state. In the ordinary SC phase the eigenvalue Y = 0 connected to the largest SC bonding channel van-4ishes, and the saddle-point value ¯ h of the correspondingHS field is finite and can be taken real. The eigenvalueconnected to the antibonding SC channel is always pos-itive, so that the corresponding HS field is always zero.In this way, one sees that antibonding channels do notcontribute to the SC state. A TRSB phase in the three-band model occurs when a second bonding SC channelbecomes active, so that the corresponding eigenvalue, say Y , vanishes. In this situation the corresponding imagi-nary part of the HS fields ¯ h acquires a finite saddle-pointvalue, and the gaps are intrinsically complex: Y = 0 ⇒ TRS − SC , ¯ h = R (65) Y = Y = 0 ⇒ TRSB − SC , ¯ h = R , ¯ h = iI (66)(2) The low-energy behavior of the collective phasefluctuations is uniquely determined by the mean-fieldequations. More specifically, one sees that at q = 0 thefluctuations in the TRS phase sectors are described as S PF L ( q = 0) ∼ X l Y l ( Imh l ) (67)This has several implications: (i) the fluctuations of theordering field Imh are trivially massless ( Y = 0 ) sincethis is the Goldstone mode of the SC transition; (ii) forthe two-band case the fluctuations on the antibondingHS fields do not identify any collective mode, so that forinterband-dominate pairing the Leggett mode is absent,in contrast to ordinary intraband-dominated supercon-ductors as MgB ; (iii) in the three-band case the transi-tion to a TRSB phase ( Y = 0 ) is uniquely associated to massless phase fluctuations described by the imaginarypart of the ordering field h .The absence of the Leggett mode in a two-band modelfor pnictides can be understood on physical grounds byhaving in mind the marked difference between a two-bandsuperconductor with dominant intra- or inter-band pair-ing. In the former case one has two bonding SC channels:the SC transition is controlled as usual by the one givingrise to a larger condensation energy, i.e. larger gaps. Onthe other hand, there exists a second possible solutionwith smaller gap values: the Leggett mode describes in-deed deviations from the ground state in the direction ofthis second possible solution. For example, when the in-terband pairing is positive the two gaps ∆ , ∆ have thesame sign, i.e. the same saddle-point phases ¯ ϑ = ¯ ϑ .In this situation, the Leggett mode identify phase fluc-tuations of opposite sign in the two bands, that wouldlead the system towards the solution ¯ ϑ = − ¯ ϑ withhigher energy, as reflected in the massive character of theLeggett mode. In this respect, antiphase oscillations canbe put in resonance with the additional SC channel, sothat they identify a collective mode. On the other hand,when the coupling is predominantly interband there is noadditional SC channel in the problem, so that the samekind of oscillations do not identify any proper mode ofthe system.For the three-band case, it must be emphasized thatthe massless character of the Legget phase mode found in item (2) above holds regardless the specific nature ofthe two SC bonding channels which become degenerateat the TRSB transition. Thus, it can be applied as wellalso to other (multiband or multichannel) systems wherea TRSB phase as been predicted, as e.g. highly dopedgraphene , water-intercaled sodium cobaltates andlocally noncentrosymmetric SrPtAs. The last system isparticularly promising in this respect, since recent muonspin rotation ( µSR ) measurements could be actually in-terpreted as evidence of a TRSB phase. Indeed, themassless character of the phase mode at the transition isa unique signature of the TRSB transition, that can beused to rule out other possible interpretation of the µSR data.In the specific case of pnictides, by performing an ex-plicit numerical calculation of the collective modes wealso established some additional results. In particular,we found that below T T RSB the massless mode becomesmassive again, while a second low-energy mode can ap-pear in some range of parameters. For the case of pnic-tides, by scanning in temperature a given sample whichis either in case (a) or (b) of Fig. (3) the appearance of amode below the gap threshold will signal the emergenceof the TRSB state. Since phase fluctuations couple todensity fluctuations, the most direct probe ofthis mode is via Raman scattering, in analogy with theresult found for MgB . . Even though a full calcula-tion of the Raman response is beyond the scope of thepresent manuscript, we expect that in the realistic case ofpnictides the temperature evolution of the Raman spec-tra can offer a powerful mean to the identification of aTRSB state, shedding then new light on the theoreticalunderstanding of the pairing mechanism itself in theseunconventional superconductors. V. ACKNOWLEDGEMENTS
We thank S. Caprara for useful discssions and sug-gestions. The authors acknowledge financial sup-port by the Italian MIUR under the project FIRB-HybridNanoDev-RBFR1236VV and by the Spanish Min-isterio de Economía y Competitividad (MINECO) underthe project FIS2011-29680.
Appendix A: Equivalence between the Gaussianaction derived in cartesian or polar coordinates
In this Appendix we show that in the single-band casethe Gaussian action for amplitude and phase fluctuationscan be equally derived by using a cartesian (real andimaginary part) or polar (amplitude and phase) descrip-tion of the SC fluctuations. In the multiband case, aswe discussed in the present manuscript, one chooses asconvenient variables (i.e., as HS fields) proper combina-tions of the gaps in the various bands. Nonetheless, whenall the the SC channels are bonding one can still intro-5duce a set of HS fields associated to the fluctuations ineach band. Indeed, in this case the equivalent of the re-lations (33)-(34) between the physical fields and the HSfields do not contain imaginary units. Thus, they arenot only valid for the average values, but they representa change of variables for bosonic fields in the functionalintegral defining the partition function. This case willbe discussed indeed in Appendix B, where we explicitlyapply our formalism to the two-band case with predomi-nant intraband pairing. However, as soon as at least onechannel is antibonding the only possible route is to usecollective HS fields properly defined in each channel, aswe have done in the present manuscript. Still, the gen-eral relations between the fermionic bubbles that we willestablish below can be used to interpret the low-energybehavior of the collective modes. In particular, it will beuseful to compare the derivation of collective modes inthe TRSB phase given in Refs. [13,15], based on a polardescription, and the one presented here, based on the useof cartesian coordinates, equivalent also to the approachof Refs. [16,17].Let us start again from Eq. (1) written for a single-band superconductor, and let us decouple the interactionterm by means of the HS decoupling (5), by introducingan HS field ∆( x ) . The equivalent of Eq. (12) will thenread: S = S + Z dτ d x | ∆( x ) | g − (∆ ∗ c ↓ c ↑ + h.c. ) (A1)After integration of the fermions the saddle-point value ∆ ( chosen to be real) of the HS field will appear in the mean-field Green’s function ¯ G , while its fluctuations willbe decomposed as ∆ q = Re ∆ q + iIm ∆ q , so that Eqs.(14) and (15) will be explicitly given by: ¯ G − k = (cid:18) iω n − ξ k ∆∆ iω n + ξ k (cid:19) , (A2) Σ q = k − k ′ = r TV ( Re ∆ q σ + Im ∆ q σ ) , (A3)where ∆ is the solution of the self-consistency equation: Π = 1 g ⇒ V X k tanh βE k E k = 1 g , (A4)where Π is the Cooper bubble (21) and E k = p ξ k + ∆ .The explicit connection established by Eq. (A3) abovebetween the real and imaginary part of the fluctuatingfield and the σ , σ Pauli matrices, respectively, allowsone to derive immediately the coefficients of the action S F L (18) at Gaussian order: S CarF L = 12 X q η T − q (cid:18) Λ ( q ) + g Λ ( q )Λ ( q ) Λ ( q ) + g (cid:19) η q , (A5)where in analogy with Eq. (27) we defined η Tq =( Re ∆ q , Im ∆ q ) . The fermionic bubbles are defined as: Λ ij ( q ) = TV X k Tr (cid:2) ¯ G k + q σ i ¯ G k σ j (cid:3) . (A6)More specifically we have: Λ ij ( q ) = 1 V X k (cid:20) ( uu ) ij E ′ k − E k − i Ω m + ( vv ) ij E ′ k − E k + i Ω m (cid:21) [ f ( E ′ k ) − f ( E k )] ++ (cid:20) ( uv ) ij E ′ k + E k − i Ω m + ( vu ) ij E ′ k + E k + i Ω m (cid:21) [ f ( E ′ k ) − f ( − E k )] (A7)where E ′ k = E k + q and the coherence factors are givenby: ( uu ) = ( vv ) = 12 (cid:18) − ξ ′ k ξ k − ∆ E ′ k E k (cid:19) (A8) ( uv ) = ( vu ) = 12 (cid:18) ξ ′ k ξ k − ∆ E ′ k E k (cid:19) (A9) ( uu ) = ( vv ) = 12 (cid:18) − ξ ′ k ξ k + ∆ E ′ k E k (cid:19) (A10) ( uv ) = ( vu ) = 12 (cid:18) ξ ′ k ξ k + ∆ E ′ k E k (cid:19) (A11) ( uu ) = − ( vv ) = 12 (cid:18) ξ k E k − ξ ′ k E ′ k (cid:19) (A12) ( uv ) = − ( vu ) = − (cid:18) ξ ′ k E ′ k + ξ ′ k E ′ k (cid:19) (A13)As one can easily check, Λ ( q ) ∼ Ω m O ( q ) , so that inthe static limit it can be neglected, leading to the effectivedecoupling between the amplitude and phase fluctuationsthat has been used in Sec. IIB. Moreover, from Eq. (A7)6and (A10)-(A11) above it follows that: Λ (0) = − V X k tanh βE k E k = −
2Π = − g , (A14)where we used the self-consistency equation (A4) above.Thus, one immediately recovers the massless character ofthe Im ∆ q fluctuations from Eq. (A5). Notice that sincewe did not introduce explicitly the density fluctuations,other phase modes like the Carlson-Goldman one cannot be explicitly obtained. On the other hand, thesesound-like modes are usually relevant only at high tem-perature or strong disorder, unless the gap has nodes ,that are not the cases relevant for the present discussion.Let us now discuss the derivation of the Gaussian fluc-tuations within the polar-coordinate scheme. In thiscase, before integrating our the fermions in Eq. (A1),one can make explicit the dependence on the phase θ ofthe HS field by means of a Gauge transformation on thefermionic fields, c σ ( x ) → c σ ( x ) e iθ/ . As a consequence,while ¯ G k is unchanged, the self-energy Σ kk ′ describingthe SC fluctuations will be expressed in terms of the vari-ables | ∆ | q , θ q : Σ kk ′ = TV | ∆ | k − k ′ σ + r TV i m ( k − k ′ ) · ( k + k ′ ) θ k − k ′ σ ++ r TV ω k − k ′ θ k − k ′ σ ++ TV X s ( k − s ) · ( s − k ′ )8 m θ k − s θ s − k ′ σ (A15)As one can see, in this case the self-energy contains onlyspatial and time derivatives of the phase, associated re-spectively to the Pauli matrices ∼ k σ and σ , whichdescribe in Nambu formalism the fermionic current anddensity. Thus, using Eq. (A15) in the expansion (18) onecan write the effective Gaussian action as: S P olF L = 12 X q ζ T − q (cid:18) Λ ( q ) + g − i q µ Λ µJ − i q µ Λ µ J q µ q ν ˜Λ µνJJ ( q ) (cid:19) ζ q , (A16)where ζ q ≡ ( | ∆ | q , θ q ) . Here we used the quadrivector no-tation q µ = ( i Ω m , q ) , q µ = ( i Ω m , − q ) and we introducedthe generalized current-current bubbles ˜Λ µνJJ = − nm η µν (1 − η ν ) + Λ µνJJ , (A17) Λ µνJJ = TV X k Tr (cid:2) ¯ G k + q γ µ ( k, k + q ) ¯ G k γ ν ( k + q, k ) (cid:3) , (A18) Λ µJi = TV X k Tr (cid:2) ¯ G k + q γ µ ( k, k + q ) ¯ G k σ i (cid:3) , (A19)where η µν = diag(1 , − , − and the current vertex is γ µ ( k, k + q ) = (cid:18) σ , k + q / m σ (cid:19) . (A20) Notice that the above definitions (A6) and (A18)-(A19)are slightly redundant, since e.g. Λ JJ ≡ Λ . Neverthe-less, Eqs. (A18)-(A19) allow for a transparent interpre-tation of the phase mode in Eq. (A16), whose dispersionis given explicitly by: q µ q ν ˜Λ µνJJ ( q ) ≡ − Ω m ˜Λ JJ + q i q j ˜Λ ijJJ − i Ω m q i ˜Λ iJJ . (A21)Indeed, since only time or spatial derivatives of the phasefield enter the self-energy (A15), the hydrodynamic limitof the phase mode is easily obtained by taking the q = 0 limit of the current-current fermionic bubbles which ap-pear as coefficients in Eq. (A21). More specifically, since Λ JJ (0) = − κ , where κ is the compressibility, Λ ijJJ (0) =( ρ s /m ) δ ij and Λ iJJ (0) = 0 , one finds immediately that inthe long-wavelength limit the phase mode has the well-known sound-like dispersion q µ q ν ˜Λ µνJJ ( q ) ≃ − ω κ + q ρ s m (A22)which characterizes the Bogoliubov-Anderson mode. In-stead, in the cartesian notation of Eq. (A5) the disper-sion of the phase mode must be obtained by performingthe low- q expansion of the Λ ( q ) bubble. An alternativeroot is to exploit the equivalence between the two deriva-tions (A5) and (A16), that must be valid at all orders in q . Here we prove explicitly this equivalence by using thedefinitions of the fermionic bubbles and the identity : ¯ G − k + q σ − σ ¯ G − k = q µ γ µ ( k + q, k ) − i ¯∆ σ . (A23)One can then prove the three equalities: q µ Λ µJ ( q ) = 2 i ¯∆ (cid:20) Λ ( q ) + 2 g (cid:21) , (A24) q µ ˜Λ µνJJ ( q ) = 2 i ¯∆Λ ν J ( q ) , (A25) q µ ˜Λ µJ = 2 i ¯∆Λ . (A26)Let us show for example the demonstration of Eq. (A24).By means of the definitions (A6) and (A19) and theequivalence (A23) we have: q µ Λ µJ ( q ) − i ¯∆Λ == TV X k Tr (cid:2) ¯ G k + q ( q µ γ µ ( k, k + q ) − i ¯∆ σ ) ¯ G k σ (cid:3) == TV X k Tr (cid:2) σ ¯ G k σ (cid:3) − TV X k Tr (cid:2) ¯ G k + q σ σ (cid:3) == 2 i TV X k Tr (cid:2) ¯ G k σ (cid:3) = 4 i ¯∆Π = 4 i ¯∆ g (A27)where we used the definition of the Π bubble, Π ≡ T V P k Tr (cid:2) ¯ G k σ (cid:3) and the self-consistency equation (A4) Π = 1 /g . The remaining equalities (A25)-(A26) can beobtained with a similar procedure. By means of Eq.s(A24)-(A25), and using Λ µ J ( q ) = Λ µJ ( − q ) = − Λ µJ wethen have: q µ q ν ˜Λ µνJJ ( q ) = − i q ν Λ νJ ( q ) = ¯∆ (cid:20) Λ ( q ) + 2 g (cid:21) (A28)7so that, since Im ∆ q = ¯∆ θ q , we recover the equivalencebetween the phase-fluctuation propagator in Eqs. (A5)and (A16). Analogously, by means of Eq. (A26) we re-cover the equivalence in Eq.s (A5) and (A16) betweenthe off-diagonal terms in the amplitude-phase fluctua-tions, completing the demonstration of the full equiva-lence of the two procedures. Notice that the above re-lations (A22) and (A28) allow one to easily derive thelow-momentum expansion of the Λ bubble, that hasbeen used in Eq. (38) of the main text. Indeed, we havethat: Λ ( q ) = −
2Π + 14∆ q µ q ν ˜Λ µνJJ ( q ) ≃ = −
2Π + 14∆ (cid:16) Ω m κ + q ρ s m (cid:17) (A29)Notice that in Eq. (A29) above the coefficient of the Ω m term, i.e. the density-density correlation function Λ JJ ( q ) , has been taken in the static limit, where it givesthe compressibility. On the other hand, as it is wellknown, at finite T the dynamic limit of Λ JJ ( q ) , whichis the one relevant to compute the collective modes at q = 0 in Sec. II and III, differs from κ . On the otherhand, the low- q expansion of the Λ ( q ) can still be con-nected to the generalized current-current bubbles, thatis the relation needed to recover the equivalence betweenthe two derivations of the collective modes.It is worth noting that in the single-band case the de-scription in terms of cartesian or polar coordinates isequivalent since one usually chooses a gauge where thesaddle-point gap value ¯∆ is real. Thus, the two descrip-tions correspond to the same choice of fluctuations di-rections and are then trivially equivalent. On the otherhand, if one had chosen a finite saddle-point phase for thegap then the cartesian description of fluctuations wouldbe very inconvenient, since in this case the Λ ( q = 0) bubble which couples real and imaginary parts in Eq.(A5) would be non zero, as we already emphasized inEq. (56) above for the TRSB. However, in the single-band case one could still make a proper rotation to polarcoordinates that would lead again to decoupled ampli-tude and phase fluctuations. The multiband TRSB casediscussed in Sec. IIIB is instead different. Indeed, inthe TRSB state each gap acquires a non-trivial saddle-point value for the phase. In this situation, even thoughtthe choice of polar coordinates for each band could stillmake the part of fluctuations described by fermionic bub-bles ortogonal, the coupling matrix ˆ g lm in Eq. (58) willnot be diagonalized by this rotation, making amplitudeand phase fluctuations always intrinsically mixed.Finally, let us derive for the sake of completeness therelations between the average values of the HS field ∆ q and the physical correlation functions expressed in termsof the fermionic operators φ q defined in Eq. (2) above. Let us discuss it for the case of cartesian coordinates:starting from Eq. (A1) we add a source field Ψ q which couples to φ q such that: Z = Z D c σ D c † σ D ∆ D ∆ † e − S + P q ( φ † q Ψ q + h.c. ) h φ q i = ∂ ln Z∂ Ψ ∗ q (cid:12)(cid:12)(cid:12)(cid:12) Ψ=0 , (A30) h φ † q φ − q i = ∂ ln Z∂ Ψ q Ψ ∗− q (cid:12)(cid:12)(cid:12)(cid:12) Ψ=0 . (A31)To perform explicitly the derivatives on the right-side ofEqs. (A30)-(A31) we can notice that the total action inEq. (A30) can be written as: S ′ = S − X q ( φ † q Ψ q + h.c. ) == X q | ∆ q | g − X q (cid:2) (Ψ ∗ q + ∆ ∗ q ) φ q + h.c. (cid:3) == X q | ˜∆ q | g − X q h ˜∆ ∗ q φ q + h.c. i ++ X q | Ψ q | g − ( ˜∆ ∗ q Ψ q + h.c. ) g , (A32)where we shifted Ψ q + ∆ q = ˜∆ q . One can then easilyderive the relations: ∂ ln Z∂ Ψ ∗ q (cid:12)(cid:12)(cid:12)(cid:12) Ψ=0 = 1 g h ∆ q i = h φ q i , (A33) ∂ ln Z∂ Ψ q Ψ ∗− q (cid:12)(cid:12)(cid:12)(cid:12) Ψ=0 = − g + 1 g h ∆ ∗ q ∆ − q i = h φ † q φ − q i , (A34)where we also used the fact that at Ψ = 0 the averagesof the ∆ q and ˜∆ q fields coincide. By direct inspection onthe Gaussian action (A5) for the HS-field fluctuations, werecover the well-known result that the correlator for theHS field ∆ q corresponds to the RPA resummation of thepotential, while the correlator of the physical field givesthe RPA resummation on the corresponding fermionicsusceptibility. For example for the real components wehave: h Re ∆ − q Re ∆ q i = g g Λ ( q ) , (A35) h Reφ − q Reφ q i = − Λ ( q ) /
21 + g Λ ( q ) . (A36) Appendix B: Derivation of the Leggett mode fordominant intraband pairing
In this Appendix we show explicitly how the general-ized transformation (9) can be used to obtain the Leggettmode in the two-band case with dominant intraband pair-ing, as it would be appropriate for example to MgB . .As it has been shown in Ref. [20] in this case one couldeasily obtain the Leggett’s mode dispersion by using a8straightforward generalization to a two-band case of thederivation (A16) reviewed above in terms of polar coor-dinates. By introducing a HS field for each band, withphases ( θ ( q ) , θ ( q )) , the long-wavelength q = 0 phasefluctuations (decoupled from the amplitude ones) are de-scribed by the matrix: ˆ S PF L = 18 (cid:18) − N ω + A − A − A − N ω + A (cid:19) (B1)where A is a constant connected to the matrix ˆ g of theSC couplings and to the saddle-point gap values ¯∆ l ineach band A = 8 g ¯∆ ¯∆ det ˆ g . (B2)In Eq. (B1) we recognize the ω expansion derived in Eq.(A21) above, with the compressibility κ l in each bandapproximated by the corresponding density of states N l .The phase collective modes are found as usual as solu-tions of the equation det ˆ S PF L = 0 . The first solution ω = 0 corresponds as to the Bogoliubov-Anderson (BA)mode, while a second solution exists corresponding to theLeggett mode : ω = ω L = A N + N N N . (B3)Notice that, as already observed by Leggett in his orig-inal paper, this solution only exists when A > , i.e.when det ˆ g > (since sign ( ¯∆ ¯∆ ) = sign g ), which cor-responds to intraband -dominated coupling. One can alsoeasily verify that the BA mode corresponds to fluctua-tions having θ = θ , while the Leggett mode correspondsto ( − N ω L + A ) θ − Aθ = 0 ⇒ θ θ = − N N (B4)i.e. to antiphase oscillations in the two bands, weightedwith the respective DOS. As usual, the Leggett mode (aswell as the Carlson-Goldman one) can be equally foundin linear-response theory. Let us rewrite instead the effective phase-only actionafter using the generalized transformation T . In this case,since both the eigenvalues Λ > Λ > of ˆ g are positive,the hyperbolic matrix in Eq. (7) will be replaced by anordinary rotation matrix, which preserves the structure diag (1 , present in this case. The T matrix will againbe used to decouple the mean-field equations for the HSfields: (cid:18) − P l T l Π l − P l T l Π l (cid:19) (cid:18) ¯ h ¯ h (cid:19) = Y ¯ h + Y ¯ h = 0 . (B5)where now no imaginary unit is associated to ¯ h since itdecouples an ordinary bonding SC channel, and also Y could in principle vanish. However, as soon as Y = 0 and one enters the SC state Y can never vanish (see Ap-pendix C), so the ¯ h will still be the only order param-eter of the SC transition. The effective action will have the same structure of Eq. (36) derived above, with theremarkable difference that now the HF field h used todecouple the Λ channel will not carry out an additional i unit, so that the − / Λ term in Eq. (36) is replaced by +1 / Λ : ˆ S PF L ( q ) = (cid:18) P l Λ l ( q ) T l + P l Λ l T l T l P l Λ l T l T l P l Λ l ( q ) T l + (cid:19) (B6)To compute the above Eq. (B6) at q = 0 and small ω we use the expansion (38) of the Λ bubbles, so that weobtain: ˆ S PF L = − (cid:18) Bω Cω Cω Dω − m (cid:19) (B7)where B = 18 X l N l ∆ l T l (B8) D = 18 X l N l ∆ l T l (B9) C = 18 X l N l ∆ l T l T l (B10) m = 1Λ − X l Π l T l (B11)Notice that m in Eq. (B11) above in nothing else thanthe second eigenvalue Y of the matrix of mean-field equa-tions (B5) above, which is always non-zero below T c fora system with finite interband coupling. It is then clearthat also the matrix (B7) leads to two solutions. Thefirst one at ω = 0 corresponds to the BA mode: it in-volves only fluctuations of the Imh ,q field, which thanksto the gap definitions (24) is indeed an uniform phase ro-tation for the gaps in both bands, see Eq. (40) above.The second solution is found at the frequency: ω = Bm BD − C . (B12)We will now show that Eq. (B12) coincides with the ex-pression (B3) above, by deriving the explicit expressionsof the T matrix from the three conditions established inSec. II: det T = 1 , Eq. (22) P l T l Π l T l = 0 and thesaddle-point equation (23), which defines also the mean-field gaps ∆ l in Eq. (24). One can then easily show that T = 1 pP l Π l ∆ l (cid:18) ∆ / √ Λ ∆ / √ Λ −√ Λ Π ∆ √ Λ Π ∆ (cid:19) , (B13)where we also have that the saddle-point value of the HSordering field ¯ h is given by ¯ h = Λ X l Π l ∆ l (B14)9By means of Eqs. (B13)-(B14) we can then express the co-efficents (B8)-(B11) in terms of the gap values ∆ l and ofthe eigenvalues, connected to the matrix ˆ g of the SC cou-plings. With lengthly but straightforward calculations wethen have: B = 18¯ h X l N l , (B15) D = Λ N Π ∆ + N Π ∆ ∆ ∆ , (B16) C = Λ h − N Π ∆ + N Π ∆ ∆ ∆ , (B17) m = Λ ¯ h (1 − Π Π detˆ g ) = g detˆ g ∆ ∆ . (B18)As a consequence, we get in Eq. (B12) that ( BD − C ) /B = N N / ∆ ( N + N ) and we then recoverthe expression (B3) for the frequency of the second eigen- mode. By means of the same relations one can also provethat the eigenvector corresponding to the (B12) solution,i.e. Imh = − ( C/B ) Imh describes the antiphase fluc-tuations (B4) identified above for the Leggett mode. Appendix C: TRSB transition in a three-band model
In this Appendix we discuss the TRSB transition in thethree-band case in terms of the action for the HS fields. Ingeneral, once given the matrix (47) of the SC couplings,we are interested to the case where there are two bondingeigenvalues Λ , Λ and one antibonding one − Λ . The T = P α,β,ϕ R transformation in Eq. (9) is defined throughthe rotation R which diagonalizes g and the matrix P α,β,ϕ consisting in a 3D Poincaré transformation in the h -spacewith h , being the spatial ( x, y ) dimensions and h thetime ( t ) one: P α,β,ϕ = √ Λ √ Λ
00 0 √ Λ R xyα H ~v xy ( β ) ,tϕ √ Λ √ Λ
00 0 √ Λ α, β ∈ [0 , π ] ϕ ∈ R (C1)Here R xyα is an ordinary rotation of angle α (with re-spect to the x axis) in the ( x, y ) plane, while H ~v xy ( β ) ,tϕ isa hyperbolic rotation of angle ϕ in the plane identifiedby the t direction and by the versor ~v xy ( β ) of the ( x, y ) plane, β being the angle with respect to x . After the HSdecoupling the equivalent of the action (12) will read: S = S + Z dτ d x | h ( x ) | Λ + | h ( x ) | Λ + | h ( x ) | | Λ |− (cid:16) h ∗ ψ + h ∗ ψ + h.c. (cid:17) − i (cid:16) h ∗ ψ + h.c. (cid:17) (C2)We can then proceed as in the two-band case, havingin mind that now the HS field h associated to the an-tibonding channel will enter with an imaginary unit inboth the saddle-point Green’s function (14) and the self-energy (15), playing then the role of h for the two-bandcase. We observe that the T matrix depends on threeparameters, i.e. the rotation angles α, β, ϕ of the ma-trix (C1) above: they are fixed (self-consistently) by thethree conditions (48) above which are used to decouplethe saddle-point equations for the HS fields. In the ordi- nary SC phase, when only the h field has a finite saddle-point value, the derivation of the action is a straightfor-ward extension of the calculations presented in Sec. II.Now we prove briefly that, if exists T T RSB such thatEq. (50) holds, then the system undergoes a second-orderphase transition to a TRSB phase. In particular, ¯ h willemerge purely imaginary. First of all we recall that the T transformation is used to put the saddle-point equa-tions (25) in diagonal form, see Eq. (26). In the ordinaryTRS phase only one element of the matrix T ˆΠ T T − ˆΛ − vanishes, while at T ≤ T T RSB
Eq. (50) holds and a sec-ond element vanishes. In this situation the h and h spaces are degenerate and any additional rotation α inthe transformation matrix P α,β,ϕ will leave the result un-changed. Hence, if we define in general ¯ h i = R i + iI i wecan use the parameter α , along with the U(1) gauge sym-metry, to impose I = R = 0 . Indeed, even if at some T < T
T RSB one has I , R = 0 we can pass to an othersolution ( ~R ′ , ~I ′ ) with vanishing I ′ and R ′ by means ofthe transformation ˆ | Λ | − / cos α − sin α α cos α α − sin α α cos α ˆ | Λ | / cos ϑ − sin ϑ
00 cos ϑ − sin ϑ sin ϑ ϑ
00 sin ϑ ϑ R R I I = R ′ I ′ (C3)0where ϑ is the U(1) angle. Once established the possibil-ity to choose R = 0 we should prove that at T < T
T RSB the ground state favors indeed a finite value of the imag-inary part of ¯ h . Indeed, even if at T = T T RSB
Eq. (50)is satisfied, at lower temperatures there are still threepossibilities: (i) I remains zero and Eq. (50) does nothold anymore, so that R reamins the only order param-eter; (ii) ¯ h opens with a real component R only or (iii) ¯ h opens with an finite imaginary component I , and aTRSB phase is established. To show that the case (iii) isthe ground state we make use of the fact that the imag-inary fluctuations of h at q = 0 become massless at T = T T RSB , as proven in Eq. (52) above. Indeed, let uswrite down the expansion of the action at a temperature T . T T RSB with respect to the mean-field action ˜ S MF computed with the solution (i), i.e. ~R = ( R , , , ~I = ~ .By using in the fluctuation action the T matrix and thefermionic bubble evaluated at the expansion point andthe results of Sec. III we have S = ˜ S MF + 12 δ ~R T ( T ˆΛ T T + 2 ˆΛ − ) δ ~R ++ 12 δ~I T ( ˆ T ˆΛ ˆ T T + 2 ˆΛ − ) δ~I + O ( R , I , RI , IR ) (C4)where δ ~R, δ~I are the displacements with respect to theexpansion point, and δ~I includes the imaginary factor i of the antibonding channel, in accordance with the def-inition given above Eq. (52). Observe that in Eq. (C4)the linear terms do not appear since S (1) MF is a stationarypoint for the action, and the phase-amplitude couplingsare absent since above T T RSB we have chosen the gaugewhere all the gaps are real. Using the relations (37)-(38)for the q = 0 values of the fermionic bubble we can thensee that δ ~R fluctuations are always costly, so that case ( ii ) leads to an increase of the energy. On the other hand,thanks the identity ˆΛ ( q = 0) = − the δ~I fluctuationshave explicitly the form: δ~I T diag (0 , − a, − b ) δ~I = a ′ ( T − T T RSB ) I + bI , a, a ′ , b > (C5)where we used the fact that at T = T T RSB
Eq. (50)holds and δI fluctuations are massless. As one can see, δI fluctuation increase the energy while δI fluctuationsdecrease it, making the TRS phase unstable towards aphase with a finite I , which leads to non-trivial phasesfor the gaps (51) and then to a TRSB phase. This in-stability will be of course compensated by higher-orderterms in the expansion (C4), that can also lead to a fi-nite δI and δR along with a finite δI . However, as wediscussed above, these components can be eliminated bythe trasformation (C3), making the definition (51) fullygeneral.We have then proven that the vanishing of a secondeigenvalue of the matrix ˆΠ − ˆ g − , i.e. Eq. (50), is asufficient condition for a TRSB phase, since when thishappens the ¯ h field acquires a finite imaginary part. We will now show that this is also a necessary condi-tion for having a TRSB phase. Let us go back to the setof self-consistency equations (25), and let us decomposethe matrix ˆΠ − ˆ g − in its eigenvectors at a temperature T ≤ T c where the ordinary SC state is established: ˆΠ − ˆ g − = λ ( T ) u T u + λ ( T ) u T u (C6)where λ i ( T ) and u i are real, since the matrix ˆΠ − ˆ g − is real and symmetric. Here we used the fact that below T c one eigenvalue vanishes, allowing for a finite solution ~ ∆ . In general, ~ ∆ is a vector of complex numbers, thatsatisfy the self-consistency equation (25), i.e.: λ ( T )( Re~ ∆ · u ) u + λ ( T )( Re~ ∆ · u ) u = 0 (C7) λ ( T )( Im~ ∆ · u ) u + λ ( T )( Im~ ∆ · u ) u = 0 (C8)Since u and u are orthogonal, the above equations im-ply that the vectors Re~ ∆ and Im~ ∆ are also orthogonalto both u and u . Thus, either one of the two vanishes,or they are paralell to u . In all these cases the gapshave all the same phases, and then the state is TRS. Onthe other hand, when one additional eigenvalue vanishesin Eq. (C6), say λ = 0 , then Re~ ∆ and Im~ ∆ belong tothe two-dimensional subspace spanned by u and u , sothat their phases can be complex. In this respect, it canalso be instructive to show how simple geometrical argu-ments can be used to establish if a TRSB phase exists a T = 0 for a generic coupling matrix. Let us start fromEq. (C6) with λ = , and let us redefine V = √ λ u , sothat ˆΠ − ˆ g − is written explicitly as a projector: ˆΠ − ˆ g − = V T V . (C9)This equation allows one to determine V in terms onlyof the couplings: indeed, it gives explicitly: g − = − V − Π V V V V V V V − Π V V V V V V V − Π (C10)Thus, temperature or bands parameters, which only en-ter via the Cooper bubbles, do not affect the forbiddendirection V , that is fully determined as (let ˆ G be theinverse of ˆ g ): V = − G G G , (C11) V = − G G G , (C12) V = − G G G . (C13)Observe also that V can be identically zero only if all theinterband couplings vanish. Thus is any real multibandsystem, where at least one interband coupling is finite,there must exist one forbidden direction. In the two-band case this guarantees that the second eigenvalue of1the matrix ˆ π − ˆ g − never vanishes. Once V is known,the self-consistency equations for the gap amplitudes arealso determined by the diagonal terms in Eq. (C10), thatgive explicitly: Π i = G ii + V i = G ii − G ik G ji G kj ( ∀ i, k = j = i ) . (C14)As one can see, in the TRSB phase the equations forthe gap amplitudes in the various bands decouple, andthey all reduce to a single-band BCS equation with dif-ferent effective couplings. This allows us also to expressin the self-consistency equations (C7)-(C8) the gap am-plitudes as a function of the Cooper bubbles, by invertingthe relation Π i ( T = 0) = N i asinh( ω / | ∆ | i ) . 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