s+is superconductivity with incipient bands: doping dependence and STM signatures
Jakob Boeker, Pavel A. Volkov, Konstantin B. Efetov, Ilya Eremin
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l s + is Superconductivity with incipient bands: doping dependence and STMsignatures
Jakob B¨oker , Pavel A. Volkov , Konstantin B. Efetov , , and Ilya Eremin (Dated: October 6, 2018)Motivated by the recent observations of small Fermi energies and comparatively large supercon-ducting gaps, present also on bands not crossing the Fermi energy ( incipient bands) in iron-basedsuperconductors, we analyze the doping evolution of superconductivity in a four-band model acrossthe Lifshitz transition including BCS-BEC crossover effects on the shallow bands. Similar to theBCS case, we find that with hole doping the phase difference between superconducting order pa-rameters of the hole bands change from 0 to π through an intermediate s + is state, breakingtime-reversal symmetry (TRS). The transition, however, occurs in the region where electron bandsare incipient and chemical potential renormalization in the superconducting state leads to a signifi-cant broadening of the s + is region. We further present the qualitative features of the s + is statethat can be observed in scanning tunneling microscopy (STM) experiments, also taking incipientbands into account. I. INTRODUCTION
The discovery of unconventional and high- T c super-conductivity in Fe pnictides and chalcogenides openedup several new directions in the study of nonphononicmechanisms of Cooper pairing in multiband correlatedelectron systems . One of the interesting issues thathas attracted considerable attention is the observationof finite superconducting gaps on bands not crossing theFermi level . Furthermore, photoemission and quan-tum oscillation experiments have shown that in manyFe-based superconductors either electron or hole pock-ets are smaller than previously thought and the corre-sponding bands barely cross the Fermi level or are lo-cated fully below or above it . Interestingly, a recentquasiparticle interference (QPI) study of FeSe has re-vealed that the ratio of the superconducting gap to theFermi energy on the shallow electron band is very large∆ /E F e ∼ /E F ≈ . x Te − x . This has been followed upby observation of strong superconducting fluctuations inFeSe far above T c .The large values of ∆ /E F observed initially in FeSehave been interpreted as a signature of a crossover fromconventional Cooper pairing (BCS) to Bose-Einstein con-densation (BEC) regime . In the latter case, Cooperpairs are formed well above superconducting T c andshould manifest themselves as unusually strong super-conducting fluctuations.A great deal of current understanding of the BCS-BEC crossover physics comes from the remarkable ex-periments on systems of ultracold fermionic atoms .These systems offer the great advantage of an experi-mentally tunable interparticle attraction between con-stituents via Feshbach resonances, allowing us to performexperiments throughout the crossover region. Of partic-ular relevance are experiments on quasi two-dimensional(quasi-2D) systems . More recent experiments re-port the observation of a Berezinskii-Kosterlitz-Thouless (BKT) transition to the superfluid state as well as thepresence of pairing far above the critical temperature .Similar ideas have been recently applied theoreticallyto study the potential BCS-BEC crossover in multibandsystems with small Fermi energies . One of the mostimportant conclusions of these studies is that for smallFermi energies the chemical potential of the system isstrongly renormalized in the superconducting state evenin situations when the temperature of the pair forma-tion roughly agrees with the superconducting transition.This renormalization of the chemical potential is espe-cially important given the variety of the superconduct-ing and magnetic states, which appear in the iron-basedsuperconductors upon changing the control parameterssuch as disorder, pressure and doping.Another interesting situation where related physics canappear is near a Lifshitz transition , where one of thebands continuously moves away from the Fermi level as afunction of an external parameter (e.g. pressure, doping,external magnetic field or nanostructuring ). Theo-retical proposals have been put forward that strong T c enhancement can be achieved close to Lifshitz transitionin striped and layered systems . Superconductivityin two-band models close to a Lifshitz transition is alsobeing studied . Recent experiments on monolayerFeSe do suggest enhancement of superconductivity bya Lifshitz transition.A peculiar example of an iron-based supercon-ductor with a Lifshitz transition is Ba − x K x Fe As .Angle-resolved photoemission (ARPES) andthermopower measurements point toward the exis-tence of such a transition in the overdoped compoundwith x ∼ . − .
9. Intriguingly, in the same dopingrange, the structure of the superconducting gaps under-goes dramatic changes, seemingly inconsistent with atwo-band description.Multiple experiments such as ARPES , neutronscattering and thermal conductivity measurements support a nodeless s + − gap structure around the opti-mal doping x ≈ . s -wave pairing symmetry with accidental nodes where theorder parameter changes sign between the two remain-ing hole pockets or d -wave-pairing symmetry withwell-pronounced line nodes . Moreover, in the inter-mediate doping region, frustration between the two su-perconducting channels has been theoretically predictedto result in a time-reversal symmetry-breaking s + is state or s + id state . In these states, the phase dif-ference φ between the order parameters at the two holebands is not equal to a multiple of π with the φ ↔ − φ symmetry being spontaneously broken. s + is superconductors have been theoretically pre-dicted to possess many unconventional properties.Josephson critical current of a constriction junction be-tween s + is superconductors has been found to be anoma-lously suppressed . As a result of simultaneous break-ing of U (1) and Z vortex fractionalization and unusualvortex cluster states have been predicted . Addition-ally, excitations inside the vortex cores in the s + is state have been found to lead to a new mechanism ofvortex viscosity . Collective excitations of the phasedifferences between order parameters of different bands(Leggett modes) in the s + is state have peculiar phase-density nature and have been predicted to soften at the s + is critical points .The time-reversal symmetry-breaking in the s + is state is most directly manifested in spontaneous currentsaround nonmagnetic impurities or quench-induced do-main walls . The currents result in local magnetic fieldsin the superconducting phase and provide a signature ofthe s + is state. This idea has been implemented in recent µ Sr experiments on Ba − x K x Fe As aiming at detectionof the s + is state. While the first report providedno evidence, recent results are consistent with s + is state at x = 0 .
73, close to the region where the Lifshitztransition is considered to occur. Other possible exper-imental signatures of s + is superconductivity have alsobeen suggested . Crucially, the data presented in encourages one to consider s + is state in more detailwith application to Ba − x K x Fe As and propose com-plementary techniques to study its properties.In this paper, motivated by the recent observations ofcomparatively small Fermi energies and possible close-ness of s + is state to the Lifshitz transition, we analyzethe doping evolution of superconductivity in a four-bandmodel for the iron pnictides including the effects of BCS-BEC crossover physics. Similar to the BCS case, we findthat with hole doping the phase difference between super-conducting order parameters of the hole bands changesfrom 0 to π through an intermediate s + is state break- ing TRS. However, in contrast to the BCS treatment, wefind that the region of the s + is state is considerably ex-panded in the phase diagram due to additional renormal-ization of the chemical potential. In addition, the s + is state is shown to extend to the region where the electronbands are already above the Fermi level, which agreeswith recent experiments . Finally, we consider possiblesignatures of the s + is state in the STM quasiparticle in-terference patterns using a recent proposal , extendingthe formalism also to the incipient band case. The paperis organized as follows: In Sec. II we present the model,the superconducting phase diagram with degenerate andnon-degenerate hole bands is studied in Sec. III. SectionIV shows the qualitative effects of the s + is state forquasiparticle interference. We present the conclusions inSec. V. In Appendix A, we show that our solutions cor-respond to minima of the effective action by analyzingits second variation matrix. The details of the calcula-tions of the corrections to the local density of states dueto impurities for the s + is state and incipient bands arepresented in Appendix B. II. MODEL
We consider a two-dimensional model with two holebands centered at the Γ point and two identical elec-tron bands around the M point of the Brillouin zone(see Fig.1), reproducing the qualitative features ofBa − x K x Fe As band structure. For simplicity, themasses of electrons and holes are taken to be isotropicand equal. We define the Fermi energies E F h , E F h ofhole bands as the difference between the band top en-ergy and the chemical potential at zero temperature inthe absence of interactions µ and E F e = E F e for elec-tron bands as the difference of µ and the band bottomenergy. Thus if the Fermi energy of any band defined thisway becomes negative, then the corresponding pocket ofthe Fermi surface vanishes and the band is incipient (seeFig. 2).We consider superconductivity to be driven by the re-pulsive inter-band interactions between the electron andthe hole bands and between the two hole bands. More-over, we assume s -wave symmetry throughout the phasediagram and ignore anisotropy within each band. The in-traband interactions are taken to be repulsive but weakerthan the interband ones, and as we are mostly interestedin the hole-doped case we do not take the interaction be-tween the electron bands into account. We assume theinteractions to be frequency- and momentum indepen-dent up to a spin fluctuation cutoff energy Λ assumedto exceed the Fermi energies of the bands . This canbe considered as a minimal approximation for the iron-based superconductors, where generally anisotropy canbe relevant. However, as the s + is state arises solely dueto competition between two different A g -symmetric su-perconducting channels, momentum- and frequency- in-dependent interactions are sufficient to examine its prop- k E Fh2 E Fh1
E >0 Fe e k L - L ( ) e-doped E <0
Fh2
E <0
Fh1
E <0 Fe ( ) h- doped FIG. 1. Band structure of the considered 2D model. Twoparabolic hole and electron bands are around the Γ and M points, respectively. E F h , E F h ( E F e ) correspond to the Fermienergies of hole (electron) bands. The position of the k axismarks the overall chemical potential of the system. Threecases are illustrated: Undoped or moderate doping (black):both hole and electron bands cross the Fermi level. E F hi and E F e are positive. Hole-doped (red dashed): hole bands crossthe Fermi level while electron bands are incipient, E F e is nega-tive. Electron-doped (blue dashed-dotted) both hole bands areincipient. E F hi negative. Energy difference E D = E F h + E F e is independent from doping. Interactions are assumed to befrequency- and momentum- independent up to an upper en-ergy cutoff Λ. erties.We study our model within a mean field ap-proach where the chemical potential is found self-consistently as a function of the superconducting gapand temperature . This approach has been found towork well for single-band systems throughout in both theBCS and BEC cases at sufficiently low temperatures .At elevated temperatures, pairing fluctuations start toplay a very prominent role as the system goes over tothe BEC regime . On the other hand, for two-bandsystems there are indications that if one of the bands isstrongly in BCS regime E F ≫ | ∆ | then the behavior ofsystem as a whole is closer to BCS type . For multibandsuperconductors allowing for the s + is state to occur,fluctuations have been shown to lead to new phases and re-entrant phase transitions . Moreover, softeningof Leggett modes suggests that close to the s + is state boundaries fluctuations become progressively moreimportant. In our work, we shall not address the issue offluctuations further, concentrating on the effects of thechemical potential renormalization on the s + is state. G (b) M G M (a) -- - - + - ++ FIG. 2. k x , k y -cut through the Fermi surface of the foldedBrillouin-zone at (a) moderate doping and (b) large hole dop-ing. In panel (b), Fermi surface consists of hole pockets onlywhile electron bands are incipient. Plus and minus signs rep-resent the pairing symmetry. Additionally, we shall not consider corrections topairing interaction arising in the non-adiabatic limitΛ ≫ E F . These corrections result in different pre-exponential factors; however, as the s + is region occupiesa relatively small part of the phase diagram, we assumethat these corrections are independent of doping and canbe effectively absorbed into a redefinition of the factors E e and E h in Eq. (8) without altering any of the qual-itative results.The Hamiltonian of the system is then H = X k ,α,σ ξ α k c α + k σ c α k σ + X k , k ′ ,α,γ h U αγ c α + k ↑ c α + − k , ↓ c γ − k ′ ↓ c γ k ′ , ↑ + H.c. i , where { α, γ } ∈ { h , h , e , e } label the bands, and ξ h i k = − ǫ k + µ h i , (1) ξ e k = ǫ k − µ e i (2)are the bare energy dispersions for holes and electronswith ǫ k = k m , µ e i = E F ei − ( µ − µ ), and µ h i = E F hi + ( µ − µ ). We further set U h h = U h h = U h , U e e = U e e = U e , U h h = U h h = U hh , and U h i e j = U e j h i = U he . The interaction term can be writ-ten then in a convenient matrix form: H int = X k , k ′ ˆ b † k ˆ U ˆ b k ′ + U e c e − † c e − , ˆ U = U h U hh U eh U hh U h U eh U eh U eh U e / , (3)where ˆ b k = ( c h − k ↓ c h k , ↑ ; c h − k ↓ c h k , ↑ ; c e − k ↓ c e k , ↑ + c e − k ↓ c e k , ↑ ) and c e − = c e − k ↓ c e k , ↑ − c e − k ↓ c e k , ↑ . As the quantities µ α are notindependent, we shall use the following relations: µ h + µ e = E F h + E F e ≡ E D (4) µ h − µ h = E F h − E F h ≡ E h h . (5)Hole doping corresponds to decreasing E F e while keeping E h h and E D constant.Introducing the order parameters ∆ h , ∆ h , ∆ e , and∆ e , it is easy to see from (3) that there the ∆ e = − ∆ e channel is decoupled from the others and has no attrac-tion. Thus we can set ∆ e = ∆ e = ∆ e and obtainthe following set of self-consistent equations (a rigor-ous derivation can be obtained by Hubbard-Stratonovich(HS) decoupling of the field (∆ e + ∆ e ) / h = − ϑ h h L h − λ hh h L h − ∆ e λ eh L e , ∆ h = − ϑ h h L h − λ hh h L h − ∆ e λ eh L e , (6)∆ e = − ϑ e e L e − λ eh h L h + ∆ h L h ) , where we have introduced the dimensionless quantities ϑ α = N U α , λ αγ = N U αγ ,L α = Z Λ0 dǫ k tanh (cid:16) E α k T (cid:17) E α k , with N = mS π being the 2D density of states where S is the area of the 2D system and E α k = p ( ξ α k ) + | ∆ α | is the quasiparticle spectrum. Note that the system ofequations (6) is not a closed one as µ α (the chemicalpotential) depend on the total number of particles N . N is given by the amount of electrons minus that of holes.As E F α are defined for normal state at T = 0, we canexpress N through E F e and use it as a doping variable.In the superconducting state, we obtain N =4 E F e Θ( E F e ) − (cid:0) E F h + E F h (cid:1) = − Z ∞ dǫ k ( ξ e k tanh (cid:16) E e k T (cid:17) E e k + ξ h k tanh (cid:18) E h k T (cid:19) E h k + ξ h k tanh (cid:18) E h k T (cid:19) E h k ) (7) III. DOPING DEPENDENCE OF THE S + IS STATE
One of the main results of our analysis is that inclu-sion of the additional equation for the renormalizationof the chemical potential [Eq.7] has a profound effect onthe s + is superconducting phase. In Fig.3, we present atypical phase diagram obtained from numerical solutionsof the self-consistency equations (6) and (7) with an iter-ative procedure. We have also checked that the solutionsobtained do correspond to minima of the free energy byanalyzing the second variation matrix of the action, as discussed in detail in Appendix A. Encoded in color isthe phase difference φ between ∆ h and ∆ h modulo π/ E e := 2Λe (cid:18) − λ ehλhh − ϑe (cid:19) − ,E h := 2Λe − / ( λ hh − ϑ h ) . (8) pp p p LCP(T=0)UCP(T=0)
FIG. 3. Phase diagram as function of temperature ( T ) anddoping ( E F e ). Color encodes the phase difference φ betweenthe order parameters of hole bands modulo π/
2. Black dashedline marks the s + is state boundaries if one ignores (7) andtakes µ i = E F i . The parameters used are: λ hh /λ eh = 0 . λ eh = 0 . ϑ α = 0, E h h = 0, Λ = 3000, E D = 110, E e = 10 . E h = 4 . × − in arbitrary energy units. One can see that there exists a region where φ is not amultiple of π , corresponding to the s + is state. In thisstate the initial Z symmetry φ ↔ − φ of the system isbroken and a second order phase transition is expected atthe onset of φ = 0 , π . A clear thermodynamic signatureof the s + is state would then be the presence of a specificheat discontinuity below T c . To verify this, we calculate C V = − T (cid:16) d Ω dT (cid:17) µ,V , approximating the grand canonicalpotential by the value of the action at the saddle point(A2). One obtains C V = C regV ( T ) + X α ∂ | ∆ α | ∂T Z ∞− µ α N dξ T cosh √ ξ + | ∆ α | T , (9)where C regV ( T ) is a continuous function of tempera-ture. In Fig.4, we present [ C V − C regV ]( T ) calculated for E F e /E e = − . s + is transi-tion is present but its value is much smaller than theone at T c . The general reason for that can be de-duced directly from the expression (9). Below T c thevalue of the integral in the right-hand side. is sup-pressed by a factor ∼ e −| ∆ α ( T ) | /T with respect to its T/E ( C V - C V r e g ) / ( N E e ) T s+is T c FIG. 4. The discontinuous contribution to the specific heat C V − C regV [Eq. 9] as a function of temperature for E F e /E e = − . T c and the s + is state( T s + is ). The parameters of the model are taken to be thesame as in Fig.3. value at T c . Consequently, one can expect that for T s + is well below T c the discontinuity in the specific heat tobe relatively small. Together with inevitable sampleinhomogeneity , which can significantly smear T s + is due to local variation of the doping level, this provides apossible explanation for the absence of features in C V at T s + is in the recent experimental data .For every temperature the s + is region is bounded bya lower (right in Fig. (3)) and an upper (left in Fig. (3))critical points (LCP, UCP) with respect to E F e , the re-gion being largest at zero temperature and shrinking intoa single point for T = T c . The latter feature is actuallydue to E h h = 0 and in general case the s + is regionis bound from above by a temperature lower than T c asis shown in Sec.III A 4. Nevertheless, the onset tempera-tures of the s + is state in Fig.3 are still well separatedfrom T c for most dopings, consistent with the experimen-tal observations of Ref. 61. Note also that the electronbands are incipient in the s + is region; i.e.; it is locatedafter the Lifshitz transition.The black dashed line marks the phase boundary ofthe s + is state if the chemical potential renormalizationdue to (7) is neglected, i.e., µ α = E F α . It is evidentfrom Fig. 3 that the renormalization of the chemicalpotential by the superconducting gap leads to a broad-ening of the s + is region. Peculiarly, while the UCPshifts considerably when including (7), the LCP remainspractically the same. However, for T → T c LCP andUCP converge seemingly to the same point, regardless ofthe chemical potential renormalization being included ornot. To understand these qualitative features, we studythe positions of UCP and LCP analytically at T = 0 and T = T c . A. Analytical calculation for the s + is ordercritical points Our calculation will follow a route, similar to that inRef. , where LCP and UCP for variable λ eh have beenfound. Let us rewrite the system (6) in the followingform:∆ h + ∆ h = − λ hh + ϑ h h L h + ∆ h L h ) − e λ eh L e , (10)∆ h − ∆ h = − λ hh − ϑ h h L h − ∆ h L h ) , (11)∆ e = − ϑ e e L e − λ eh h L h + ∆ h L h . ) , (12)We choose the global phase of the order parameters suchthat ∆ e is real. Considering then the imaginary and realparts of the system [10-12] separately yieldssin( φ ) | ∆ h | + sin( φ ) | ∆ h | = 0 , (13) L h = L h = 2 λ hh − ϑ h , (14) L e = (cid:18) λ eh λ hh − ϑ e (cid:19) − , (15) | ∆ h | cos( ϕ ) + | ∆ h | cos( ϕ ) = − κ ∆ e , (16)where we define the dimensionless constant κκ := λ eh λ hh − ϑ h λ eh − ϑ e λ hh . (17)Note that relations (14 and 15) hold for all temperatures.
1. UCP and LCP at T=0
At zero temperature, we find for (7) and L α N =4 E F e Θ( E F e ) − (cid:0) E F h + E F h (cid:1) =2 (cid:16) µ e + p µ e + | ∆ e | (cid:17) − µ h − q µ h + | ∆ h | − µ h − q µ h + | ∆ h | , (18) L α = ln p µ α + | ∆ α | − µ α ! , (19)and thus from (14) and (15) we get | ∆ h | = q E h + 2 µ h E h , | ∆ h | = q E h + 2( µ h − E h h ) E h ,µ e = | ∆ e | − E e E e . (20)Using the relations above, we can also rewrite Eq.(18) as N = 4 E F e Θ( E F e ) − (cid:0) E F h + E F h (cid:1) = 2 | ∆ e | E e − µ h − E h − µ h − E h , leading to E F e = 2 µ e + E e − E h . (21)
2. Identical hole bands
To study the effect of chemical potential renormaliza-tion let us first consider the hole bands to be identical( E h h = 0). It is evident from (20) that | ∆ h | = | ∆ h | and from (13) we find φ h = − φ h ≡ φ/
2. The system[10-12] has then three types of solutions:1. s + − h h : ∆ h = − ∆ h ( φ = π );2. s + is : | ∆ h | = | ∆ h | φ = 0 , π ;3. s ++ h h : ∆ h = ∆ h ( φ = 0).LCP and UCP are then found by matching the s + is solution with s + − h h and s ++ h h one, respectively. LCP : s + is state coincides with s + − h h when φ = π . Itfollows then, that ∆ e = 0 and from (20) and (21) we get: µ e min ≡ − E e ,E minF e ≡ − E e − E h . Thus, LCP is shifted to larger hole dopings due tothe chemical potential renormalization. However, for E h ≪ E e the correction to the LCP is insignificant,which is consistent with Fig. 3. It is important to notethat according to this result the electron bands arealways incipient at the LCP. U CP : The transition from s + is to s ++ h h takes placefor φ = 0. From (16) it follows that there is a sign changeof the order parameter between hole and electron pock-ets. Inserting then the values of | ∆ h | , | ∆ e | obtained from(14), (15), into (16), we arrive at1 = κ p E e + 2 µ e E e p E h + 2( E D − µ e ) E h , (22)where we used µ h = E D − µ e . We can now find thecorresponding µ e : µ e max ≡ E h + 2 E D E h − κ E e κ E e + 2 E h .E max F e ≡ (1 − κ ) E e E h + E h + 4 E D E h − κ E e κ E e + 2 E h . Collecting the results for µ LCPe and µ UCPe we find thatthe s + is state is confined to the region: − E e < µ e < E h + 2 E D E h − κ E e κ E e + 2 E h . (23)Note that for E D ≫ κ E e /E h , E F e is positive atUCP, and s + is state extends into the moderately dopedregion where electron bands cross the Fermi energy. InFig.5, we present the evolution of µ e min , µ e max , E min F e , and E max F e for ϑ α = 0 as functions of the ratio λ hh /λ eh . FIG. 5. The lower and the upper boundaries of the s + is re-gion for chemical potential µ e min and µ e max and doping E maxF e and E minF e as functions of the ratio λ hh /λ eh taking ϑ α = 0. One can see, that the s + is region shrinks with decreas-ing λ hh . As for small values of λ hh we have E h ≪ E e ,and we can expand µ e max in E h E e : µ e max ≈ − E e + 12 κ E h E e (2 E D + E e ) ≥ µ e min . (24)Equation (24) shows that µ e max > µ e min holds up to ar-bitrary small values of λ hh . As a consequence, whateverare the coupling constants, the transition from s ++ h h to s + − h h always occurs through the s + is state at T = 0.However, the width of the s + is region is sensitive to thevalues of the coupling constants.Let us now compare the effect Eq.(7) has on LCPand UCP. For the true doping variable E F e , we need totake into account the chemical potential renormalizationcoming from Eq. (7). In the physically relevant limit E h ≪ E e , we have µ e min − E h < E F e ≤ µ e max − E h
2+ 12 κ E h E e (2 E D + E e ) . (25)The observation that the correction to the upper com-pared to the lower boundary is much more pronouncedcan be explained by Eq.(25). Using the values taken inFig. 3, we see that the second term in the upper bound-ary shift clearly dominates over − E h /
3. Effect of hole band offset on the s + is state Now we consider the case E h h = E F h − E F h = 0.It is seen from (21) that the corrections to E F e due to E h h are simply: δE F e = 2 δµ e = δ | ∆ e | E e .LCP : We have φ = φ + π and it immediately fol-lows from (13) that φ is either 0 or π and from (16)that | ∆ e | = κ ( | ∆ h | − | ∆ h | ) ≈ E h h E h κ ∆ h . Note thatin the case E h ≪ ∆ h (BCS limit for hole bands) thisexpression is valid even for E h h ∼ ∆ h . We obtain then δE LCPF e ≈ κ E h h E h E e (2 E D + E h + E e ) . We remark that this result remains valid even for mod-erate values of E h h , if E h ≪ E e , E D . U CP : We have φ = φ and it immediately followsfrom (13) that φ is either 0 or π and from (16) that | ∆ e | = κ ( | ∆ h | + | ∆ h | ). The result is δE UCPF e ≈ − κ E h h E h E e . This correction E h h is not suppressed by E D , but com-paring with (25) we see that it is still much smaller thanthe effect of the chemical potential renormalization.Overall, the effect of the hole band offset at T = 0 is toshrink the s + is region; however, for E D ≫ E e , E h thiseffect is insignificant in comparison to the one introducedby the chemical potential renormalization.
4. Critical doping at T = T c At T = T c we can linearize the self-consistency equa-tions (14) and (15) and obtain (assuming | µ e,h ,h | ≫ T c ):ln (cid:18) . Λ µ h T c (cid:19) = ln (cid:18) . Λ µ h T c (cid:19) = 2 λ hh − ϑ h ,L e = ln (cid:18) Λ | µ e | (cid:19) = (cid:18) λ eh λ hh − ϑ e (cid:19) − , with the neglected terms being exponentially small ∼ exp {−| µ e,h ,h | /T c } . It follows then that s + is state per-sists up to T c only if µ h = µ h . Moreover, it is confinedto a single point: µ min e = µ max e := − E e . (26) T c is equal to 0 . √ E h µ h and one can see that for theparameters used in Fig.3 T c is indeed smaller than both µ h and µ e . From (7) taken at T c we find that: E F e = µ e + O (cid:16) T c e −| µe | Tc (cid:17) (27)and thus E min F e = E max F e := − E e O (cid:16) T c e −| µα | Tc (cid:17) . (28)We see that the s + is region shrinks to a single pointat T c for equal hole bands. The doping level for thispoint is close to − E e / E h ≪ E e the s + is point at T c and LCP at zero temperature should be close to eachother, as is the case in Fig. 3. For unequal hole bandswe find that s + is state onset is below T c for all dopings.Such a separation between the s + is state onset and T c is actually observed in a recent experiment ). IV. STM-SIGNATURES OF S + IS -STATE Recently it has been proposed that the sign structureof the order parameter in a multiband system can be ex-tracted from the Fourier transform of the local densityof states (QPI pattern near an impurity in the super-conducting state). The QPI intensity integrated overthe wave vectors corresponding to scattering betweentwo bands has been shown to have a dependence onenergy very different for s + − and s ++ scenarios, lead-ing to a strong enhancement of the integrated responsefor the s + − but not the s ++ case. This method hasbeen recently successfully applied to confirm the sign-changing nature of the order parameter in FeSe , wherethe superconducting gaps are also extremely anisotropic.More recently, the method has been also applied to(Li − x Fe x )OHFe − y Zn y Se with only electron Fermi sur-face pockets . For the 122 doped systems, such an ex-periment has not been yet performed; however, one wouldexpect that such a test is feasible in these compoundsas well, given the availability of the high-quality STMdata .Here we show that the QPI patterns contain informa-tion on the phase difference between the order parame-ters of different bands for the case when it is not 0 or π and even if one of the bands is incipient. Based onthe results, we provide several methods for detection ofthe s + is state. As the discussion here has qualitativecharacter, we present the results obtained in the Born ap-proximation assuming weak impurity potential at T = 0and ignore anisotropy. Here we consider nonmagneticand Andreev impurities only, as the magnetic ones donot contribute to the density of states in the Born ap-proximation. Let us first concentrate on the scatteringbetween the two hole bands in the s + is state. The calcu-lations are similar to those in Ref. and we present thedetails in Appendix B. It is important to notice that fora single wavevector q corresponding to interband scat-tering, there are contributions from both h → h h → h t τ , with τ i being matrices in Nambu space. One obtains: δρ h h ch ( ω ) = 2 t πρ h ρ h sgn( ω ) × Im 2 ω − | ∆ h || ∆ | cos( φ − φ ) p ω − | ∆ h | + iδ p ω − | ∆ h | + iδ . (29) d r ( w ) rr c h3h1h2/th1h2 FIG. 6. Interband density of states δρ h h ch ( ω ) /t ρ h ρ h [(29)with δ = 10 − ] for scattering between the hole bands. Insetshows φ − φ as a function of E F /E e for T = 0 obtainedfrom (6)and (7) using the same parameters as in Fig.3 exceptfor E h h = 90 in arbitrary units. Arrows mark the dopingsfor which δρ curves are presented. In Fig.6 we present the evolution of ρ h h ch ( ω ) throughthe s + is state for ω >
0. The doping dependence ofthe order parameters has been obtained from Eqs. (6)and (7). The parameters of the model have been takento yield | ∆ h / ∆ h | ≈ . As . One can see thatthe result evolves continuously from s ++ -like before the s + is region to s + − -like after it. Obtaining the valueof φ − φ requires then, in principle, fitting the wholecurve, which can be rather complicated provided thatthe exact impurity potential is not known. On the otherhand, the edges of the region where the contribution isnonzero (corresponding to ω = | ∆ h | , ω = | ∆ h | ) arepractically unchanged throughout the s + is state, whilethe form of the curve in between changes dramatically,meaning that QPI pattern is quite sensitive to the valueof φ − φ . Moreover, as φ − φ depends on tempera-ture (see Fig.3), abrupt changes in the QPI pattern withtemperature could be also considered as a signature ofthe s + is state.Local suppression of superconductivity by animpurity or an individual vortex in a disorderedvortex lattice constitutes an Andreev scatterer. Theimpurity potential in this case is given by t A τ for thecase when order parameters can be taken real. In the s + is state this is not so, so we use the general form t A ( ατ + βτ ) with real α, β and fix their values so thatthe answer is gauge invariant and goes over to t A τ inthe s ++ h h limit. The result is then: δρ h h A ( ω ) = 4 t A πρ ρ Im | ω | ( | ∆ | + | ∆ | ) cos[( ϕ − ϕ ) / p ω − | ∆ | + iδ p ω − | ∆ | + iδ . (30)An interesting feature of this result is that the responsevanishes in the s + − h h case but is finite throughout the s + is state. This leads to a sufficient qualitative criterionfor the s + is state detection: Observation of an s + − -likepattern near a charge impurity together with a nonzero δρ inter A ( ω ) near an Andreev scatterer at the same dopingsuggests the presence of the s + is state.Let us now move onto e − h scattering. As the electronbands in the s + is state are likely to be incipient or have | µ e | ∼ | ∆ e | , the BCS-like expressions for the momentum-integrated Green’s function are no longer valid. However,in 2D the momentum integration can be performed ex-actly (assuming a large energy cutoff scale Λ). For acharge impurity, only the component odd in frequencycontains the phase information. The result is ρ eh ch ( ω ) odd = − t ρ e ρ h sgn( ω )Im ( iπ + log " − µ e + p ω − | ∆ e | + iδµ e + p ω − | ∆ e | + iδ i (2 ω − | ∆ h || ∆ e | cos( φ h − φ e ))2 p ω − | ∆ h | + iδ p ω − | ∆ e | + iδ ) + t ρ ρ sgn( ω ) π Im (cid:26) log (cid:20) Λ µ h − ω + | ∆ h | − iδ (cid:21) log (cid:20) Λ µ e − ω + | ∆ e | − iδ (cid:21)(cid:27) . (31)The results for different doping levels are presentedin Fig.7 for the scattering between the electron and thesmaller hole (h2) bands. One can see that the square-root singularity for µ e > ω = | ∆ e | ceases to exist in the incipient case µ e <
0. However, a weaker logarith-mic singularity is present then at ω = p | ∆ e | + µ e . Thecurve evolves throughout s + is state in a similar mannerto h1-h2 scattering. The results for h1-e scattering are d r ( w ) rr c h3h2e/teh FIG. 7. Doping evolution of the interband density of states δρ h ech ( ω ) /t ρ h ρ e [(31) with δ = 10 − ] for scattering betweenthe electron and the smaller hole (h2) bands. (a) Dopingsclose to the Lifshitz transition for the electron band. (b)Dopings around the s + is region. Red dashed line is forthe same doping as line 4 but with ∆ e = 0. Inset in pabel (a)shows φ − φ as a function of E F /E e obtained from (6) and(7) using the same parameters as in Fig.6. Arrows mark thedopings for which δρ curves are presented comparable, except that at dopings beyond the s + is re-gion the phase difference between the order parametersat h1 and e is equal to π instead of 0. This suggests acomplementary way to measure φ h and φ h by extract-ing φ h − φ e and φ h − φ e from the e-h interband densityof states.One could ask if a presence of a superconducting gapat the incipient electron band is at all distinguishablein the result. In Fig.7(b) we compare the patterns forh2-e scattering at the same doping level with finite ∆ e obtained from self-consistency equations (solid blue line)and with ∆ e = 0. The qualitative difference is due tothe order parameter on h2 having the same sign as theone on electron band leading to a sign-changing s ++ -likepattern. Taking ∆ e = 0 however leads to an answernot crossing zero and thus the singularity at ω = ∆ h changes sign. Thus STM-QPI can be also a useful tool fordetecting superconducting gaps at incipient bands, whichis a rather nontrivial task in, e.g., ARPES experiments. V. CONCLUSION
We have studied superconducting pairing in a four-band model as a function of hole doping, where the elec-tron bands can become incipient after a Lifshitz transi-tion. The interactions have been chosen to study the in-terplay between two superconducting channels of s -wavesymmetry: With a sign change between the hole andelectron pockets ( s + − eh ) [phases at the two hole (electron)pockets being the same], dominant close to optimal dop-ing, and with a sign change between the hole pockets( s + − h h ), leading in the overdoped region.The phase diagram of the model has been analyzedin the mean-field approximation including BCS-BECeffects . We have found that the crossover from s + − eh to s + − h h at low temperatures always occurs via an inter- mediate s + is state, characterized by a phase differencebetween the order parameters at the two hole pocketsnot equal to a multiple of π . We have shown that the s + is state always extends into the region where electronbands are incipient. Additionally, heat capacity anoma-lies at the transition to the s + is state have been found tobe suppressed at temperatures below T c . These findingsare in line with the recent experimental data suggestingthe presence of s + is state in overdoped Ba − x K x Fe As at x = 0 . , close to the doping where the Lifshitztransition is expected to occur .A large broadening of the s + is -region has been founddue to the chemical potential renormalization in the su-perconducting state (BCS-BEC corrections), previouslyunaccounted for. This is encouraging for future exper-iments on Ba − x K x Fe As , as the doping is hard tocontrol precisely due to clustering of K atoms in thelattice . An energy offset between the Γ-centered holepockets, on the contrary, narrows the s + is -region, andmakes it disappear around T c . The narrowing effect hasbeen found, however, to be strongly suppressed for suf-ficiently large enough hole pockets. These observationsgive additional support to the presence of the s + is statein the hole-overdoped pnictides.Finally, we have studied the interband density of statesin the s + is state and to proposed several complemen-tary approaches to detect this state in future STM-QPIexperiments. Our results also suggest a general methodto detect superconducting gaps on incipient bands. ACKNOWLEDGMENTS
We acknowledge the discussions with A.V. Chubukov,and P.J. Hirschfeld. J.B. and I.E. were supported by thejoint DFG-ANR Project (ER 463/8-1). K.B.E. acknowl-edges the financial support of the Ministry of Educationand Science of the Russian Federation in the frameworkof Increase Competitiveness Program of NUST “MISiS”(Nr. K2-2014-015).J.B. and P.A.V. contributed equally to this work.
Appendix A: STABILITY OF THE s + is SOLUTION
To show that the obtained solutions of Eqs.(6) and (7)are thermodynamically stable, one needs to show thatthe second variation matrix of the action is positive defi-nite. An expression for S can be obtained by decouplingthe interaction (3) in the path integral. A problem arisesthen because the conventional Hubbard-Stratonovich de-coupling yields a diverging integral in the case of repul-sive interaction, and the matrix in (3) has one positive(repulsive) eigenvalue (let us consider for simplicity thecase U e = 0, where the ∆ e = − ∆ e channel does notarise). This has been noted previously in Ref. and amethod to take the repulsive channel into account has0been proposed for a two-band case with interband inter-action in Ref. . We use a different procedure, similar tothe one used in Ref. , easily generalizable for the multi-band case and arbitrary interactions. First of all, one canbring the interaction Hamiltonian to the form P i λ i ˆ e † i ˆ e i ,where ˆ e i is an eigenvector of matrix ˆ U composed of op-erators ˆ b i . The negative eigenvalues are then decoupledas usual, while for positive ones we use the following re- lation: e − λb ∗ b = 1 e Z Z D ∆ ∗ D ∆ e − | ∆ | λ − ib ∗ ∆ − i ∆ ∗ b , e Z = Z D ∆ ∗ D ∆ e − (∆ ∗ + ib ∗ λ ) λ (∆+ iλb ) , where R D ∆ ∗ D ∆ ≡ R D [Re∆] D [Im∆]. e Z can be shownto be independent of b by shifting the integration contourinto the complex plane (Re∆ → Re∆ − iλ Re b , Im∆ → Im∆ − iλ Im b ): e Z = Z D ∆ ∗ D ∆ e − | ∆ | λ . Consequently, the decoupling for the full 3 × e − ~ R ~ β dτb † ( τ ) Ub ( τ ) = 1 Z Z D ˜∆ D ˜∆ ∗ exp ~ Z ~ β dτ − b † ( τ ) R T i ˜∆ − ˜∆ † i Rb ( τ ) + ˜∆ † /λ /λ − /λ ˜∆ ,Z = Z D ˜∆ D ˜∆ ∗ exp T ˜∆ † /λ /λ − /λ ˜∆ , (A1)where λ , λ < λ > U and R is an orthogonal matrix that satisfies: RU R T = λ λ λ . In principle, one can next integrate the action over thefermionic fields to obtain an action depending only on˜∆ and ˜∆ ∗ . The convergence of the resulting integral issolely determined by the last term in the exponential inthe right-hand side. of Eq.A1 and one can evidently seethat the described procedure yields a converging integral.The resulting saddle-point equations are rather cumber-some, but their solution is actually related to the one ofEq.(6): ˜∆ = − i R ∆ , ˜∆ ∗ = − i R ∆ ∗ , where ∆ = (∆ h , ∆ h , ∆ e ) T . It is evident that ˜∆ ∗ =( ˜∆ ) ∗ , meaning that the saddle point of the action is situated in the complex continuation of the integrationrange. However, as the integral over ˜∆ , ˜∆ ∗ convergesat ∞ , one can shift the integration contour such that itincludes the point ( ˜∆ , ˜∆ ∗ ) without altering the result.The value of the action at the saddle point is most easilyexpressed through the quantities ∆ α and is given by S = − ∆ † U − ∆ − T X ω n , k ,α ln (cid:2) ω n + ( ξ α k ) + | ∆ α | (cid:3) , (A2)where α ∈ { h , h , e , e } and ω n = (2 n + 1) πT .We shall evaluate the second variation matrix of the ac-tion with respect to real and imaginary parts of the HSfields ˜∆ r , ˜∆ im defined through ˜∆ = ˜∆ r + i ˜∆ im ; ˜∆ ∗ =˜∆ r − i ˜∆ im . Once again, it is easier to evaluate the deriva-tives with respect to ∆ ai and use the transformation ∂ S∂ ˜∆ ai ∂ ˜∆ bj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜∆= ˜∆ = X kl ∂ ∆ ak ∂ ˜∆ ai ∂ S∂ ∆ ak ∂ ∆ bl (cid:12)(cid:12)(cid:12)(cid:12) ∆=∆ ∂ ∆ bl ∂ ˜∆ bj , where a, b = { r, im } . For the considered model, thisyields1 E /E
Fe 0e
E /E
Fe 0e h i / N h i / N a) b) FIG. 8. Panels (a) and (b): Eigenvalues of the second variation matrix of the action ∂ S∂ ∆ ai ∂ ∆ bj at T = 0. Vertical dashed linesmark the boundaries of the s + is state. ∂ S∂ ˜∆ ai ∂ ˜∆ bj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = i R ∂ S∂ ∆ ai ∂ ∆ bj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R T i ,∂ S∂ ∆ ri ∂ ∆ rj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − U − − N L h + (∆ rh ) L ′ h | ∆ h | L h + (∆ rh ) L ′ h | ∆ h | L e + 2 (∆ re ) L ′ e | ∆ e | ,∂ S∂ ∆ ri ∂ ∆ imj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − N ∆ rh ∆ imh L ′ h | ∆ h | ∆ rh ∆ imh L ′ h | ∆ h | ∆ re ∆ ime L ′ e | ∆ e | ,∂ S∂ ∆ imi ∂ ∆ imj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − U − − N L h + (∆ imh ) L ′ h | ∆ h | L h + (∆ imh ) L ′ h | ∆ h | L e + 2 (∆ ime ) L ′ e | ∆ e | (A3)where L ′ α = ∂L α /∂ | ∆ α | and | means value taken atthe extremum. In Figs. 8(a) and 8(b) we present theeigenvalues η i of the resulting 6 × s + is state correspondingto a soft Leggett mode. All other eigenvalues are clearlypositive with the largest ones (black dashed lines) in Fig.8being due to the repulsive s ++++ channel. Additionally,while the eigenvectors are complex, they can be normal- ized, and the corresponding Jacobian is real and has anabsolute value of 1.Overall, apart from the overall phase mode, present inall superconductors and Leggett mode, softening at theboundaries of the s + is state, we have shown that thesecond variation matrix of the action is positive definiteand the resulting Gaussian integrals converge, provingthat the solutions discussed in the main text correspondto a minimum of the free energy due to (cid:16) d Fd ∆ ai d ∆ bj (cid:17) T,N = (cid:16) ∂ Ω ∂ ∆ ai ∂ ∆ bj (cid:17) T,µ = µ ( T ) .2 Appendix B: LOCAL DENSITY OF STATES NEAR AN IMPURITY FOR s + is STATE AND ROLE OFINCIPIENT BANDS We consider the spatially resolved density of states for a multiband superconducting system with a single impurityat T = 0. Generally one can write the coherent part of Green’s function as [we do not take the effects of interactionsuch as quasiparticle residue ( Z ) renormalization and finite quasiparticle lifetime into account] G ( r , r ′ , ω ) = − i Z d ( t − t ′ ) e iω ( t − t ′ ) h T Ψ( r , t )Ψ † ( r ′ , t ′ ) i = X n ψ n ( r ) ψ ∗ n ( r ′ ) ω − ǫ n + iδ sgn( ǫ n ) , where ψ n ( r ) are eigenfunctions of the Hamiltonian including the impurity potential and the spin indices are suppressed.The density of states is given by ρ ( r , ω ) = X n | ψ n ( r ) | δ ( ω − ǫ n ) = − sgn( ω ) π Im[ G ( r , r , ω )] . (B1)One can relate the Fourier transform of ρ ( r , ω ) to the Green’s functions in momentum space G ( k , k ′ , ω ): ρ ( q , ω ) = Z d r e i qr ρ ( r , ω ) = − sgn( ω ) π Z d r e i qr Im (cid:20)Z d k d k ′ (2 π ) d e i ( k − k ′ ) r G ( k , k ′ , ω ) (cid:21) = − sgn( ω ) π Z d k (2 π ) d Im[ G ( k , k + q , ω )] , (B2)where we have assumed inversion symmetric scattering G ( k , k + q , ω ) = G ( k , k − q , ω ).
1. Green’s functions for SC state
Introducing Gor’kov-Nambu spinors ˆΨ = (cid:18) Ψ k , ↑ Ψ †− k ,ω, ↓ (cid:19) we haveˆ G ( k , ω ) = − i Z d ( t − t ′ ) d ( r − r ′ ) e iω ( t − t ′ ) − i k ( r − r ′ ) h T ˆΨ( r , t ) ˆΨ † ( r ′ , t ′ ) i = (cid:18) ω + ξ k ∆∆ ∗ ω − ξ k (cid:19) ω − p ξ k + | ∆ | + iδ )( ω + p ξ k + | ∆ | − iδ ) . (B3)Another useful quantity is the momentum-integrated Green’s function P k G ( k , ω ). Assuming BCS limit ( µ ≈ E F ≫ ∆), we linearize the spectrum near Fermi surface. One obtains X k ˆ G ( k , ω ) = ρ Z ∞−∞ dξ (cid:18) ω ± ξ ∆∆ ∗ ω ∓ ξ (cid:19) ω − ξ − | ∆ | + iδ = − iπρ (cid:18) ω ∆∆ ∗ ω (cid:19)p ω − | ∆ | + iδ , (B4)where ± is for electron-hole bands. Note that the iδ in the denominator determines the sign of the imaginary partof the square root for ω < ∆. Let us use the expressions (B2) to calculate the density of states for a uniformsuperconducting system: ρ ↑ (0 , ω ) = ρ ↓ (0 , ω ) = − sgn( ω ) π Im[ ˆ G ( ω )] = ρ Im " i | ω | p ω − | ∆ | + iδ = ( | ω | √ ω −| ∆ | if | ω | > | ∆ | | ω | < | ∆ | .
2. Incipient band and e-h assymetry
We evaluate now the momentum-integrated Green’s function for a quadratic band in 2D exactly (assuming onlyΛ ≫ ∆ , ω, | µ | ) to account for the smallness of the Fermi energy: X k ˆ G ( k , ω ) = − ρ log " Λ − µ − p ω − | ∆ | + iδ − log " Λ − µ + p ω − | ∆ | + iδ ω ∆∆ ∗ ω (cid:19) p ω − | ∆ | + iδ ∓ ρ (cid:20) Λ µ − ω + | ∆ | − iδ (cid:21) τ . Here one should take care with the sum of logarithms of complex argument. Before joining the logarithms, one needsto ensure that the difference between the phases of the two arguments lies in the interval ( − π, π ). This is not so forlarge µ > π . One can work around this with a following trick:log " Λ − µ − p ω − | ∆ | + iδ − log " Λ − µ + p ω − | ∆ | + iδ == iπ + log " Λ µ + p ω − | ∆ | + iδ − log " Λ − µ + p ω − | ∆ | + iδ = iπ + log " − µ + p ω − | ∆ | + iδµ + p ω − | ∆ | + iδ . One can see that in this case the correct answer 2 iπ is recovered for µ → + ∞ . However, for an incipient band µ < π . One obtains the result X k ˆ G ( k , ω ) µ< = − ρ log " | µ | + p ω − | ∆ | + iδ | µ | − p ω − | ∆ | + iδ ω ∆∆ ∗ ω (cid:19) p ω − | ∆ | + iδ ∓ ρ (cid:20) Λ µ − ω + | ∆ | − iδ (cid:21) τ . (B5)The square root singularity at ω = | ∆ | is replaced here by a logarithmic one at ω = p | ∆ | + µ . Such a singularityalso appears in the µ > µ we shall simply useBCS answer for those.
3. q-integrated δρ and band space The correction to the Green’s function due to the impurity potential takes the form: δ ˆ G ( k , k + q , ω ) = ˆ G ( k , ω ) ˆ T ( k , k + q , ω ) ˆ G ( k + q , ω ) , (B6)where ˆ T ( k , k + q , ω ) satisfiesˆ T ( k , k + q , ω ) = ˆ V ( q ) + X k ˆ V ( k − k ) ˆ G ( k , ω ) ˆ T ( k , k + q , ω ) , (B7)ˆ V ( q ) being the Fourier component of the impurity potential. The Green’s functions ˆ G ( k , ω ) are concentrated in k space near the Fermi surface at values | ξ k | < p ω + | ∆ | (B4) or | ξ k | < Γ, where Γ is the inverse scattering time dueto self-energy corrections. One can use this fact to discriminate between inter- and intraband scattering. Using (B2)one has δρ ↑ ( q , ω ) = − sgn( ω ) π Z d k (2 π ) d Im[ ˆ G ( k , ω ) ˆ T ( k , k + q , ω ) ˆ G ( k + q , ω )] . Note that in the case when impurity is spinless δρ ↑ ( q , ω ) = δρ ↓ ( q , ω ), which is easy to see by taking Gor’kov-Nambuspinors with a different choice of spins, which leads to exactly the same Green’s functions.Let us consider the situation when q corresponds to the distance between two bands on the Fermi surface. If theinterband scattering wave vector is larger than the momentum extension of the Green’s functions within each band4 FIG. 9. An example for q -integration for a centrosymmetric two-band system. Green regions represent the extents of Green’sfunction in momentum space, and k corresponds to the band b
1, then one can take ˆ G ( k , ω ) → ˆ G b ( k , ω ) and ˆ G ( k + q , ω ) → ˆ G b ( k + q , ω ).Consequently, integrating the result over q such that k + q is in b
2, one has X q ≈ q inter δρ ↑ ( q , ω ) ≈ − sgn( ω ) π Im[ ˆ G b ( ω ) ˆ T ( k , k , ω ) ˆ G b ( ω ) + ˆ G b ( ω ) ˆ T ( k , k , ω ) ˆ G b ( ω )] , where k and k are approximate positions of the two bands in momentuum space. Note that because the integrationover k is performed over all the Brillouin zone and because same set of q ’s gives both b → b b → b q ’s (see also Fig.B 3). In the same way,one can treat the case when q correspond to intra-band scattering. Then, assuming that the dependence of ˆ V ( q ) andˆ T ( k , k + q , ω ) can be neglected for variations of q of the order of the Fermi pocket size, one can perform the integralin (B7) to obtain T ( ω ) = [1 − V G ( ω )] − V , (B8)where all the quantities are now matrices also in band space with V µν = ˆ V ( q µν ) and G ( ω ) µν = δ µν ˆ G µ ( ω ). Let usnow present the final expression for centrosymmetric δρ for interband scattering for impurities not acting on spin: δρ inter µν ( ω ) ≈ − ω ) π Im[ ˆ G µ ( ω ) T ( ω ) µν ˆ G ν ( ω ) + ˆ G ν ( ω ) T ( ω ) µν ˆ G µ ( ω )] = − ω ) π Im Tr (cid:20) τ + τ (cid:16) ˆ G µ ( ω ) { T ( ω ) } µν ˆ G ν ( ω ) + ˆ G ν ( ω ) { T ( ω ) } µν ˆ G µ ( ω ) (cid:17)(cid:21) , (B9)where trace is taken over the Gor’kov-Nambu indices and summation over band indices µ, ν is not implied.
4. Born approximation
To consider the qualitative features in the interband DOS, we consider the lowest-order approximation where { T ( ω ) } µν = ˆ V ( q µν ). We consider impurities with different structure in Nambu space corresponding to different typesof scatterers.5 a. Charge impurity Here we consider the impurity acting on the charge density P q ,σ V ( q )Ψ † k ,σ Ψ k ,σ → t τ . One has ρ inter ( ω ) = − t sgn( ω ) π Im Tr (cid:20) τ + τ (cid:16) ˆ G ( ω ) τ ˆ G ( ω ) + ˆ G ( ω ) τ ˆ G ( ω ) (cid:17)(cid:21) . Evaluating the traces yields: ρ inter ch ( ω ) = 2 t πρ ρ sgn( ω )Im 2 ω − ∆ ∆ ∗ − ∆ ∆ ∗ p ω − | ∆ | + iδ p ω − | ∆ | + iδ (B10)Let us also consider scattering between an incipient electron band and a hole band. We also include the e-hasymmetry effects for both but take µ h ≫ ∆ , ω . One obtains using the Green’s functions (B5), ρ eh ch ( ω ) = − t ρ e ρ h sgn( ω )Im ( log " | µ e | + p ω − | ∆ e | + iδ | µ e | − p ω − | ∆ e | + iδ i (2 ω − ∆ h ∆ ∗ e − ∆ e ∆ ∗ h )2 p ω − | ∆ h | + iδ p ω − | ∆ e | + iδ ) + t ρ ρ sgn( ω ) π Im (cid:26) log (cid:20) Λ µ h − ω + | ∆ h | − iδ (cid:21) log (cid:20) Λ µ e − ω + | ∆ e | − iδ (cid:21)(cid:27) − t ρ ρ Im i | ω | log h Λ µ e − ω + | ∆ e | − iδ ip ω − | ∆ h | + iδ − log " | µ e | + p ω − | ∆ e | + iδ | µ e | − p ω − | ∆ e | + iδ | ω | log h Λ µ h − ω + | ∆ h | − iδ i p ω − | ∆ e | + iδ . The last term is even in ω and can be thus omitted by considering ρ odd = [ ρ ( ω ) − ρ ( − ω )] /
2. It contains onlyinformation on the absolute value of the order parameters. The second term remains in the odd-frequency part, butis zero for ω < p | ∆ e | + µ e . It should affect the curve shape for ω > p | ∆ e | + µ e though. The answer for the oddpart is then: ρ eh ch ( ω ) odd = − t ρ e ρ h sgn( ω )Im ( log " | µ e | + p ω − | ∆ e | + iδ | µ e | − p ω − | ∆ e | + iδ i (2 ω − ∆ h ∆ ∗ e − ∆ e ∆ ∗ h )2 p ω − | ∆ h | + iδ p ω − | ∆ e | + iδ ) ++ t ρ ρ sgn( ω ) π Im (cid:26) log (cid:20) Λ µ h − ω + | ∆ h | − iδ (cid:21) log (cid:20) Λ µ e − ω + | ∆ e | − iδ (cid:21)(cid:27) . (B11) b. Andreev impurity Andreev impurity is ∼ t τ if order parameters on both bands can be taken real. In s + is state this is not so, soto ensure gauge invariance we perform the calculation for ατ + βτ = (cid:18) Z ∗ Z (cid:19) , where Z = α + iβ and α, β arereal. Evaluating the traces, one obtains ρ inter A ( ω ) = 2 πρ ρ | ω | Z ∆ + Z ∗ ∆ ∗ + Z ∆ + Z ∗ ∆ ∗ p ω − | ∆ | + iδ p ω − | ∆ | + iδ . To ensure gauge invariance, the answer should be invariant with respect to ∆ → ∆ e iϕ , ∆ → ∆ e iϕ . Taking Z = t A exp {− ( ϕ + ϕ ) / } leads then to a gauge-invariant answer with the correct limit Z → s ++ state. Theresulting expression is ρ inter A ( ω ) = 4 t A πρ ρ Im | ω | ( | ∆ | + | ∆ | ) cos[( ϕ − ϕ ) / p ω − | ∆ | + iδ p ω − | ∆ | + iδ (B12) c. Spin impurity To consider a spin impurity ˆ V ∼ m · σ , we introduce the four-component Balian-Werthammer spinors:ˆΨ = Ψ k , ↑ Ψ k , ↓ Ψ †− k ,ω, ↓ − Ψ †− k ,ω, ↑ . × δρ isquite similar: ρ inter s ( ω ) = − t s sgn( ω ) π Im Tr (cid:20) τ + τ (cid:16) ˆ G ( ω ) m · σ ˆ G ( ω ) + ˆ G ( ω ) m · σ ˆ G ( ω ) (cid:17)(cid:21) , only instead of 2 in front we evaluate an additional trace over spin indices. It is clear now, that the answer for fullDOS correction is 0; however, one can also calculate the spin-resolved one: ρ inter ↑ ( ↓ ) ( ω ) s = − t s sgn( ω ) π Im Tr (cid:20) σ ± σ τ + τ (cid:16) ˆ G ( ω ) m · σ ˆ G ( ω ) + ˆ G ( ω ) m · σ ˆ G ( ω ) (cid:17)(cid:21) = ∓ m z t s sgn( ω ) π Im Tr (cid:20) τ + τ (cid:16) ˆ G ( ω ) ˆ G ( ω ) + ˆ G ( ω ) ˆ G ( ω ) (cid:17)(cid:21) = ± m z t s πρ ρ sgn( ω )Im 2 ω + ∆ ∆ ∗ + ∆ ∆ ∗ p ω − | ∆ | + iδ p ω − | ∆ | + iδ , where only m z has entered the expression due to the choice of the quantization axis. Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono,J. Am. Chem. Soc. , 3296 (2008). D. C. Johnston, Adv. Phys. , 803 (2010). J. Paglione and R. L. Greene, Nat. Phys. , 645 (2010). P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep.Prog. Phys. , 124508 (2011). H. Hosono and K. Kuroki, Phys. C , 399 (2015). H. Miao, T. Qian, X. Shi, P. Richard, T. K. Kim,M. Hoesch, L. Y. Xing, X.-C. Wang, C.-Q. Jin, J.-P. Hu,and H. Ding, Nat. Commun. , 6056 (2015). K. Okazaki, Y. Ito, Y. K. Ota, T. Y. Shimojima, T. Kiss,S. Watanabe, C.-T. Chen, S. Niitaka, T. Hanaguri, H. Tak-agi, A. Chainani, and S. Shin, Sci. Rep. , 4109 (2014). A. Charnukha, S. Thirupathaiah, V. B. Zabolotnyy,B. B¨uchner, N. D. Zhigadlo, B. Batlogg, A. N. Yaresko,and S. V. Borisenko, Sci. Rep. , 10392 (2015). T. Terashima, N. Kikugawa, A. Kiswandhi, E.-S. Choi,J. S. Brooks, S. Kasahara, T. Watashige, H. Ikeda,T. Shibauchi, Y. Matsuda, T. Wolf, A. E. B¨ohmer,F. Hardy, C. Meingast, H. v. L¨ohneysen, M.-T. Suzuki,R. Arita, and S. Uji, Phys. Rev. B , 144517 (2014). M. D. Watson, T. K. Kim, A. A. Haghighirad,N. R. Davies, A. McCollam, A. Narayanan, S. F.Blake, Y. L. Chen, S. Ghannadzadeh, A. J. Schofield,M. Hoesch, C. Meingast, T. Wolf, and A. I. Coldea,Phys. Rev. B , 155106 (2015). S. Kasahara, T. Watashige, T. Hanaguri, Y. Kohsaka,T. Yamashita, Y. Shimoyama, Y. Mizukami, R. Endo,H. Ikeda, K. Aoyama, T. Terashima, S. Uji, T. Wolf,H. v. L¨ohneysen, T. Shibauchi, and Y. Matsuda,Proc. Nat. Acad. Sci. USA , 16309 (2014). Y. Lubashevsky, E. Lahoud, K. Chashka, D. Podolsky, andA. Kanigel, Nat. Phys. , 309 (2012). S. Rinott, K. B. Chashka, A. Ribak, E. D. L. Rienks,A. Taleb-Ibrahimi, P. Le Fevre, F. Bertran, M. Randeria,and A. Kanigel, Sci. Adv. (2017). S. Kasahara, T. Yamashita, A. Shi, R. Kobayashi, Y. Shi-moyama, T. Watashige, K. Ishida, T. Terashima, T. Wolf, F. Hardy, C. Meingast, H. v. L¨ohneysen, A. Levchenko,S. T., and Y. Matsuda, Nat. Commun. , 12843 (2016). Y. G. Naidyuk, G. Fuchs, D. A. Chareev, and A. N.Vasiliev, Phys. Rev. B , 144515 (2016). A. A. Sinchenko, P. D. Grigoriev, A. P. Orlov, A. V. Frolov,A. Shakin, D. A. Chareev, O. S. Volkova, and A. N.Vasiliev, Phys. Rev. B , 165120 (2017). M. Randeria and E. Taylor,Annu. Rev. Condens. Matter Phys. , 209 (2014). C. A. Regal, M. Greiner, and D. S. Jin,Phys. Rev. Lett. , 040403 (2004). M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim,C. Chin, J. H. Denschlag, and R. Grimm,Phys. Rev. Lett. , 203201 (2004). W. Ketterle and M. W. Zwierlein,Riv. Nuovo Cimento , 247 (2008). W. Zwerger,
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