Multiband mechanism of the pair fluctuation screening
MMultiband mechanism of the pair fluctuation screening
T. T. Saraiva, A. A. Shanenko, A. Vagov, and A. S. Vasenko
1, 3 National Research University Higher School of Economics, 101000, Moscow, Russia Institut f¨ur Theoretische Physik III, Bayreuth Universit¨at, Bayreuth 95440, Germany I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute,Russian Academy of Sciences, 119991 Moscow, Russia (Dated: February 11, 2021)Recent chain-like structured materials have shown a robust superconducting phase. These ma-terials exhibit the presence of quasi-one-dimensional bands (q1D) coupled to conventional higher-dimensional bands. On the mean-field level such systems have a high critical temperature whenthe chemical potential is close to the edge of a q1D band and the related Lifshitz transition isapproached. However, the impact of the pair fluctuations compromises the mean-field results. Re-cently it has been demonstrated that these fluctuations can be suppressed (screened) by a specificmultiband mechanism based on the pair-exchange coupling of the q1D condensate to a stable higher-dimensional one. In the present work we demonstrate that strikingly enough, this mechanism is notvery sensitive to the basic parameters of the stable condensate such as its strength and dimension-ality. For example, even the presence of a passive higher-dimensional band, which does not exhibitany superconducting correlations when taken as a separate superconductor, results in suppressionof the pair fluctuations.
I. INTRODUCTION
Since their first experimental detection in MgB , multiband superconductors have shown a rich phe-nomenology improving our understanding and knowledgeof superconductivity . The fundamental difference ofthe multiband superconductors from the conventionalsingle-band superconducting materials is that the inter-ference of multiple contributing condensates can result insignificant deviations from the single-condensate physics.Recently it has been demonstrated that such interfer-ence affects the superconducting fluctuations, leading tothe multiband fluctuation screening mechanism. Thepair exchange coupling between the multiple condensatescan wash out the fluctuations of the order parameterand thus amplify the critical temperature of the system.It has been revealed that the severe fluctuations of thequasi-one-dimensional (q1D) condensate are suppressedby an almost negligible pair-exchange coupling to the sta-ble BCS condensate. However, it was not investigatedhow the multiband screening mechanism depends on theparameters of the stable higher-dimensional condensatesuch as its strength and dimensionality. Here, motivatedby on-going experiments with the multiband q1D super-conductors A Cr As (A = K, Rb, Cs) and similarorganic materials , we are going to fill this gap. Asa prototype of chain like structured multiband supercon-ducting materials, we consider a two-band superconduc-tor with a q1D band coupled to a stable 2D/3D conden-sate and investigate the dependence of the fluctuationshifted critical temperature on the system parameters. II. THEORETICAL APPROACH
We consider the standard multiband generalization ofthe BCS model with a pair exchange coupling be- tween the two contributing bands. The coupling matrix g νν (cid:48) ( ν, ν (cid:48) = 1 ,
2) is symmetric, where ν = 1 stays forthe q1D band and g > g , g , e.g., the q1D band is astronger one. The weaker band corresponds to ν = 2 and,taken in its passive limit, it has g = 0. In this case, thesecond gap function is nonzero due to the pair exchangecoupling between bands 1 and 2. We choose the spher-ical Fermi surfaces for both bands with the dispersions(absorbing the chemical potential µ ) ξ (1) k = (cid:126) k z m − µ and ξ (2) k = ε + (cid:126) k m − µ, (1)where m , are the effective electronic masses for eachband, the q1D single-electron energy varies only in thez-direction (depending on k z ) and the single-electronenergy in the higher-dimensional band depends on the2D/3D wavevector k . The condensate in band 2 is sta-ble so that the band is deep enough with ε < | ε | (cid:29) µ , as sketched in Fig. 1.The Hamiltonian reads H = (cid:90) d r (cid:40) (cid:88) ν =1 , (cid:34) (cid:88) σ = ↑ , ↓ ˆ ψ † νσ ( r ) T ν ( r ) ˆ ψ νσ ( r )+ (cid:16) ˆ ψ † ν ↑ ( r ) ˆ ψ † ν ↓ ( r )∆ ν ( r ) + h . c . (cid:17) (cid:35) + (cid:104) (cid:126) ∆ , ˇ g − (cid:126) ∆ (cid:105) (cid:111) (2)where ˆ ψ † νσ ( r ) and ˆ ψ νσ ( r ) are the field operators for thecarriers in band ν , T ν ( r ) is the single-particle Hamilto-nian corresponding to with the single-particle energiesgiven by Eq. (1), and ∆ ν ( r ) is the gap function for band ν . We also use a vector notation (cid:126) ∆ = (∆ , ∆ ) with (cid:104) ., . (cid:105) the corresponding inner product, and ˇ g − is the inverseof the coupling matrix.The Hamiltonian is solved together with the self-consistency equation written in terms of the anomalous a r X i v : . [ c ond - m a t . s up r- c on ] F e b Energy ξ k ( ) ξ k ( ) μ k a ) ε Energy D O S b ) Lifshitz
Point ↘
2D q1D + Energy D O S c ) LifshitzPoint ↘
3D q1D + FIG. 1. a) Sketch of the dispersion relations of the shallow anddeep bands, ξ (1) k and ξ (2) k respectively. The distance betweenthe bottom of the bands is ε . To the right, in plots b) and c)we show sketches of the DOSs in the 2D and in the 3D casesin the deep band regime, respectively, combined to the q1Dband. Blue and Red (purple) lines represent partial (total)DOS. Green functions R ν ( r ) = (cid:104) ψ ν ↑ ( r ) ψ ν ↓ ( r ) (cid:105) as∆ ν ( r ) = (cid:88) ν (cid:48) =1 , g νν (cid:48) R ν (cid:48) ( r ) . (3)From Eqs. (2) and (3) one derives the linearized gapequation (cid:88) ν (cid:48) =1 , γ νν (cid:48) ∆ ν (cid:48) = A ν ∆ ν ⇒ ˘ L(cid:126) ∆ = 0 , (4)where the auxiliary matrix L νν (cid:48) = γ νν (cid:48) − A ν δ νν (cid:48) is intro-duced in terms of the inverse coupling matrix ˘ γ = ˘ g − and the coefficients A ν given by (see Appendix A): A = N (cid:90) − ˜ µ dy tanh( y/ T c ) y √ y + ˜ µ (5) A = N ln (cid:18) e Γ π ˜ T c (cid:19) , (6)where Γ ≈ .
577 is the Euler-Mascheroni constant, T c is the mean-field critical temperature of the system andquantities marked by a tilde are normalized by the cutoffenergy (cid:126) ω c . The parameter N = σ ( xy ) (cid:112) m z / π (cid:126) hasunits of DOS, but the divergent part is kept inside theintegral as shown in Eq. (5). The term σ ( xy ) accountsfor the DOS in the x and y directions. For the higher-dimensional band we have N (2 D )2 = σ ( x ) m / (cid:126) ( σ ( x ) accounts for the DOS in the x direction) in the 2D caseand N (3 D )2 = m k F / π (cid:126) in the 3D case. The DOSs aresketched in Fig. 1.In fact, the couplings and the partial DOSs can be com-bined in a smaller set of parameters when one expressesthe system in terms of the dimensionless couplings λ = g N , λ = g N , λ = g (cid:112) N N . (7) From Eq. (4), one can obtain the equation for the mean-field critical temperature of the two-band system, T c :( g − G A ) ( g − G A ) − g = 0 , (8)or( λ A −
1) ( λ A − − λ = 0 , (9)where G = g g − g . Taking the highest from bothsolutions of Eq. (8) as the critical temperature of thesystem, it becomes bounded from bellow by the solutionof the isolated single-band systems. The solutions for T c as function of the chemical potential (both in unitsof (cid:126) ω c ) for different interband couplings are displayedin color plots of Fig. 2. By choosing the q1D band asthe strongest band, the effect of introducing the second(deep) band is to increase T c but very weakly for λ (cid:29) λ → L(cid:126)η + = (cid:18) S (cid:19) , (cid:126)η − = (cid:18) − S (cid:19) , (10)where S = g A = λ A /χ / . The equation ˘ L(cid:126) ∆ = 0states that ˘ L must have at least one null eigenvector.Considering the non-degenerate case, the eigenvector (cid:126)η + must be such that (cid:126) ∆ = Ψ( r ) (cid:126)η + . (11)The function Ψ( r ) is the GL order parameter of the sys-tem and it obeys the single-component GL equation (seeAppendix A and Refs. 6 and 7) a Ψ + b Ψ + (cid:88) i = x,y,z K i ∇ i Ψ = 0 (12)with the coefficients given by a = a + a S , (13) b = b + b S , (14) K i = K i + K i S , (15) - - μ / ℏω c T c0 ℏω c a ) λ = λ = λ = λ = λ = λ → - - μ / ℏω c T c0 ℏω c b ) λ = λ = λ = λ = λ = λ → FIG. 2. Mean-field critical temperature, T c , as functionof the chemical potential (both quantities are expressed inunits of the Debye energy). In plot a) the weaker band hascoupling λ = 0 .
01, which is The dimensionless coupling ofthe stronger band is λ = 0 . λ , shown in the figures. The dashed linerepresents T c in the limit of uncoupled bands, λ → where a = − N T c (cid:90) − ˜ µ dy sech ( y/ T c ) √ y + ˜ µ , (16) b = N (cid:126) ω c (cid:90) − ˜ µ dy sech ( y/ T c ) y √ y + ˜ µ × (cid:20) sinh (cid:18) y ˜ T c (cid:19) − y ˜ T c (cid:21) , (17) K z = (cid:126) v F N (cid:126) ω c (cid:90) − ˜ µ dy √ y + ˜ µy sech ( y/ T c ) × (cid:20) sinh (cid:18) y ˜ T c (cid:19) − y ˜ T c (cid:21) (18)and the coefficients for the terms from the deep bandare widely known a = N , b = N ζ (3)8 π T c and K = N ζ (3)8 π T c (cid:126) v F D , ( D = 2 , N , but as this constantcan be hidden in the dimensionless coupling, we are ableto perform a joint analysis for both cases. Note that K x = K y = 0 for the q1D band and K x = 0 for the2D variant of the stable band, due to very large effec- tive electronic masses along these directions. Finally, theGL free energy for the composite systems q1D+2D orq1D+3D has actually a single-component order param-eter, Ψ( r ), because of the symmetry of the gap vector.In principle, fluctuations could enable a non-zero com-ponent also in the second eigenvector (cid:126)η − , but these fluc-tuations are non-critical and thus they can safely notbe considered. Furthermore, the resulting free energyis effectively of a q1D+2D system and the correspond-ing Ginzburg-Levanyuk parameter (or Ginzburg number)can be expressed as Gi = Gi D b b + S π | S | (cid:16) a a + S (cid:17) (cid:113) K x K + S (19)where Gi D = T c b πa K . (20)And similarly for the q1D+3D case the Ginzburg-Levanyuk parameter becomes Gi = Gi D (cid:16) b b + S (cid:17) (cid:16) a a + S (cid:17) (cid:16) K x K x + S (cid:17) S , (21)where Gi D = 132 π T c b a K . (22)In the simple case where v T /v F →
0, i.e. the flat-bandregime, we have Gi = Gi D b b + S πS (cid:16) a a + S (cid:17) (23)and Gi = Gi D (cid:16) b b + S (cid:17) (cid:16) a a + S (cid:17) S . (24)for the q1D+2D and q1D+3D, respectively. Finaly, theshift over the critical temperature can be written in termsof Gi as T c − T c T c = 8 π Gi / , (25)and T c − T c T c = 4 Gi (26)as the shift of the critical temperature due to theBerezinski-Kosterlitz-Thouless (BKT) transition for the2D case. - - μ / ℏω c T c ℏω c a ) q1D + + λ = × - - - μ / ℏω c T c ℏω c b ) q1D + + λ = × - - - μ / ℏω c T c ℏω c c ) q1D + + λ = × - - - μ / ℏω c T c ℏω c d ) q1D + + λ = × - - - μ / ℏω c T c ℏω c e ) q1D + + λ = × - - - μ / ℏω c T c ℏω c f ) q1D + + λ = × - FIG. 3. Renormalized critical temperatures due to fluctuations for q1D+2D (blue) and q1D+3D (red) systems for theparameters λ = 0 . λ = 0 .
01 upper row, plots a), b) and c) and λ = 0 for the lower row, plots d), e) and f). The dottedblack line is the mean-field critical temperature. We used the deep band typical Ginzburg number values for 2D and 3D deepbands Gi (2 D ) = 10 − and Gi (3 D ) = 10 − , respectively. III. RESULTS AND DISCUSSION
The theoretical derivation shown in the previous sec-tion shows that, at the mean-field level, having a weaker2D or a 3D deep bands should induce similar changes inthe critical temperature. We show in Fig. 2 the mean-field critical temperature as a function of the chemicalpotential, both normalized by (cid:126) ω c . In both plots, wecan see that there is a very strong increase of the crit-ical temperature above µ/ (cid:126) ω c ≈ − .
2, where the sys-tem goes through the so-called Lifshitz transition. In thefirst plot, a), the second band is much weaker but hasnonzero coupling ( λ = 0 . µ/ (cid:126) ω c < .
2, where there should not be pairs in theq1D band, once it is bellow the Lifshitz point. There aredifferent values for the interband coupling from a verystrong value, λ = λ = 0 .
2, down to the limit of un-coupled bands λ →
0. For a non-passive second band( λ > λ , the higher is T c and one can see thatthe curves have a maximum around µ/ (cid:126) ω c ≈ .
4, afterthe point of divergence in the DOS for the q1D bandwhere there is a sudden increase in T c . The dependenceon λ is very moderate and one can say that the mainparameter in this system is the coupling in the shallowband, λ , because the deep band is taken with a smallintraband coupling.Finally, we demonstrate how the introduction of thesecond passive band induces the screening of fluctuationseven in the extreme case of just a passive band. As can beseen in Figs. 3 a) and d), for small interband couplings thefluctuations take over the superconducting phase and therenormalized critical temperatures can get much smaller than the mean-field solution. Now, the cases b), c) eand f), the stronger values of the interband couplingare enough to produce critical temperatures closer to themean-field values. In the plot c), the value λ = 6 × − is almost two orders of magnitude smaller than the cou-pling in the stronger band, λ = 0 .
2. As can be seen, thedifference between the upper and lower plots are neg-ligible and therefore one concludes that the mechanismwhich we described is very robust and can work as a pro-totype for novel High-Tc materials.
IV. CONCLUSION
We showed a simple mechanism to stabilize fluctua-tions in a q1D superconductor where the q1D band isstronger and coupled to another weaker band with twoor more dimensions. This second band can be even justbe a passive band where the Coopar pairs are formed inthe stronger band and is exchanged to the weaker band.The mean field solutions for the critical temperature in asingle-band q1D system might be very high due to diver-gence of the DOS next to the bottom of the band, the Lif-shitz point, but the shift of the critical temperature dueto fluctuations is huge, making the superconducting statepractically impossible in this case. In the case of a two-band system with a second band with a higher dimen-sional Fermi surface, this makes the material essentiallyhigher dimensional which drastically reduces the renor-malization of the critical temperature. Also, we showedhow this mechanism is very robust once the second bandcan even be just a passive band, i.e. it would not besuperconductor by itself. This mechanism captures bothinteresting qualities from the q1D, 2D and 3D systems:possible high critical temperatures next to the Lifshitzpoint and it shows little effect of fluctuations.
Appendix A: Calculation of the mean-field criticaltemperature, T c0 , and the GL coefficients Following the Green function formalism developed inRef’s. 19 and 20, the Hamiltonian given in Eq. (2) allowsus to construct Dyson-like equations for the anomalousaverages in terms of the normal-state temperature Greenfunctions G (0) νω ( x , y ) and ¯ G (0) νω ( x , y ): R ν [∆ ν ] = (cid:90) d y K νa ( x , y )∆ ν ( y )+ (cid:90) (cid:89) l =1 d y l K νb ( x , y , y , y ) × ∆ ν ( y )∆ ∗ ν ( y )∆ ν ( y ) , (A1)where the kernels are given by K νa ( x , y ) = − gT (cid:88) ω G (0) νω ( x , y ) ¯ G (0) νω ( y , x ) (A2)and K νb ( x , y , y , y ) = − gT (cid:88) ω G (0) νω ( x , y ) × ¯ G (0) νω ( y , y ) G (0) νω ( y , y ) ¯ G (0) νω ( y , x ) . (A3)The normal-state temperature Green functions are de-fined in terms of the band-dependent single electron en-ergies, ξ ( ν ) k , as G (0) νω ( x , y ) = (cid:90) d k (2 π ) e − i k ( x − y ) i (cid:126) ω − ξ ( ν ) k (A4)and ¯ G (0) νω ( x , y ) = −G (0) ν, − ω ( y , x ). The integral kernels in-volve, as usual, the summation over the fermionic Mat-subara frequencies ω n = πT (2 n + 1) / (cid:126) (here the Boltz-mann constant k B is set to 1).The effective dimensions of the Fermi sheets are con-sidered in the regime when the dispersion relation hasvery large effective electronic masses in some directions,say, m y , m z (cid:29) m x (for the q1D case) or m z (cid:29) m x , m y (for the 2D case). Then the related single-particle energybecomes ξ k = (cid:88) i =1 (cid:126) k i m i − µ ≈ (cid:126) k x m x − µ (1D) (cid:126) k x m x + (cid:126) k y m y − µ (2D) (cid:126) k m − µ (3D) (A5)In the so called deep band regime, one can shift the bot-tom of the 2D or 3D bands by the constant ε , as statedin Sec. II.Let us begin with the linearized version of Eq. (A1) fora system with a q1D band and a second band with higher number of dimensions D = 2 or 3 and contract with thematrix g νν (cid:48) :∆ ν (cid:48) = (cid:88) ν =1 , g νν (cid:48) R ν = − T c (cid:88) ν =1 , g νν (cid:48) ∆ ν × (cid:88) ω (cid:90) d z d k d k (cid:48) (2 π ) exp[ − i ( k − k (cid:48) ) · z ] (cid:104) i (cid:126) ω − ξ ( ν ) k (cid:105) (cid:104) i (cid:126) ω + ξ ( ν ) k (cid:48) (cid:105) (A6)where z = x − y . The two coefficients of ∆ ν in Eq. (A6)can be rewritten as − g ν (cid:48) T c (cid:88) ω σ ( z ) (cid:90) dk π i (cid:126) ω − ξ (1) k )( i (cid:126) ω + ξ (1) k ) . (A7)for a 2D band. − g ν (cid:48) T c (cid:88) ω σ ( yz ) (cid:90) dk x π i (cid:126) ω − ξ k )( i (cid:126) ω + ξ (1) k ) , (A8)Here we defined the auxiliary parameters σ ( z ) = (cid:90) dk z π = k ( z ) F π , (A9) σ ( yz ) = (cid:90) dk y π dk z π = (cid:90) dk π = k ( yz ) F π . (A10)Furthermore, considering rotation symmetry ( m x = m y = m ), we have: dξ = (cid:126) k x m x dk x ⇒ dk x → (cid:114) m x (cid:126) dξ √ µ + ξ , (A11) dξ = (cid:126) km dk ⇒ dk x dk y (2 π ) = kdk π → m (cid:126) dξ. (A12)This means that the integrals can be written as T c gσ ( yz ) (cid:114) m x π (cid:126) (cid:88) ω (cid:90) (cid:126) ω c − µ dξ ( µ + ξ ) − / (cid:126) ω + ξ = 1 (A13) T c gσ ( z ) m (cid:126) (cid:88) ω (cid:90) (cid:126) ω c − µ dξ (cid:126) ω + ξ = 1 (A14)and that the DOS for the q1D band becomes (by intro-ducing the excitation energy independent from the chem-ical potential E = ξ + µ ): N d ( E ) = σ ( yz ) (cid:114) m x π (cid:126) E (A15)while the DOS for the 2D system is a constant N d = σ ( z ) m (cid:126) . (A16)The summation over the Matsubara frequencies is known (cid:88) ω (cid:126) ω + ξ = tanh( ξ/ T )2 T ξ (A17)and then gσ ( yz ) (cid:114) m x π (cid:126) (cid:126) ω c + µ (cid:90) dE tanh[( E − µ ) / T c ]( E − µ ) E / = 1 , (A18) gσ ( z ) m (cid:126) (cid:126) ω c + µ (cid:90) ε dE tanh[( E − µ ) / T c ] E − µ = 1 , (A19)where it was introduced the cutoff energy (cid:126) ω c . Althoughthe integral appearing in Eq. (A18) is not divergent, weintroduce a physical cutoff as was done in the 3D case.It is natural to introduce the dimensionless couplings λ d = g σ ( yz ) (cid:114) m x π (cid:126) ω c (A20) λ d = g σ ( z ) m (cid:126) (A21)and, writing the relevant quantities in units of (cid:126) ω c , theequations for T c becomes: λ d (cid:90) µ dx tanh[( x − ˜ µ ) / T c ]( x − ˜ µ ) x / = 1 (A22) λ d (cid:90) µ dx tanh[( x − ˜ µ ) / T c ] x − ˜ µ = 1 . (A23)In the deep band regime, the equation for the 2D bandbecomes: λ d ∞ (cid:90) dx tanh( x/ T c ) x = 1 ⇒ ˜ T c = 2 e Γ π exp ( − /λ d ) (A24)In order to include the effect of fluctuations of the gap,we consider the deviation from the critical temperature, τ = 1 − T /T c , and we will consider the first gradientterms of the Taylor expansion of the gap in the linearterm (cid:90) d zK a ( z )∆( z ) ≈ (cid:90) d zK a ( z ) (cid:34) ∆( x ) + ( z · (cid:126) ∇ ) x ) (cid:35) . (A25)We can obtain the first GL coefficients, a d and a d , byderiving with respect to τ the hyperbolic tangent: a d = N d √ (cid:126) ω c T c (cid:126) ω c + µ (cid:90) d E sech [( E − µ ) / T c ] E / (A26) a d = N d T c (cid:126) ω c + µ (cid:90) dE sech [( E − µ ) / T c ]= N d [1 + tanh( µ/ T c )] (A27) The second term is composed by integrals such as I ( i,j ) K = − T (cid:88) ω (cid:90) d z d k (2 π ) d k (cid:48) (2 π ) z i z j ×× exp[ − i ( k − k (cid:48) ) · z ]( i (cid:126) ω − ξ k )( i (cid:126) ω + ξ k (cid:48) ) , (A28)but due to the symmetry of the integrands, I ( i,j ) a = 0for i (cid:54) = j . The terms z i G (0) ω ( z ) can be replaced by thederivative with respect to k i in the k − space and, again,the volume integration over z produces δ ( k (cid:48) − k ) and then I ( i,i ) K = − T c (cid:88) ω (cid:90) d k (2 π ) (cid:18) ∂ k i i (cid:126) ω − ξ k (cid:19) ×× (cid:18) ∂ k i i (cid:126) ω + ξ k (cid:19) . (A29)It is trivial that I ( i,i ) a = 0 for i = y, z in the q1D caseand for i = z in the 2D case. Finally, in the q1D case for i = x , we have I ( x,x ) K = − T c (cid:88) ω (cid:90) d k (2 π ) − (cid:16) (cid:126) k x m x (cid:17) ( (cid:126) ω + ξ k ) (A30)= (cid:126) m x T c (cid:88) ω (cid:90) d k (2 π ) ξ k + µ ( (cid:126) ω + ξ k ) (A31)= (cid:126) m x T c σ ( xy ) (cid:88) ω (cid:90) d k x π ξ k + µ ( (cid:126) ω + ξ k ) (A32)and for i = x, y in the 2D case, we have (again considering m x = m y = m ) I ( i,i ) K = − T c (cid:88) ω (cid:90) d k (2 π ) − (cid:16) (cid:126) k i m (cid:17) ( (cid:126) ω + ξ k ) (A33)= (cid:126) m T c (cid:88) ω (cid:90) d k (2 π ) ( ξ k + µ ) / (cid:126) ω + ξ k ) (A34)= (cid:126) m T c σ ( z ) (cid:88) ω (cid:90) k d k π ξ k + µ ( (cid:126) ω + ξ k ) (A35)and here we use the tabled infinite summation over Mat-subara frequencies (cid:88) ω (cid:126) ω + ξ k ) = [ T sinh ( ξ k /T ) − ξ k ] sech (cid:16) ξ k T (cid:17) ξ k T (A36)Then: K ( x )1 d = (cid:126) m x T c σ ( yz ) (cid:114) m x π (cid:126) (cid:126) ω c + µ (cid:90) d EE / × sech [( E − µ ) / T c ]8( E − µ ) T c × (cid:20) sinh (cid:18) E − µT c (cid:19) − E − µT c (cid:21) (A37)= (cid:126) m x N d √ (cid:126) ω c (cid:126) ω c + µ (cid:90) d EE / sech [( E − µ ) / T c ]( E − µ ) × (cid:20) sinh (cid:18) E − µT c (cid:19) − E − µT c (cid:21) . (A38)For the 2D case, one has the shallow band version: K ( i )2 d = 14 π T c σ ( z ) (cid:126) ω c + µ (cid:90) d E sech [( E − µ ) / T c ]8( E − µ ) T c ×× (cid:20) sinh (cid:18) E − µT c (cid:19) − E − µT c (cid:21) (A39)= (cid:126) m N d π (cid:126) ω c + µ (cid:90) d E sech [( E − µ ) / T c ]( E − µ ) × (cid:20) sinh (cid:18) E − µT c (cid:19) − E − µT c (cid:21) (A40)and the stiffness of the gap parameter along the otherorthogonal directions is zero. In the deep band regime, K ( i )2 d = (cid:126) µm T c σ ( z ) µ m (cid:126) (cid:88) ω ∞ (cid:90) −∞ dξ (cid:126) ω + ξ k ) = (cid:126) v F T c N d (cid:88) ω | (cid:126) ω | ∞ (cid:90) dx x ) = (cid:126) v F N d π π T c ∞ (cid:88) n =0 n + 1) = (cid:126) v F ζ (3) N d π T c (A41)The last term is given by the cubic contribution fromEq. (A1). It is enough to consider only the zero-ordercontribution of the gap in the Taylor expansion on thecoordinates (i.e. it becomes independent of the gap) andthus the integral becomes b = (cid:90) (cid:89) l =1 d y l K b ( x , y , y , y ) (A42)= − T (cid:88) ω (cid:90) d k (2 π ) i (cid:126) ω − ξ k ) ( i (cid:126) ω + ξ k ) , (A43) where we used the convolution theorem to find theFourier transform of the product of unperturbed Greenfunctions. Next we apply the summation given byEq. (A36) and the final expression for the coefficient b for the q1D case becomes b d = T c σ ( xy ) (cid:114) m x π (cid:126) (cid:126) ω D + µ (cid:90) d E sech [( E − µ ) / T c ]8 T c E / ( E − µ ) ×× (cid:20) T c sinh (cid:18) E − µT c (cid:19) − ( E − µ ) (cid:21) (A44)= N d √ (cid:126) ω c (cid:126) ω c + µ (cid:90) d E sech [( E − µ ) / T c ] E / ( E − µ ) ×× (cid:20) sinh (cid:18) E − µT c (cid:19) − E − µT c (cid:21) (A45)and for the 2D case, in the shallow band regime it be-comes b d = T c σ ( z ) m (cid:126) (cid:126) ω c + µ (cid:90) d E sech[( E − µ ) / T c ]8 T c ( E − µ ) ×× (cid:20) T c sinh (cid:18) E − µT c (cid:19) − ( E − µ ) (cid:21) (A46)= N d (cid:126) ω c + µ (cid:90) d E sech[( E − µ ) / T c ]( E − µ ) ×× (cid:20) sinh (cid:18) E − µT c (cid:19) − E − µT c (cid:21) (A47)and in the deep band regime it becomes b d = T c σ ( z ) m (cid:126) (cid:88) ω | (cid:126) ω | ∞ (cid:90) dx x ) = T c N d π (cid:88) ω | (cid:126) ω | = N d π T c ∞ (cid:88) n =0 n ) = 7 ζ (3) N d π T c (A48)One can simplify Eq’s. (A26), (A40) and (A45) by ex-pressing all energies in units of T c after noticing that theupper limit (cid:126) ω c + µT c (cid:38) → ∞ (A49)for all the range of parameters we have used and that allthe terms integrated in these equations are fast-decayingbecause of the term sech [( E − µ ) / T c ]. These expres-sions are a d = N d √ (cid:126) ω T / c ∞ (cid:90) d x sech [( x − µ (cid:48) ) / x / , (A50) b d = N d √ (cid:126) ω c T / c ∞ (cid:90) d x sech [( x − µ (cid:48) ) / x / ( x − µ (cid:48) ) (cid:20) sinh( x − µ (cid:48) ) x − µ (cid:48) − (cid:21) , (A51) K d = (cid:126) m x N s √ (cid:126) ω c T / c ∞ (cid:90) d x x / sech [( x − µ (cid:48) ) / x − µ (cid:48) ) ×× (cid:20) sinh( x − µ (cid:48) ) x − µ (cid:48) − (cid:21) , (A52) where µ (cid:48) = µ/T c . The only reason for the appearance of (cid:126) ω c in the equations is to maintain the term N d constantand with units of DOS. 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