Im(z) RARA ARRA ARAR

Transcription

1 Supplementary Fgures q/+p q/+p' q/+p q/+p' = γp + q/+p U J q/+p'' q/+p' U J Lq J U J U Lq γp γp' γp' γp γp'' γp'' γp' q/-p q/-p' q/-p q/-p' q/-p q/-p'' q/-p' Supplementary Fgure : The Bethe-Salpeter equaton for the Leggett mode propagator. L q s the Leggett mode propagator, γ p s the d-wave symmetry factor, the matrx wth elements U and V s the bare Cooper scatterng n the subband space. p, p, p and q denote both momentum and frequency e.g. q = q, q. Imz RARA Imz= Imz=-Ω ARRA ARAR Rez Supplementary Fgure : Contour ntegraton n the complex z-plane. Black dots are the fermonc frequences to be summed over, bold black lnes are the branch cuts of the ntegrand where Ω s a bosonc frequency. R/A denotes retarded/advanced Green s functons n the ntegrand of Supplementary Equaton 37.

2 Supplementary Note : Dervaton of the Leggett mode Fluctuaton regon As mentoned n the man text, the propagator for superconductng fluctuatons s derved by the followng Bethe- Salpeter equaton see Supplementary Fgure U J U L pp q = γ p L q γ p = γ p γ J U p γ p J π q = βv p ω J U πa q L π b q q γ p, q q γp G + p, q + ω G p, ω, where q = q, q, and smlarly for p and p. Note that U < and J < are both attractve n our conventon. The notaton p means the summaton over momentum space s restrcted to the th subband. Takng nto account the strong dampng of the quaspartcles n the underdoped regme, we assume a phenomenologcal form of the electron Green s functon G p, ω = ω ɛ p + Γ e sgnω, Γ e = at + bt, where a quadratc temperature dependence of the quaspartcle dampng rate s used accordng to prevous transport studes [. To calculate π, we follow the general procedure of superconductng fluctuaton theory [, 3. For any bosonc frequency ω k >, we have π q, ω k N γp v F p q n n + + ω k 4πT + Γe 4πT 3. 3 πt n + + ω k n n + + ω k 4πT + Γe πt 4πT + Γe πt Here N s the densty of states per spn at the Ferm level n one subband, v F p s the Ferm velocty at momentum p. We use... to denote the average over the Ferm surface of subband. A normalzed d-wave symmetry factor γ p = cos θ p s chosen, where θ p s the angle measured from the center of the arc and γp =. Introducng a large bosonc frequency ω c = n max πt to cut off the summaton of the frst term whch s logarthmc dvergent, t follows n max = ψ + ω k 4πT + Γe πt + ω c ψ πt + ω k 4πT + Γe, πt n n + + ω k 4πT + Γe πt 3 = ψ + ω k 4πT + Γe πt where the specal functon ψ x = d [log Γ x /dx s the dgamma functon. frequency s taken by ω k ω, whch leads to π R q, ω = ψ N ω 4πT + a + bt + ω c ψ π πt + 4πT ψ ω 4πT + a + bt π, 4 The analytc contnuaton to real ω 4πT + a + bt π γp v F p q. 5 The superconductng orderng temperature T c, at whch the fluctuaton propagator L R q =, ω = dverges, s gven by T c = ω [ c π exp ψ + a + bt c exp. 6 π N U + J

3 Note that fnte dampng a, b > suppresses T c. When a = b =, the above equaton reduces to the conventonal BCS equaton T c =.3ω c exp [ / N V max, where V max s the most attractve nteracton, n our two subband case t s the egenvalue U + J of the bare nteracton vertex. In terms of T c, we could rewrte the bubble π as N π R q, ω N U + J ɛ + 4πT ψ + a + bt c π + 4πT ψ + a + bt π [ω bt T T c γp v F p q, 7 where ɛ = log T/T c s the reduced temperature. Solvng the matrx Bethe-Salpeter equaton for L R Supplementary Equaton, one gets U + U J πb R q, ω J L R J U + U J π R a q, ω q, ω = U J πa R q, ω πb R q, ω + U [ πa R q, ω + πb R q, ω +. 8 If we make a further sotropc smplfcaton by averagng over the drecton of q such that πa R = πb R = πr, then U + U J π R J L R J U + U J [ π R P + P = [U + J π R + [U J π R = + π R + + U+J π R +, 9 U J 3 where the two projectors n the nomnators are P ± = ± σ x. They satsfy P ± = P ± and P + P =. Physcally, P + projects nto the subspace of n-phase fluctuatons, and P nto that of out-of-phase fluctuatons. The poston of the pole n P + s located at q = ω = when T = T c. Thus t s the fluctuatng verson of the Anderson-mode pole wth n-phase nature and gapless nature. In the presence of a local superconductng order parameter, the n-phase phase fluctuaton at long wavelengths s assocated wth the total charge densty modulaton, whch s prohbted by the long-range Coulomb nteracton. As a result we are manly nterested n the out-of-phase fluctuatons.e. the fluctuatng verson of the Leggett mode wth nomnator P L R q, ω = T cn P ω τ Dq, where the constant c = [ 4π ψ + π a + bt c. The above propagator has the form of an overdamped bosonc mode. The total relaxaton tme τ and dffuson constant D are defned by the followng equatons τ = T c log T + bt T T c + T J T c cn U J, Dq = ψ [ + π a + bt 8πT ψ [ γ + π a + bt c p v F p q ψ [ + π a + bt = v F 6πT ψ [ + π a + bt c q, where n the expresson for D, the symbol... means averagng over both the drectons of p and q, and the result s ndependent on the subband ndex. From now on we denote the total dampng of Leggett mode to be whch leads to the smple expresson used n the man text L R q, ω = Γ LM q = τ + Dq, 3 T cn ω Γ LM q σ x. 4

4 4 Ordered regon T = In ths case, the structure of the Bethe-Salpeter equaton s unchanged, but the electronc bubble receves another contrbuton from the anomalous Green s functon due to the superconductng [ order [4 6. Also [ the Green s functons change form n the superconductng phase, G p = p + ɛ p / p Ep and F p = p / p Ep, wth E p = ɛ p + p the Bogolubov quaspartcle energy. We generalze the dervaton by Leggett for s-wave superconductors [4 to the d-wave case. In the followng equatons, the q-dependent quanttes are ɛ p± q, E p± q, and p± q. Because q π s small and p ɛ p p p, we set p± q p = γ p. Then t follows π q = βv N p p γ p dɛ p γ p tanh Ep T E p [ q q q q G + p, q + p G p, p + F + p, q + p F p, p N q v F p q Here we have taken the leadng order term of q and used the dentty dɛ p p p E p E p tanh Ep T dɛ p. 5 E p q + p + ɛ p [ [ =, 6 q + p E p p+ q E p q because the ntegrand s odd under ɛ p ɛ p and p p q. In zero temperature, the frst term n the last lne of Supplementary Equaton 5 becomes γ p N dɛ p = E p U + J, 7 whch follows from the gap equaton. The energy ntegral n the second term s p E p E p dɛ p = E p 4 dɛ p p E 3 p = 4 dɛ p ɛ p After a Wck rotaton q ω and averagng the drecton of q, the zero temperature bubble reads π q, ω = U + J + N 4 ɛp E p =. 8 ω vfq. 9 Then L q, ω = π q, ω + U J σ x = 4 N ω ωq + δ σ x, where the dsperson of the Leggett mode s ω q = ω + v Fq, ω = 4 N J U J.

5 5 Supplementary Note : Calculaton of the phonon self energy Summaton of subband ndces In the man text, the full expresson for the phonon self energy s wrtten as Π R Q, Ω = 4 [α Q 4 [ B R Q, Ω I R Ω, I R Ω = V TrI R q, k, Ω. q,k The meanng of the trace Tr n Supplementary Equaton s explaned as follows. We are nterested n the case of Q beng large enough to lead to nter-subband transtons. In ths case the ncomng and outgong Cooper pars n the dagram for B are from two subbands, thus B s consdered to be off-dagonal n the subband space. The fluctuaton propagators, on the other hand, contan ntra-subband and nter-subband components. As a result, all possbltes of the arrangements of subband ndces sum up to a trace over the product of the two fluctuaton propagators Tr [ B B L ntra L nter L nter L ntra B B L ntra L nter L nter L ntra = B Tr [ L ntra L nter L nter L ntra L ntra L nter L nter L ntra. 3 Intermedate state n fluctuaton regon We use the spectral decomposton for the fluctuaton propagator L and the bare phonon Green s functon D as follows L q, ω = D ω = dx π dx π ImL R q, x, ω x ImD R x, 4 ω x where the phonon s assumed to be dspersonless for smplcty, whch s justfed because we are only nterested n large phonon momentum. The frequency summaton of the ntermedate state becomes I q, k, Ω = β ω,ν = π 3 β ω,ν L q, ω L q k, ω ν D Ω ν dxdydz ImL R q, x ImL R q k, y ImD R z ω x ω ν y where the last lne could be carred out explctly as follows β ω,ν ω x ω ν y Ω ν z = 4 Ω ν z, 5 coth x T + coth y T coth x y T Ω x + y z + coth z T. 6 We then perform the analytc contnuaton of the external frequency Ω Ω + δ, and take the magnary part of the whole expresson. Now the ntegral contans two delta-functons Im Ω x + y z + δ = πδ Ω x + y z, ImD R z = πδ z Ω + πδ z + Ω, 7 where Ω s the phonon frequency. ImI R q, k, Ω could now be wrtten as a one-dmensonal ntegral

6 6 ImI R q, k, Ω = dx ImL R q, x ImL R q k, x Ω + Ω π coth x 4 T + coth x Ω + Ω coth Ω Ω + coth Ω T T T {Ω Ω }. 8 We are nterested n the regon where Ω Ω T, thus we denote δω = Ω Ω. The frequences for the two Leggett modes n the ntegrand are x and x δω. Because the Leggett mode has spectral weght only n a small frequency range near x =, we gnore the {Ω Ω } term n the last lne. When Ω Ω, coth Ω Ω T coth Ω T, so we also neglect the coth Ω T term. After these smplfcatons, we arrve at ImI R q, k, δω = δω coth 4π T dx ImL R q, x ImL R q k, x δω coth x T The above result s to be summed over q and k, whch allows us to make shfts, so that TrImI R q, k, δω δω coth dx Tr 4π T T T cn Γ LM q+ k = T 4 c N [ImL R q + k, x + δω + Γ LM q k ImL R q k, x δω δω + Γ LM q+ k τ + D q + k /4 { [τ + D q + k /4 D q k cos θ + Γ LM q k coth x + δω T x δω coth. 9 T coth x δω T } 3 {δω + 4 [τ + D q + k }. /4 In the dervaton we have used coth x /x when x <. Here θ s the angle between q and k. The momentum ntegral then follows τ + D q + k /4 { } V [τ + D q + k /4 D q k cos θ {δω + 4 [τ + D q + k /4 } = π = π = πτ D q,k dx dx dx π π dy dθ π +x x τ + D x + y dθ { } [τ + D x + y 4D xy cos θ {δω + 4 [τ + D x + y } τ + Dx dy { τ + Dx D x y cos θ} {δω + 4 τ + Dx } [ arctan dθ cos θ Defnng a nonnegatve weght functon we then have p x = cos θ x +x x cos θ [ + x x cos θ π [ arctan dθ cos θ + x τ δω x. 3 cos θ x +x x cos θ [ + x x cos θ, 3 ImI R δω = π T 4 D c N dx p x Im δω + +x τ. 33

7 Our result s vald for δω T. If we extrapolate the above result to the whole range of δω, then the Kramers-Krong relatons lead to I R δω = π D T 4 c N dx p x δω + +x τ Intermedate state n ordered regon T = At T =, the Anderson mode s pushed towards the plasma frequency and s decoupled from the low-energy physcs. The only remanng low-lyng collectve mode s then the Leggett mode. We rewrte the zero-temperature form of the Leggett mode and the phonon Green s functon as follows L q, ω = 4 N ω q D Ω = ω ω q + δ ω + ω q δ σ x, Ω Ω + δ Ω + Ω δ. 35 We then carry out the frequency ntegral of the ntermedate state by the method of resdues TrI q, k, Ω dω dν = Tr L q, ω L q k, ω ν D Ω ν π π = 4 4 dω dν N ω q ω q k π π ω ω q + δ ω + ω q δ ω ν ω q k + δ ω ν + ω q k δ [ = 4 4 N ω q ω q k Ω ν Ω + δ Ω Ω + ω q + ω q k + δ Ω + Ω + ω q + ω q k δ Ω ν + Ω δ. 36 Numercal calculaton for B n the fluctuaton regon Wth the dampng effect, the electron Green s functon G k, ω z has a branch-cut on the real-axs Imz =. In ths case, we do the frequency summaton by expandng the contour ntegral of z around the magnary axs to nfnty, but avodng the branch-cuts of the ntegrand as follows Supplementary Fgure β G k, ω G k, ω G k + Q, ω + Ω G k Q, ω Ω ω = 4π = 4π 4π + 4π 4π dz tanh z G k, z G k, z G k + Q, z + Ω G k + Q, z Ω T T GR k, x G A k, x G R k + Q, x + Ω G A k + Q, x Ω T GA k, x G R k, x G R k + Q, x + Ω G A k + Q, x Ω T GA k, x Ω G R k, x + Ω G R k + Q, x G A k + Q, x T GA k, x Ω G R k, x + Ω G A k + Q, x G R k + Q, x. 37 In the above equatons, z s a complex number and x s a real number. The analytc contnuaton of the phonon frequency Ω Ω + δ could then be performed to lead to the followng form

8 8 B R Q, Ω = dk 4π π γ kγ k+q dk 4π π γ kγ k+q + dk 4π π γ kγ k+q dk 4π π γ kγ k+q T GR k, x G A k, x G R k + Q, x + Ω G A k + Q, x Ω T GA k, x G R k, x G R k + Q, x + Ω G A k + Q, x Ω + Ω T GA k, x G R k, x G R k + Q, x + Ω G A k + Q, x Ω + Ω T GA k, x G R k, x G A k + Q, x + Ω G R k + Q, x Ω, 38 where the ntegral regon of k s restrcted to one of the subbands. The fnal expresson s to be calculated numercally. We use the Cuba lbrary [7 to perform the numerc ntegral. Parameters and results are shown n the man text. Supplementary References [ Buhmann, J. M., Ossadnk, M., Rce, T. M. & Sgrst, M. Numercal study of charge transport of overdoped La xsr xcuo 4 wthn semclasscal Boltzmann transport theory. Phys. Rev. B 87, [ Aslamasov, L. & Larkn, A. The nfluence of fluctuaton parng of electrons on the conductvty of normal metal. Phys. Lett. A 6, [3 Larkn, A. & Varlamov, A. Theory of Fluctuatons n Superconductors Oxford Scence Publcatons, Oxford, 5. [4 Leggett, A. J. Number-phase fluctuatons n two-band superconductors. Prog. Theor. Phys. 36, [5 Leggett, A. J. Theory of a superflud Ferm lqud. II. collectve oscllatons. Phys. Rev. 47, [6 Leggett, A. J. Theory of a superflud Ferm lqud. I. general formalsm and statc propertes. Phys. Rev. 4, A869 A [7 Hahn, T. Cuba a lbrary for multdmensonal numercal ntegraton. Comput. Phys. Commun. 68,