Universal thermal and electrical transport near the superconductor-metal quantum phase transition in nanowires
Adrian del Maestro, Bernd Rosenow, Nayana Shah, Subir Sachdev
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug Universal thermal and electrical transport near thesuperconductor-metal quantum phase transition in nanowires
Adrian Del Maestro, Bernd Rosenow, Nayana Shah, and Subir Sachdev Department of Physics, Harvard University, Cambridge, MA 02138 Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green St, Urbana, IL 61801 (Dated: November 5, 2018)We describe the thermal ( κ ) and electrical ( σ ) conductivities of quasi-one dimensional wires, acrossa quantum phase transition from a superconductor to a metal induced by pairbreaking perturbations.Fluctuation corrections to BCS theory motivate a field theory for quantum criticality. We describedeviations in the Wiedemann-Franz ratio κ/σT (where T is the temperature) from the Lorenznumber ( π / k B /e ) , which can act as sensitive tests of the theory. We also describe the crossoversout of the quantum critical region into the metallic and superconducting phases. The Wiedemann-Franz law relates the low tempera-ture ( T ) limit of the ratio W ≡ κ/ ( σT ) of the thermal ( κ )and electrical ( σ ) conductivities of metals to the univer-sal Lorenz number L = ( π / k B /e ) . This remarkablerelationship is independent of the strength of the inter-actions between the electrons, relates macroscopic trans-port properties to fundamental constants of nature, anddepends only upon the Fermi statistics and charge of theelementary quasiparticle excitations of the metal. It hasbeen experimentally verified to high precision in a widerange of metals [1], and realizes a sensitive macroscopictest of the quantum statistics of the charge carriers.It is interesting to note the value of the Wiedemann-Franz ratio in some other important strongly interact-ing quantum systems. In superconductors, which havelow energy bosonic quasiparticle excitations, σ is infinitefor a range of T >
0, while κ is finite in the presenceof impurities [2], and so W = 0. At quantum phasetransitions described by relativistic field theories, such asthe superfluid-insulator transition in the Bose Hubbardmodel, the low energy excitations are strongly coupledand quasiparticles are not well defined; in such theoriesthe conservation of the relativistic stress-energy tensorimplies that κ is infinite, and so W = ∞ [3]. Li andOrignac [4] computed W in disordered Luttinger liquids,and found deviations from L , and found a non-zero uni-versal value for W at the metal-insulator transition forspinless fermions.The present paper will focus on the quantum phasetransition between a superconductor and a metal (aSMT). We will consider quasi-one dimensional nanowireswith a large number of transverse channels (so that theelectronic localization length is much larger than themean free path ( ℓ )) which can model numerous recentexperiments [5, 6, 7, 8, 9, 10, 11, 12, 13]. We will de-scribe universal deviations in the value of W from L ,which can serve as sensitive tests of the theory in futureexperiments.The mean-field theory for the SMT goes back to theearly work [14] of Abrikosov and Gorkov (AG): in oneof the earliest discussions of a quantum phase transition, they showed that a large enough concentration of mag-netic impurities could induce a SMT at T = 0. It hassince been shown that such a theory applies in a largevariety of situations with ‘pair-breaking’ perturbations:anisotropic superconductors with non-magnetic impuri-ties [15], lower-dimensional superconductors with mag-netic fields oriented in a direction parallel to the Cooperpair motion [16], and s -wave superconductors with inho-mogeneity in the strength of the attractive BCS inter-action [17]. Indeed, it is expected that pair-breaking ispresent in any experimentally realizable SMT at T = 0:in the nanowire experiments, explicit evidence for pair-breaking magnetic moments on the wire surface was pre-sented recently by Rogachev et al. [13].Fluctuations about the AG theory have been con-sidered [16, 18, 19] in the metallic state, and lead tothe well-known Aslamazov-Larkin (AL), Maki-Thomson(MT) and Density of States (DoS) corrections to the con-ductivity. At the SMT, field-theoretic analyses [20, 21]show that the AG theory, along with the AL, MT andDoS corrections, is inadequate in spatial dimension d ≤ d defines the dimensionality of theCooper pair motion, while the metallic fermionic quasi-particles retain a three-dimensional character; therefore,the confining dimension, R , is larger than the inverseFermi wavevector, but smaller than a superconductingcoherence length or Cooper pair size, ξ . The behaviorof W has been considered in this field-theoretic frame-work [21], and it was found that there were logarithmiccorrections to the Lorenz number in d = 2. Here wewill examine the d = 1 case in some detail: the tran-sition is described by a strongly-coupled field theory ofbosonic Cooper pairs, overdamped by their coupling tothe fermionic quasiparticles. Remarkably, all importantcouplings between the bosons and the fermions scale touniversal values, and consequently the Wiedemann-Franzratio of this theory also approaches a universal constantwhich we compute in a 1 /N expansion (the physical case α T α c Metal
AL, MT, DoS fluctuationsPhase fluctuations realize the Mooij-Schön mode
Quantum criticalLAMHFluctuating superconductor
FIG. 1: (Color online) Crossover phase diagram of thesuperconductor-metal transition in a quasi-one dimensionalsuperconductor. The “Metal” is described by the perturba-tive theory of Ref. 16. The “Quantum critical” region is de-scribed by S and realizes our result for W in Eq. (1). TheMooij-Sch¨on mode is present everywhere, but couples stronglyto superconducting fluctuations only in the “Fluctuating su-perconductor” regime, where it is described by Eq. (7); notethat S does not apply here. The dashed lines are crossoverboundaries which, by Eq. (4) occur at T ∼ | α − α c | zν . is N = 1) W = (cid:18) k B e (cid:19) (cid:18) . . N (cid:19) . (1)Our present computation of W ignores the influence ofdisorder on quantum criticality, and this may require theclean limit ξ ≪ ℓ [22].We also discuss the nature of the crossovers from thisuniversal quantum critical physics to previously studiedregimes at low T about the superconducting and metallicphases: these are summarized in Fig. 1. On the metallicside, there is a crossover to a low T regime described bythe theory [16] of AL+MT+DoS corrections in d = 1. Onthe superconducting side, there is a regime of intermedi-ate temperatures where the classical phase slip theory ofLanger, Ambegaokar, McCumber, and Halperin (LAMH)applies [23, 24], and eventually another crossover at stilllower temperatures to a phase fluctuating regime whosedescription requires a non-linear σ -model of fermion pairfluctuations coupled to the superconducting order [25].Here the phase fluctuations are essentially equivalent toa plasma charge oscillation, which in d = 1 is the Mooij-Sch¨on mode [26]. A number of works [27, 28, 29, 30] haveexamined the destruction of superconductivity due toquantum phase slips in such a phase fluctuating regime:we maintain that the phase with phase slip prolifera-tion in such models is an insulator at T = 0, and sosuch theories describe a superconductor-insulator quan-tum transition, and may be appropriate in inhomogenous systems. Our theory includes amplitude and phase fluc-tuations on an equal footing, along with strong dampingfrom the fermionic modes, and describes the transitioninto a metallic phase at T = 0: this is the case eventhough physics of the Mooij-Sch¨on mode is present inthe “fluctuating superconductor” regime in Fig. 1.It is useful to place our results in the context of re-cent microscopic computations in BCS theory [16] on themetallic side, with the pairbreaking parameter α largerthan critical α c of the SMT. These results were obtain inthe dirty limit ( ℓ ≪ ξ ), but we expect the same theory(the action S below) to apply in the quantum criticalregime in the both the clean and dirty limits (althoughthere are distinctions in the “fluctuating superconduc-tor” regime of Fig. 1). For the conductivity, these resultsare [16] σ = σ + e ¯ h (cid:18) k B T ¯ hD (cid:19) − / " π √ (cid:18) k B T ¯ h ( α − α c ) (cid:19) / + e ¯ h (cid:18) k B T ¯ hD (cid:19) (cid:20) c ¯ h ( α − α c ) k B T (cid:21) (2)where σ is a background metallic conductivity, c is anon-universal constant, D is the diffusion constant in themetal, and the remaining corrections from pairing fluc-tuations have been written in the form of a power of T times a factor within the square brackets which dependsonly upon the ratio ¯ h ( α − α c ) /k B T . This way of writingthe results allows us to deduce the importance of the fluc-tuations corrections, in the renormalization group sense,to the SMT. The first square bracket represents the usualAL correction, and has a prefactor of a negative powerof T , and so is a relevant perturbation; this is so eventhough this correction vanishes as T →
0. The secondsquare bracket arises from the additional AL, MT andDoS corrections: the prefactor has no divergence as apower of T , and so this correction is formally irrelevantat the SMT. Note, however, the complete second termhas a finite limit as T →
0, and so becomes larger thanthe formally relevant AL term at sufficiently low T inthe metal. We therefore identify the second term as dan-gerously irrelevant in critical phenomena parlance: i.e. important for the properties of the low T metallic re-gion, but can be safely neglected in the shaded quantumcritical region of Fig. 1.Armed with the above insights, we focus on the fluc-tuations associated with the usual AL correction. Thesehave [16] a Cooper pair propagator ( e Dq + | ω | + α ) − at wavevector q and imaginary frequency ω in the metalin both the clean and dirty limits. This motivates thequantum critical theory of Ref. 20 for a field Ψ( x, τ ) rep-resenting the local Cooper pair creation operator: S = Z dx " dτ (cid:16) e D | ∂ x Ψ | + α | Ψ | + u | Ψ | (cid:17) + Z dω π | ω || Ψ( x, ω ) | . (3)This theory will apply to quasi-one dimensional wires for R < (¯ h e D/k B T ) / . In the dirty limit, ℓ ≪ ξ , we have e D = D , but the value of e D is different in the clean case.All couplings in S are random functions of position; inparticular, randomness in α is expected to be relevant atthe quantum critical point. We neglect this randomnessin our quantitative results in the quantum critical region,and so they only apply above a T = T dis which can bemade arbitrarily small in the clean limit.From S , we obtain[20] the singular contribution to theconductivity in the vicinity of the quantum critical region σ sing = e ¯ h (cid:18) k B T ¯ h e D (cid:19) − /z Φ σ (cid:18) ¯ h ( α − α c )( k B T ) / ( zν ) (cid:19) (4)where z is the dynamic critical exponent, ν is a correla-tion length exponent, and Φ σ is a universal scaling func-tion. In a Gaussian approximation, the Kubo formulayields z = 2, ν = 1 /
2, and Φ σ ( y ) = ( π/ (12 √ y − / ,and so this result is in precise correspondence with thefirst term in Eq. (2). In the limit T ≪ ( α − α c ) zν , we havealready seen that this term is subdominant to the danger-ously irrelevant second term in Eq. (2). However, movinginto the quantum critical region where T ≫ ( α − α c ) zν ,the contribution from Eq. (4) dominates all other terms,and we have σ ∼ T − /z Φ σ (0). The microscopic analysisof Ref. 16 obtained σ ∼ T − , which is valid only at T large enough where u can be neglected. Going beyondthe Gaussian theory, the values of z and ν have beendetermined in a d = 2 − ǫ expansion [20, 31], and alsoin quantum Monte Carlo simulations [32] with excellentagreement. Here, we have obtained these exponents in atheory with N complex fields Ψ directly in d = 1 , andobtained to order 1 /Nz = 2 − . N : ν = 1 − . N , (5)to be compared with the Monte Carlo estimates of z =1 .
97 and ν = 0 .
689 [32]. The value of Φ σ (0) has a non-universal cutoff dependence associated with anomalousdimension 2 − z . Note that the results above for σ inthe metallic and quantum-critical regimes imply a non-monotonic T dependence for α > α c , possibly consistentwith the observations of Ref. 11.Similar reasoning can be applied to the thermal con-ductivity κ , which can be computed from S using a sep-arate Kubo formula [33]. The scaling form analogous to Eq. (4) is κ sing = k B T ¯ h (cid:18) k B T ¯ h e D (cid:19) − /z Φ κ (cid:18) ¯ h ( α − α c )( k B T ) / ( zν ) (cid:19) (6)with Φ κ another universal function. We have verifiedthat the Gaussian prediction from S again agrees withthe perturbative AL contribution of the microscopic the-ory at low T . Our main result for the Wiedemann-Franzratio in Eq. (1) follows from W = ( k B /e ) Φ κ (0) / Φ σ (0),as the nonuniversal prefactor cancels out in ratio of thesescaling functions; the 1 /N expansion was carried out bygeneralizing the methods of Ref. 34.We now turn to an important conceptual issue: therole of charge conservation and associated normal modes.From hydrodynamic arguments we know that a one-dimensional metal or superconductor should support agapless plasmon, or a Mooij-Sch¨on normal mode [26].In d = 1, this mode is gapless and disperses as ω ∼ q ln / (1 / ( qR )). In our theory for the quantum criticalregion, the Cooper pair field Ψ carries charge 2 e but onlyexhibits diffusive dynamics with ω ∼ ˜ Dq , and there is noMooij-Sch¨on mode in the dynamics of the action S . Theanswer to this puzzle is contained in arguments madein Refs. 35 and 36 on the role of conservation laws inthe critical fluctuations of quantum transitions in metal-lic systems for which the order parameter is overdamped(as is the case here). These early works considered theonset of spin-density wave order in a metal; in the quan-tum critical region, the spin excitations consisted of dif-fusive paramagnons whose dynamics did not conserve to-tal spin. However, Ioffe and Millis [35] argued that theWard identities associated with spin conservation onlyimposed significant constraints on the effective action at ω > ∼ q , and played little role in the ω ∼ q regime im-portant for the critical fluctuations. Essentially the sameargument can be applied here: the Mooij-Sch¨on mode ispresent only at relatively high frequencies ω ∼ q , and thecritical theory S describes the overdamped Cooper pairmodes in the distinct region of phase space with ω ∼ q .The Mooij-Sch¨on fluctuations lead to oscillations in thelocal electrochemical potential, but these remain essen-tially decoupled from the critical modes described by S [35, 36] (see however, Eq. (8) below). It must be notedthat the action S is not valid for ω ∼ q and a completedescription in terms of the underlying fermions is neces-sary to obtain the proper dynamics, which will containthe Mooij-Sch¨on mode, as required.Further insight into this issue is gained by lowering thetemperature from the quantum critical regime into the“fluctuating superconductor” regime of Fig. 1 for α < α c .When k B T < ( α c − α ) the action S does not apply forthe smallest wavevectors and frequencies. The reasonsfor this are again analogous to arguments made for thespin-density-wave ordering transition in metals, as dis-cussed in Ref. 37. For the latter case, it was argued thatwith the emergence of long-range spin density wave order,the low energy fermionic particle-hole excitations at theordering wavevector were gapped out, and so the diffu-sive paramagnon action applied only for energies largerthan this gap. At energies smaller than the gap, spin-waves with dispersion ω ∼ q emerge, as a consequenceof Ward identities associated with spin conservation. Inthe superconducting case of interest here, there is no truelong-range superconducting order at any T >