Elsevier

Annals of Physics

Volume 417, June 2020, 168120
Annals of Physics

Eliashberg equations for an electron–phonon version of the Sachdev–Ye–Kitaev model: Pair breaking in non-Fermi liquid superconductors

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Abstract

We present a theory that is a non-Fermi-liquid counterpart of the Abrikosov–Gor’kov pair-breaking theory due to paramagnetic impurities in superconductors. To this end we analyze a model of interacting electrons and phonons that is a natural generalization of the Sachdev–Ye–Kitaev-model. In the limit of large numbers of degrees of freedom, the Eliashberg equations of superconductivity become exact and emerge as saddle-point equations of a field theory with fluctuating pairing fields. In its normal state the model is governed by non-Fermi liquid behavior, characterized by universal exponents. At low temperatures a superconducting state emerges from the critical normal state. We study the role of pair-breaking on Tc, where we allow for disorder that breaks time-reversal symmetry. For small Bogoliubov quasi-particle weight, relevant for systems with strongly incoherent normal state, Tc drops rapidly as function of the pair breaking strength and reaches a small but finite value before it vanishes at a critical pair-breaking strength via an essential singularity. The latter signals a breakdown of the emergent conformal symmetry of the non-Fermi liquid normal state.

Introduction

The dynamical theory of phonon-mediated superconductivity was formulated by GerasimMatveevich Eliashberg in a pioneering tour de force of quantum many-body theory [1], [2]. Considering the regime where phonon frequencies are much smaller than the Fermi energy, electrons follow the lattice motion almost instantly. In this limit, Migdal had shown that electron–phonon vertex corrections become small [3]. Then a complicated intermediate-coupling problem suddenly becomes tractable. A closed, self-consistent dynamical theory emerges that is not limited to the regime of weak electron–phonon interactions. The Eliashberg formalism follows the Gor’kov–Nambu description of superconductivity [4], [5], reflecting the broken global U1 symmetry, associated with charge conservation. The propagation of particles and the conversion of particles into holes are described by two self energies Σω and Φω, respectively. Using the Eliashberg theory, important advances were made in understanding the physical properties of superconductors with a dimensionless electron–phonon coupling of order unity [6], [7], [8], [9], [10], [11], [12].

The Eliashberg formalism has been applied to study superconductivity in problems that go significantly beyond the original electron–phonon problem [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. When an electronic system becomes quantum critical, soft degrees of freedom emerge. The retarded nature of the coupling to such soft excitations makes an analysis in the spirit Eliashberg’s approach, with a dynamical pairing field Φω, natural. Since realistic models of quantum critical pairing usually possess no natural small parameter, a controlled approach that leads to an Eliashberg-like formalism is highly desirable.

Recently, two of us introduced and solved a model for the electron–phonon interaction in non-Fermi liquids [24]. It is a natural generalization of the Sachdev–Ye–Kitaev (SYK) model [25], [26], [27], [28], [29] and yields superconductivity due to electron–phonon interactions with quantum critical behavior in the normal state. The model becomes solvable in the limit of infinite number of degrees of freedom. The Eliashberg equations of superconductivity, with self-consistently determined electron and phonon propagators, become exact. The formalism yields rich non-Fermi liquid behavior in the normal state and gives rise to superconductivity at low temperatures. Related interesting descriptions of superconductivity in SYK-like models have also been discussed in Refs. [30], [31], [32]. While SYK models are dominated by random interactions, the belief is that the non-Fermi liquid behavior that occurs is in fact more general and may also offer insights into non-random systems.

In its normal state the model of Ref. [24] is governed by two non-Fermi liquid fixed points, characterized by distinct universal exponents. The weakened ability of such non-Fermi liquid electronic states to form Cooper pairs is offset by an increasingly singular pairing interaction, leading to coherent superconductivity in such incoherent systems. This result is closely related to the generalized Cooper theorem of quantum-critical pairing put forward by Abanov et al. in Ref. [15]. In Ref. [24] the ground state was shown to be characterized by sharp Bogoliubov quasiparticles. However, the incoherent nature of the normal-state leads to a much reduced spectral weight ZB of the Bogoliubov quasiparticles. For small values of ZB a reduction in the condensation energy occurs. At the same time, the transition temperature remains unchanged. This behavior is reminiscent of superconductivity in systems with non-pair-breaking impurities, where Anderson’s theorem guarantees an unchanged transition temperature, while the superconducting state becomes more fragile the larger the disorder strength, with e.g. a strongly reduced superfluid stiffness [33], [34], [35].

An important issue in the investigation of superconducting states is their robustness with respect to pair-breaking disorder. The topic was pioneered by Abrikosov and Gor’kov, who analyzed the role of paramagnetic impurities in conventional superconductors [36] and found a suppression of Tc determined by logTc0Tc=ψ1212πτTcψ12 with scattering rate due to paramagnetic impurities τ1 and digamma function ψ. Tc0 is the transition temperature without pair breaking. The inclusion of quantum dynamics of the impurities, unconventional pairing states, critical normal states, and strong impurity scattering are topics of ongoing theoretical investigations, see e.g. Refs. [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57] for an incomplete list of publications. In this context an interesting question is the nature by which superconductivity vanishes due to pair breaking if the normal state is quantum critical.

In this paper we generalize the electron–phonon SYK model to analyze the robustness of pairing in quantum-critical systems against pair-breaking effects due to time-reversal symmetry violation. We solve the modified Eliashberg equations and find a suppression of the transition temperature as a function of a pair-breaking parameter α, with Tc vanishing at a critical pair-breaking strength αc. While the qualitative trends are similar to the Abrikosov–Gor’kov theory [36], there are key distinctions in the overall dependence of Tc on α. Near αc0.62, we find a behavior Tcααc=TexpDαcα,where D is a non-universal constant and T an energy scale that we discuss below. This behavior is similar to the scaling near a Berezinskii–Kosterlitz–Thouless (BKT) transition [58], [59]. Such BKT-scaling was argued to be generic for systems with a transition from a conformal to a non-conformal phase [60]. Given the conformal symmetry of the SYK model [26], which is relevant to our normal state, the result Eq. (1) for the superconducting transition temperature is further confirmation of the expectation put forward in Ref. [60]. Thus, the change of the superconducting transition temperature as function of a pair-breaking impurity concentration may serve as a tool to identify whether a normal state can be effectively thought of as a critical state with an underlying conformal symmetry. A behavior like that of Eq. (1) occurs in the coupling constant dependence of the mass scale near the chiral symmetry breaking point of 2+1-dimensional quantum electrodynamics [61]. In the context of coherent versus incoherent pairing, such behavior was first seen in an Eliashberg theory near a magnetic instability in Ref. [15]. An interesting renormalization group perspective of Eq. (1) in the context of superconductivity was recently given in Ref. [20]. An appeal of our approach is that the critical coupling αc in Eq. (1) acquires a clear physical and potentially tunable interpretation as a pair-breaking parameter due to time-reversal symmetry breaking disorder.

In addition to the behavior near the critical pair-breaking strength αc we also analyze the interplay between normal-state incoherency and the robustness of superconductivity with respect to pair breaking. In the incoherent, strong coupling regime of the system, we find that Tc is already substantially suppressed for αααc, where the crossover scale αZB is proportional to the small Bogoliubov quasiparticle weight for α=0, i.e. TcαZBTc0,with Tc0=Tcα=0. Thus, a pairing state that emerges from an incoherent normal state with small ZB is particularly fragile against pair breaking. These findings are summarized in Fig. 1b.

In what follows we introduce our model, show that the solution is given by a set of coupled Eliashberg equations, and present the solution of this set of equations.

Section snippets

Eliashberg equations

We start from the Hamiltonian H=μi=1Nσ=±ciσciσ+12k=1Mπk2+ω02ϕk2+1Nij,σNkMgij,kciσcjσϕk,with electron annihilation and creation operators ciσ and ciσ, respectively, that obey ciσ,cjσ+=δij δσσ and ciσ,cjσ+=0 with spin σ=±1. In addition we have phonons ϕk with canonical momentum πk, such that ϕk,πk=iδkk. Here i,j=1N refer to electrons and k=1M to the phonons. In what follows we mostly consider the limit N=M.

The problem becomes solvable because of the fully-connected nature of the

Numerical analysis and gap equations in the scaling limit

We first present the results obtained from a complete numerical solution of the coupled Eliashberg equations. In Fig. 2 we show the superconducting transition temperature as a function of the pair-breaking parameter α for varying dimensionless coupling strength g. While Tcα is weakly g-dependent for small g, we find a strong variation of the initial suppression of the transition temperature in the strong coupling limit g>1. Tcα seems to vanish at a g-independent critical value αc. While the

Summary

We introduced and solved a model of electrons that interact with phonons via a random electron–phonon coupling. The theory is a natural generalization of the Sachdev–Ye–Kitaev model to the problem of interacting electrons and phonons and gives rise to a superconductivity. Typical for fully connected models, an exact solution becomes possible in the limit N. In our case this exact solution corresponds to the coupled Eliashberg equations of superconductivity. Since the normal state of the model

CRediT authorship contribution statement

Daniel Hauck: Methodology, Software, Validation, Formal analysis, Visualization, Writing - review & editing. Markus J. Klug: Methodology, Validation, Formal analysis, Visualization, Writing - review & editing. Ilya Esterlis: Methodology, Software, Validation, Formal analysis, Visualization, Writing - review & editing. Jörg Schmalian: Conceptualization, Methodology, Validation, Formal analysis, Visualization, Writing - original draft, Writing - review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We are grateful to Andrey V. Chubukov, Jonas Karcher, and Yoni Shattner for helpful discussions. We are particularly thankful to Andrey V. Chubukov for pointing out to us the importance of the logarithmic term in Eq. (22). IE acknowledges support from the Harvard Quantum Initiative Postdoctoral Fellowship in Science and Engineering and the DARPA DRINQS program (award D18AC00033).

References (63)

  • AndersonP.W.

    J. Phys. Chem. Solids

    (1959)
  • EliashbergG.M.

    Sov. Phys.—JETP

    (1960)
  • EliashbergG.M.

    Sov. Phys.—JETP

    (1961)
  • MigdalA.B.

    Sov. Phys.—JETP

    (1958)
  • GorkovL.P.

    Sov. Phys.—JETP

    (1958)
  • NambuY.

    Phys. Rev.

    (1960)
  • SchriefferJ.R. et al.

    Phys. Rev. Lett.

    (1963)
  • ScalapinoD.J. et al.

    Phys. Rev.

    (1966)
  • McMillanW.L.

    Phys. Rev.

    (1968)
  • McMillanW.L. et al.

    Superconductivity, Tunneling and Strong Coupling Superconductivity, Vol. 1

    (1969)
  • AllenP.B. et al.

    Phys. Rev. B

    (1975)
  • CarbotteJ.P.

    Rev. Modern Phys.

    (1990)
  • BonesteelN.E. et al.

    Phys. Rev. Lett.

    (1996)
  • SonD.T.

    Phys. Rev. D

    (1999)
  • AbanovAr. et al.

    Europhys. Lett.

    (2001)
  • Ar. AbanovAr. et al.

    Europhys. Lett.

    (2001)
  • RoussevR. et al.

    Phys. Rev. B

    (2001)
  • ChubukovA.V. et al.

    Phys. Rev. B

    (2005)
  • MetlitskiM.A. et al.

    Phys. Rev. B

    (2015)
  • RaghuS. et al.

    Phys. Rev. B

    (2015)
  • WangY. et al.

    Phys. Rev. Lett.

    (2016)
  • AbanovA. et al.

    Phys. Rev. B

    (2019)
  • WuY.-M. et al.

    Phys. Rev. B

    (2019)
  • EsterlisI. et al.

    Phys. Rev. B

    (2019)
  • SachdevS. et al.

    Phys. Rev. Lett.

    (1993)
  • GeorgesA. et al.

    Phys. Rev. Lett.

    (2000)
  • SachdevS.

    Phys. Rev. Lett.

    (2010)
  • KitaevA.

    Hidden correlations in the Hawking radiation and thermal noise

    (2015)
  • KitaevA.

    A simple model of quantum holography

    (2015)
  • PatelA.A. et al.

    Phys. Rev. Lett.

    (2018)
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    This article is part of the Special Issue: Eliashberg theory at 60.

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