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

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 saddlepoint equations of a field theory with fluctuating pairing fields. In its normal state the model is governed by two non-Fermi liquid fixed points, characterized by distinct universal exponents. At low temperatures a superconducting state emerges from the critical normal state. We study the role of pair-breaking on T c , where we allow for disorder that breaks time-reversal symmetry. For small Bogoliubov quasi-particle weight, relevant for systems with strongly incoherent normal state, T c 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.Keywords: Eliashberg Theory, Superconductivity, Non-Fermi Liquid, Sachdev-Ye-Kitaev Model, Pair Breaking IntroductionThe dynamical theory of phonon-mediated superconductivity was formulated by Gerasim Matveevich 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 of the electrons, 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, selfconsistent 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 U (1) symmetry, associated with charge conservation. The propagation of particles and the conversion of particles into holes are described by two self energies Σ (ω) and Φ (ω), 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 Φ (ω), 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....

The interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal

“…44 for arbitrary γ. A similar result has been recently found in the study of the pairing in the SYK-type model [74][75][76] (see the article by Daniel Hauck, Markus Klug, Ilya Esterlis, and Jörg Schmalian for this issue).…”

Quantum phase transition in the Yukawa-SYK model

“…Our goal is to find the fully dressed bosonic and fermionic propagators G −1 (iω) = iω + (iω) + μ and D −1 (i ) = 2 + (i ) + m 2 0 . We extended results of earlier analysis at half-filling [39][40][41] to μ = 0 and found that for M, N 1 the fermionic and bosonic selfenergies are expressed self-consistently via the Schwinger-Dyson equations…”

“…The Y-SYK model has been earlier studied at half-filling [39,40,45]. It was shown that the interaction "self-tunes" the system into a NFL, quantum-critical (QC) regime, despite that a bare bosonic mass is finite.This QC regime may in turn become unstable towards non-BCS superconductivity [39][40][41][46][47][48][49]28,50,51]. Like the SYK model, the Y-SYK model also saturates the upper bound for the onset rate of quantum chaos [52], indicating the existence a classical holographic dual.…”

“…In this communication, we consider the generalization of the SYK model, the Yukawa-SYK (Y-SYK) model [39][40][41], in which M flavors of dispersion-less fermions in a quantum dot randomly interact with N flavors of massive bosons, e.g., optical phonons or gapped collective spin or charge fluctuations. The interest to this model has been triggered by its rich and unconventional physics, and by recent experimental discoveries of strongly correlated behavior in flat band systems like magic angle twisted bilayer graphene [42,43] and d xy band in Fe-based superconductors [44].…”

Rényi entanglement entropy of Fermi and non-Fermi liquids: Sachdev-Ye-Kitaev model and dynamical mean field theories

“…SYK model [27,28] and its various extensions [30,32,[43][44][45][46][47][48][49] have emerged in recent years as a major paradigm to study phases of strongly interacting fermions, e.g., NFL, marginal and heavy Fermi liquids, strong-coupling superconductivity [32,[46][47][48][49][50][51], and quantum phase transitions [30,48,49,52,53]. Remarkably, these models can be solved exactly in a suitable large-N limit, without resorting to any kind of perturbative treatment.…”