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arXiv:2108.11392 · npj Quantum Materials

Quantum critical Eliashberg theory, the Sachdev-Ye-Kitaev superconductor and their holographic duals

Gian Andrea Inkof, Koenraad Schalm, Jörg Schmalian

Published Jun 1, 2022• 35 citations
arXivHTMLPDF

Abstract

Abstract Superconductivity is abundant near quantum critical points, where fluctuations suppress the formation of Fermi liquid quasiparticles and the BCS theory no longer applies. Two very distinct approaches have been developed to address this issue: quantum-critical Eliashberg theory and holographic superconductivity. The former includes a strongly retarded pairing interaction of ill-defined fermions, the latter is rooted in the duality of quantum field theory and gravity theory. We demonstrate that both are different perspectives of the same theory. We derive holographic superconductivity in form of a gravity theory with emergent space-time from a quantum many-body Hamiltonian—the Yukawa Sachdev-Ye-Kitaev model—where the Eliashberg formalism is exact. Exploiting the power of holography, we then determine the dynamic pairing susceptibility of the model. Our holographic map comes with the potential to use quantum gravity corrections to go beyond the Eliashberg regime.

DOI
10.1038/s41535-022-00460-8

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Abstract Superconductivity is abundant near quantum critical points, where fluctuations suppress the formation of Fermi liquid quasiparticles and the BCS theory no longer applies. Two very distinct approaches have been developed to address this issue: quantum-critical Eliashberg theory and holographic superconductivity. The former includes a strongly retarded pairing interaction of ill-defined fermions, the latter is rooted in the duality of quantum field theory and gravity theory. We demonstrate that both are different perspectives of the same theory. We derive holographic superconductivity in form of a gravity theory with emergent space-time from a quantum many-body Hamiltonian—the Yukawa Sachdev-Ye-Kitaev model—where the Eliashberg formalism is exact. Exploiting the power of holography, we then determine the dynamic pairing susceptibility of the model. Our holographic map comes with the potential to use quantum gravity corrections to go beyond the Eliashberg regime.
Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev–Ye–Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behavior, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional quantum dot model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.
Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev–Ye–Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behavior, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional quantum dot model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.
Quantum criticality plays a central role in understanding non-Fermi liquid behavior and unconventional superconductivity in strongly correlated systems. In this review, we explore the quantum critical Eliashberg theory, which extends conventional Eliashberg approaches to non-Fermi liquid regimes governed by critical fluctuations. We discuss the theoretical foundations and recent developments in the field, focusing on the interplay between electronic interactions and bosonic modes near quantum phase transitions as described in the Yukawa-coupled version of the Sachdev-Ye-Kitaev model. Special emphasis is placed on the breakdown of quasiparticle coherence, anomalous scaling behaviour, Cooper pairing without quasiparticles, and emergent universality in different physical settings. Starting from a zero-dimensional "quantum-dot" model, we discuss the generalization to higher spatial dimensions and demonstrate the connection between quantum-critical Eliashberg theory and holographic superconductivity. Our analysis provides a perspective on how quantum criticality shapes the dynamics of strongly correlated metals and superconductors.
The discovery of high-temperature superconductors has raises questions that extend beyond the scope of quasiparticles, a foundational concept in condensed matter theory. This dissertation explores new theoretical approaches to quantum matter without quasiparticle excitations, with a particular focus on quantum criticality and the associated superconductivity in correlated electron compounds. The soluble Sachdev-Ye-Kitaev (SYK) model has recently emerged as a fascinating platform to address the unusual metallic states with T-linear resistivity. We first consider differ- ent variants of random Hubbard models, which share many similarities to key aspects of the original SYK model. Using a large M analysis and a renormalization group method, we propose the existence of quantum critical points with fractionalized excitations separating two distinct phases, which provides insights into the quantum phase transitions and non-Fermi liquid behaviors observed in cuprate superconductors. In the second part of this dissertation, we examine the interplay between non-Fermi liquids and superconductivity with two different pairing mechanisms. Under the framework of random Hubbard model, we demonstrate that superconductivity emerging from non-Fermi liquids strongly deviates from the BCS theory by tuning the relative strength of hopping and exchange interactions. Furthermore, we investigate a two-dimensional Yukawa-SYK model with spatially randomn interactions coupling a Fermi surface to a scalar field associated with a quantum phase transition. Our theory agrees well with transport properties analysed in cuprates, such as a T-linear resistivity and Planckian dissipation.
We provide a quantitative and controlled analysis of the phase diagram of the Yukawa-Sachdev-Ye-Kitaev model on a lattice, in the normal and superconducting states. We analyze the entire crossover from Bardeen-Cooper-Schrieffer (BCS)/weak coupling to Eliashberg/strong coupling superconductivity, as a function of fermion-boson interaction strength and hopping parameter. Cooper pairs of sharp Fermi-liquid quasiparticles at weak coupling evolve into pairing of fully incoherent fermions in the non-Fermi liquid regime. The crossovers leave observable traces in the critical temperature, the zero-temperature and zero-energy gap, the entropy, and the phase stiffness.
The pairing of two electrons on a Fermi surface due to an infinitesimal attraction between them always results in a superconducting instability at zero temperature (T=0T=0T=0). The equivalent question of pairing instability on a Luttinger surface (LS)---190a contour of zeros of the propagator---instead leads to a quantum critical point (QCP) that separates a non-Fermi liquid and superconductor. A surprising and little understood aspect of pair fluctuations at this QCP is that their thermodynamics maps to that of the Sachdev-Ye-Kitaev (SYK) model in the strong coupling limit. Here, we offer a simple justification for this mapping by demonstrating that (i) LS models share the reparametrization symmetry of the $q\ensuremath{\rightarrow}\ensuremath{\infty}$ SYK model with qqq-body interactions close to the LS, and (ii) the enforcement of gauge invariance results in a $\frac{1}{\sqrt{\ensuremath{\tau}}}$ ($\ensuremath{\tau}\ensuremath{\sim}{T}^{\ensuremath{-}1}$) behavior of the fluctuation propagator near the QCP, as is the feature of SYK conformal Green functions, but leaves the overall form of the free energy map robust.