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arXiv:1904.07240 · Physical Review Letters

Solvable Strong-Coupling Quantum-Dot Model with a Non-Fermi-Liquid Pairing Transition

Yuxuan Wang

Published Jan 7, 2020• 88 citations
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Abstract

We show that a random interacting model exhibits solvable non-Fermi-liquid behavior and exotic pairing behavior. This model, dubbed as the Yukawa-SYK model, describes the random Yukawa coupling between M quantum dots each hosting N flavors of fermions and N^{2} bosons that self-tune to criticality at low energies. The diagrammatic expansion is controlled by 1/MN, and the results become exact in a large-M, large-N limit. We find that pairing only develops within a region of the (M,N) plane-even though the pairing interaction is strongly attractive, the incoherence of the fermions can spoil the forming of Cooper pairs, rendering the system a non-Fermi liquid down to zero temperature. By solving the Eliashberg equation and the renormalization group equation, we show that the transition into the pairing phase exhibits Kosterlitz-Thouless quantum-critical behavior.

DOI
10.1103/physrevlett.124.017002

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We show that a random interacting model exhibits solvable non-Fermi-liquid behavior and exotic pairing behavior. This model, dubbed as the Yukawa-SYK model, describes the random Yukawa coupling between M quantum dots each hosting N flavors of fermions and N^{2} bosons that self-tune to criticality at low energies. The diagrammatic expansion is controlled by 1/MN, and the results become exact in a large-M, large-N limit. We find that pairing only develops within a region of the (M,N) plane-even though the pairing interaction is strongly attractive, the incoherence of the fermions can spoil the forming of Cooper pairs, rendering the system a non-Fermi liquid down to zero temperature. By solving the Eliashberg equation and the renormalization group equation, we show that the transition into the pairing phase exhibits Kosterlitz-Thouless quantum-critical behavior.
Non-Fermi liquids (NFLs) are a class of strongly interacting gapless fermionic systems without long-lived quasiparticle excitations.An important group of NFL models feature itinerant fermions coupled to soft bosonic fluctuations near a quantum-critical point and are widely believed to capture the essential physics of many unconventional superconductors.However, numerically, the direct observation of a canonical NFL behavior in such systems, characterized by a power-law form in the Green's function, has been elusive.Here, we consider a Sachdev-Ye-Kitaev (SYK)-like model with random Yukawa interaction between critical bosons and fermions (dubbed the Yukawa-SYK model).We show that it is immune from the minus-sign problem and hence can be solved exactly via large-scale quantum Monte Carlo simulation beyond the large-N limit accessible to analytical approaches.Our simulation demonstrates that the Yukawa-SYK model features "self-tuned quantum criticality"; namely, the system is critical independent of the bosonic bare mass.We put these results to the test at finite N, and our unbiased numerics reveal clear evidence of these exotic quantum-critical NFL properties-the power-law behavior in the Green's function of fermions and bosons-which propels the theoretical understanding of critical Planckian metals and unconventional superconductors.
We study the quantum phase transition upon varying the fermionic density ν in a solvable model with random Yukawa interactions between N bosons and M fermions, dubbed the Yukawa-SYK model.We show that there are two distinct phases in the model: an incompressible state with gapped excitations and an exotic quantumcritical, non-Fermi liquid state with exponents varying with ν.We show analytically and numerically that the quantum phase transition between these two states is first-order, as for some range of ν the NFL state has a negative compressibility.In the limit N/M → ∞, the first-order transition gets weaker and asymptotically becomes second-order, with an exotic quantum-critical behavior.We show that fermions and bosons display highly unconventional spectral behavior in the transition region.
The interplay of non-Fermi liquid and superconductivity born out of strong dynamical interactions is at the heart of the physics of unconventional superconductivity. As a solvable platform of the strongly correlated superconductors, we study the pairing instabilities of the Yukawa-Sachdev-Ye-Kitaev (Yukawa-SYK) model, which describes spin-1/2 fermions coupled to bosons by the random, all-to-all, spin-independent and dependent Yukawa interactions. In contrast to the previously studied models, the random Yukawa couplings are sampled from a collection of Gaussian ensembles whose variances follow a continuous distribution rather than being fixed to a constant. By tuning the analytic behaviour of the distribution, we could control the fermion incoherence to systematically examine various normal states ranging from the Fermi liquid to non-Fermi liquids that are different from the conformal solution of the SYK model with a constant variance. Using the linearised Eliashberg theory, we show that the onset of the unconventional spin-triplet pairing is preferred with the spin-dependent interactions while all pairing channels show instabilities with the spin-independent interactions. Although the interactions shorten the lifetime of the fermions in the non-Fermi liquid, the same interactions also dress the bosons to strengthen the tendency to pair the incoherent fermions. As a consequence, the onset temperature T_c T c of the pairing is enhanced in the non-Fermi liquid compared to the case of the Fermi liquid.
We study a model of NNN fermions in a quantum dot, coupled to MMM bosons by a disorder-induced complex Yukawa coupling [Yukawa Sachdev-Ye-Kitaev (SYK) model], in order to explore the interplay between non-Fermi liquid and superconductivity in a strongly coupled, (quantum-)critical environment. We analyze the phase diagram of the model for an arbitrary complex interaction and arbitrary ratio of N/MN/MN/M, with special focus on the two regimes of non-Fermi-liquid behavior: an SYK-like behavior with a power-law frequency dependence of the fermionic self-energy and an impuritylike behavior with frequency independent self-energy. We show that the crossover between the two can be reached by varying either the strength of the fermion-boson coupling or the ratio M/NM/NM/N. We next argue that in both regimes the system is unstable to superconductivity if the strength of time-reversal-symmetry-breaking disorder is below a certain threshold. We show how the corresponding onset temperatures vary between the two regimes. We argue that the superconducting state is highly unconventional with an infinite set of minima of the condensation energy at T=0T=0T=0, corresponding to topologically different gap functions. We discuss in detail similarities and differences between this model and the model of dispersion-full fermions tuned to a metallic quantum-critical point, with an effective singular dynamical interaction $V(\mathrm{\ensuremath{\Omega}})\ensuremath{\propto}1/{|\mathrm{\ensuremath{\Omega}}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model).
We provide a quantitative and controlled analysis of the phase diagram of the the Yukawa-SYK model on a lattice, in the normal and superconducting states. We analyze the entire crossover from 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.
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.