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arXiv:2111.09906 · Physical review. B./Physical review. B

Solvable model for a charge- 4 e superconductor

N. V. Gnezdilov, Yuxuan Wang

Published Sep 15, 2022• 18 citations
arXivHTMLPDF

Abstract

A charge-$4e$ superconductor forms due to the condensation of quartets of electrons. While in previous works, the mechanism for the formation of charge-$4e$ superconductivity has been analyzed in terms of the binding of Cooper pairs in unconventional superconductors, its properties in the fermionic sector have not been studied systematically due to its inherently interacting nature even at the mean-field level. Here we propose a solvable model for a charge-$4e$ superconductor---a spinful version of the Sachdev-Ye-Kitaev model with an anomalous quartic term. We show that the ground state is gapless and resembles a heavy Fermi liquid. We analytically solve for the superfluid density and show that it is perturbative in the strength of the charge-$4e$ order parameter, in sharp contrast with a regular (charge-$2e$) superconductor. Upon lowering the temperature, we show that the correlation between charge-$4e$ order and regular interaction terms can drive a first-order phase transition to a charge-$2e$ superconducting state.

DOI
10.1103/physrevb.106.094508

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A charge-4e4e4e superconductor forms due to the condensation of quartets of electrons. While in previous works, the mechanism for the formation of charge-4e4e4e superconductivity has been analyzed in terms of the binding of Cooper pairs in unconventional superconductors, its properties in the fermionic sector have not been studied systematically due to its inherently interacting nature even at the mean-field level. Here we propose a solvable model for a charge-4e4e4e superconductor---a spinful version of the Sachdev-Ye-Kitaev model with an anomalous quartic term. We show that the ground state is gapless and resembles a heavy Fermi liquid. We analytically solve for the superfluid density and show that it is perturbative in the strength of the charge-4e4e4e order parameter, in sharp contrast with a regular (charge-2e2e2e) superconductor. Upon lowering the temperature, we show that the correlation between charge-4e4e4e order and regular interaction terms can drive a first-order phase transition to a charge-2e2e2e superconducting state.
We investigate models of electrons in the Sachdev-Ye-Kitaev class with random and all-to-all electron hopping, electron spin exchange, and Cooper-pair hopping. An attractive on-site interaction between electrons leads to superconductivity at low temperatures. Depending on the relative strengths of the hopping and spin exchange, the normal state at the critical temperature is either a Fermi-liquid or a non-Fermi liquid. We present a large-MMM [where spin symmetry is enlarged to SU(M)]\mathrm{SU}(M)]SU(M)] study of the normal state to superconductor phase transition. We describe the transition temperature, the superconducting order parameter, and the electron spectral functions. We contrast between Fermi liquid and non-Fermi liquid normal states: we find that for weaker attractive on-site interaction there is a relative enhancement of Tc{T}_{c}Tc​ when the normal state is a non-Fermi liquid, and correspondingly a strong deviation from BCS limit. Also, the phase transition in this case becomes a first-order transition for strong non-Fermi liquids. On the other hand, for stronger on-site interaction, there is no appreciable difference in Tc{T}_{c}Tc​ between whether the superconductivity emerges from a Fermi liquid or a non-Fermi liquid. Notable features of superconductivity emerging from a non-Fermi liquid are that the superconducting electron spectral function is different from the Fermi-liquid case, with additional peaks at higher energies, and there is no Hebel-Slichter peak in the NMR relaxation rate in the non-Fermi liquid case.
Characterizing the interplay between non-Fermi-liquid physics and superconductivity is an important open problem in condensed matter physics. From a theoretical standpoint, the challenge stems from the difficulty in identifying controlled models where these phenomena can be simultaneously treated on equal footing. Here, the authors propose a new model of strongly interacting electrons and phonons, in close analogy with the well-known Sachdev-Ye-Kitaev model, and, via exact solution, explore how superconductivity can emerge from quantum critical and highly incoherent non-Fermi-liquid normal states. Many of the results are in qualitative agreement with observations made in iron-based, heavy-fermion, and cuprate superconductors.
Many features of charge-4e4e4e superconductors remain unknown because even the ``mean-field Hamiltonian'' describing them is an interacting model. Here we introduce an interacting model to describe a charge-4e4e4e superconductor (SC) deep in the superconducting phase and explore its properties using quantum Monte Carlo (QMC) simulations. The QMC is sign-problem-free but only when a Majorana representation is employed. As a function of the chemical potential we observe two sharply-distinct behaviors: a ``strong'' quarteting phase in which charge-4e4e4e quartets are tightly bound (like molecules) so that charge-2e2e2e pairing does not occur even in the temperature $T\ensuremath{\rightarrow}0$ limit, and a ``weak'' quarteting phase in which a further transition to a charge-2e2e2e superconducting phase occurs at a lower critical temperature. Analogous issues arise in a putative Z4{Z}_{4}Z4​ spin liquid with a pseudo-Fermi surface and other interacting models with composite order parameters. Under certain circumstances, we also identified a stable T=0T=0T=0 charge-4e4e4e SC phase with gapless nodal quasiparticles. We further discuss possible relevance of our results to various experimental observations in 18\frac{1}{8}81​-doped LBCO.
We introduce a spinful variant of the Sachdev-Ye-Kitaev model with an effective time-reversal symmetry, which can be solved exactly in the limit of a large number NNN of degrees of freedom. At low temperature, its phase diagram includes a compressible non-Fermi liquid and a strongly correlated spin singlet superconductor that shows a tunable enhancement of the gap ratio predicted by BCS theory. These two phases are separated by a first-order transition, in the vicinity of which a gapless superconducting phase, characterized by a nonzero magnetization, is stabilized upon applying a Zeeman field. We study equilibrium transport properties of such superconductors using a lattice construction and propose a physical platform based on topological insulator flakes where they may arise from repulsive electronic interactions.
A phase of matter in which fermion quartets form a superconducting condensate, rather than the paradigmatic Cooper pairs, is a recurrent subject of experimental and theoretical studies. However, a comprehensive microscopic understanding of charge-4e4e4e superconductivity as a quantum phase is lacking. Here, we study a two-orbital tight-binding model with attractive Hubbard-type interactions. Such a model naturally provides the Bose-Einstein condensate as a limit for electron quartets and supports charge-4e4e4e superconductivity, as we show by mapping it to a spin-1/2 chain in this perturbative limit. Using density matrix renormalization group calculations for the one-dimensional case, we further establish that the ground state is indeed a superfluid phase of 4e4e4e charge carriers and that this phase can be stabilized well beyond the perturbative regime. Importantly, we demonstrate that 4e4e4e condensation dominates over 2e2e2e condensation even for nearly decoupled orbitals, which is a more likely scenario in electronic materials. Our model paves the way for both experimental and theoretical exploration of 4e4e4e superconductivity and provides a natural starting point for future studies beyond one dimension or more intricate 4e4e4e states.
The periodic Anderson model is a classic theoretical model for understanding novel physics in heavy fermion systems. Here, we modify it with the Sachdev-Ye-Kitaev interaction, (random all-to-all interaction) thus the resultant model admits an exact solution at large- N (e.g. spin flavor) limit. By analytical field theory arguments and numerical calculations, we establish that the system supports a low-temperature (heavy) Fermi liquid and more interestingly a non-Fermi liquid solution at elevated temperature/energy. For physical observable, the latter one contributes a sharp peak at Fermi energy for spectral function, a non-Fermi liquid-like resistivity and shows a robust Fano lineshape in tunneling spectrum. This system may be simulated by ultracold atom gases and can serve as a good playground for studying ubiquitous symmetry-breaking instabilities like unconventional superconductivity or topological orders in generic heavy fermion systems.