Critical metallic phase in the overdoped random $t$-$J$ model

TL;DR

This paper introduces a large-$M$ limit model of electrons with random interactions that exhibits a critical non-Fermi-liquid metallic state at high doping, characterized by linear-in-temperature resistivity and particle-hole asymmetry, relevant to cuprate superconductors.

Contribution

The study presents a novel Sachdev-Ye-Kitaev-like model for the overdoped regime of cuprates, revealing a critical metallic phase with unique symmetry and transport properties.

Findings

Existence of a critical non-Fermi-liquid metallic ground state at large doping.

Linear-in-temperature resistivity over a broad doping range.

Particle-hole asymmetry consistent with experimental observations.

Abstract

We investigate a model of electrons with random and all-to-all hopping and spin exchange interactions, with a constraint of no double occupancy. The model is studied in a Sachdev-Ye-Kitaev-like large-$M$ limit with SU($M$) spin symmetry. The saddle point equations of this model are similar to appoximate dynamic mean field equations of realistic, non-random, $t$-$J$ models. We use numerical studies on both real and imaginary frequency axes, along with asymptotic analyses, to establish the existence of a critical non-Fermi-liquid metallic ground state at large doping, with the spin correlation exponent varying with doping. This critical solution possesses a time-reparametrization symmetry, akin to SYK models, which contributes a linear-in-temperature resistivity over the full range of doping where the solution is present. It is therefore an attractive mean-field description of the overdoped region of cuprates, where experiments have observed a linear-$T$ resistivity in a broad region. The critical metal also displays a strong particle-hole asymmetry, which is relevant to Seebeck coefficient measurements. We show that the critical metal has an instability to a low-doping spin-glass phase, and compute a critical doping value. We also describe the properties of this metallic spin-glass phase.