Replica symmetry breaking in spin glasses in the replica-free Keldysh formalism

TL;DR

This paper demonstrates how ultrametric structures and replica symmetry breaking naturally emerge in the late-time dynamics of quantum spin glasses within a replica-free Keldysh formalism, linking dynamical and equilibrium descriptions.

Contribution

It introduces a novel real-time, replica-free approach to analyze spin glasses, connecting dynamical equations to ultrametricity and replica symmetry breaking phenomena.

Findings

Ultrametric matrices arise from Dyson-Keldysh equations in the glassy limit.

The formalism reproduces full replica symmetry breaking in the equilibrium limit.

Application to the spherical quantum p-spin model shows one-step RSB only.

Abstract

We show that the algebra of Parisi ultrametric matrices is recovered by the real-time, replica-free, Dyson-Keldysh equations of infinite-range quantum spin glasses in the late time glassy limit. This connects to earlier results on classical and quantum systems showing how ultrametricity emerges from the persistent slow aging dynamics of the glass phase. The stationary spin glass state thereby spontaneously breaks thermal symmetry, or the Kubo-Martin-Schwinger relation of a state in global thermal equilibrium. We describe the Keldysh path integral of the infinite-range Ising model in transverse and longitudinal fields, and in the context of the Landau expansion of the action functional, show how the long-time limit connects to the full replica symmetry breaking obtained in the equilibrium formalism. We also illustrate our formalism by applying it to the spherical quantum $p$-spin model, which only exhibits one-step replica symmetry breaking