Universality of replica-symmetry breaking in the transverse field Sherrington-Kirkpatrick model

TL;DR

This paper extends the proof of replica-symmetry breaking (RSB) in the transverse field SK model to more general interactions, showing RSB persists under certain conditions and distinguishing it from ferromagnetic order.

Contribution

It provides a rigorous extension of RSB existence to models with general random interactions using approximate integration by parts and correlation function analysis.

Findings

RSB exists in the generalized transverse field SK model under weak fields and low temperatures.

Variance of overlap between replica spins remains non-zero, indicating RSB.

RSB is distinguished from $ ext{Z}_2$-symmetry breaking by absence of ferromagnetic order.

Abstract

The existence theorem for replica-symmetry breaking (RSB) in the transverse field Sherrington-Kirkpatrick (SK) model is extended to the model with a general random exchange interactions. The relation between the expectation value of the exchange interaction energy and the Duhamel correlation function of spin operators can be obtained by an approximate integration by parts for general random interactions. In addition to the Falk-Bruch inequality, these explicit evaluations enable us to prove that the variance of overlap between two replica spin operators does not vanish under sufficiently weak transverse field in sufficiently low temperature. The absence of the ferromagnetic long range order is also shown to distinguish RSB from the $\mathbb Z_2$-symmetry breaking.