The cavity method for quantum disordered systems: from transverse random field ferromagnets to directed polymers in random media

TL;DR

This paper extends the cavity method to quantum disordered systems, linking quantum phase transitions in spin models to classical directed polymers in random media, highlighting the role of rare events near criticality.

Contribution

It introduces a quantum cavity approach with approximations to analyze spin systems with disorder, connecting quantum phase transitions to classical polymer problems.

Findings

Quantum cavity equations describe ferromagnetic-paramagnetic transition.

The quantum problem maps onto classical directed polymer in a random medium.

A Griffiths phase emerges near the quantum critical point due to rare events.

Abstract

After reviewing the basics of the cavity method in classical systems, we show how its quantum version, with some appropriate approximation scheme, can be used to study a system of spins with random ferromagnetic interactions and a random transverse field. The quantum cavity equations describing the ferromagnetic-paramagnetic phase transition can be transformed into the well-known problem of a classical directed polymer in a random medium. The glass transition of this polymer problem translates ino the existence of a `Griffith phase' close to the quantum phase transition of the quantum spin problem, where the physics is dominated by rare events.