Phase Diagram of the Quantum Random Energy Model

TL;DR

This paper rigorously proves the free energy formula for the quantum random energy model, confirming the phase transition points and providing a detailed phase diagram analysis.

Contribution

It offers a rigorous proof of Goldschmidt's formula for the quantum REM's free energy and accurately locates the phase transitions.

Findings

Verification of the first order transition location

Confirmation of the freezing transition point

Validation of the phase diagram structure

Abstract

We prove Goldschmidt's formula [Phys. Rev. B 47 (1990) 4858] for the free energy of the quantum random energy model. In particular, we verify the location of the first order and the freezing transition in the phase diagram. The proof is based on a combination of variational methods on the one hand, and percolation bounds on large-deviation configurations in combination with simple spectral bounds on the hypercube's adjacency matrix on the other hand.