Self-Duality and Phase Structure of the 4D Random-Plaquette Z_2 Gauge Model

TL;DR

This paper investigates the phase structure of the 4D random-plaquette Z_2 gauge model, highlighting its self-duality, phase transitions, and implications for quantum memory error thresholds.

Contribution

It demonstrates the self-duality of the 4D RPGM and its relation to the 2D RBIM, providing numerical phase boundary predictions relevant for quantum error correction.

Findings

Phase boundary curve matches duality predictions at high temperature.

First-order transition for small p, second-order for larger p.

Estimated error threshold p=0.110±0.002 for quantum memory.

Abstract

In the present paper, we shall study the 4-dimensional Z_2 lattice gauge model with a random gauge coupling; the random-plaquette gauge model(RPGM). The random gauge coupling at each plaquette takes the value J with the probability 1-p and -J with p. This model exhibits a confinement-Higgs phase transition. We numerically obtain a phase boundary curve in the (p-T)-plane where T is the "temperature" measured in unit of J/k_B. This model plays an important role in estimating the accuracy threshold of a quantum memory of a toric code. In this paper, we are mainly interested in its "self-duality" aspect, and the relationship with the random-bond Ising model(RBIM) in 2-dimensions. The "self-duality" argument can be applied both for RPGM and RBIM, giving the same duality equations, hence predicting the same phase boundary. The phase boundary curve obtained by our numerical simulation almost coincides with this predicted phase boundary at the high-temperature region. The phase transition is of first order for relatively small values of p < 0.08, but becomes of second order for larger p. The value of p at the intersection of the phase boundary curve and the Nishimori line is regarded as the accuracy threshold of errors in a toric quantum memory. It is estimated as p=0.110\pm0.002, which is very close to the value conjectured by Takeda and Nishimori through the "self-duality" argument.