Interplay of Topological Order and Spin Glassiness in the Toric Code under Random Magnetic Fields

TL;DR

This paper investigates how disorder affects the toric code's topological order, revealing that certain random magnetic fields can stabilize topological phases and lead to a coexistence with spin glassiness.

Contribution

It demonstrates that disorder can enhance topological order in the toric code and identifies a novel intermediate phase where topological order coexists with spin glass behavior.

Findings

Disorder favors topological phase near quantum critical point.

Topological order can be robust against strong random magnetic fields.

An intermediate phase with coexisting topological order and spin glassiness exists.

Abstract

We analyze the toric code model in the presence of quenched disorder, which is introduced via different types of random magnetic fields. In general, close to a quantum phase transition between a spin polarized phase and a topologically ordered one, we find that increasing the amount of disorder favors the topological phase. For some realizations of disorder, topological order can be robust against arbitrarily strong magnetic fields. In the case of the toric code in a random \pm h field, we show that the system exhibits a quantum phase transition to a spin glass phase in an appropriate dual variables description. The survival of topological order in the spin glass phase is directly related to the percolation properties of the rigid lattice in the Edwards-Anderson bimodal spin glass model. According to recent numerical results for this model [Phys. Rev. B 82, 214401 (2010)], it is likely that the rigid lattice does not percolate and, as a result, a new intermediate quantum phase appears in the random-field toric code. In this intermediate quantum phase, topological order coexists with spin glassiness.