The toric code under antiferromagnetic isotropic Heisenberg interactions

TL;DR

This paper studies how antiferromagnetic Heisenberg interactions affect the toric code, revealing a phase transition from topological order to a Nél phase, using neural-network quantum states and analytical methods.

Contribution

It combines neural-network quantum states with Schrieffer-Wolff transformation to analyze the breakdown of topological order under realistic local interactions.

Findings

Topological order breaks down at a critical Heisenberg coupling.

Néel phase emerges with fourfold degeneracy beyond the transition.

Schrieffer-Wolff transformation accurately describes low-energy behavior.

Abstract

We investigate the impact of an isotropic antiferromagnetic Heisenberg perturbation on the toric code, focusing on the resulting quantum phase transition and the nature of the phase that emerges beyond topological order. Using neural-network quantum states (NQS), we compute ground states over a wide range of Heisenberg couplings while fully respecting the exact symmetries of the model. In the weak-coupling regime, the numerical results are in excellent agreement with an effective low-energy description derived from a Schrieffer-Wolff (SW) transformation, providing analytic control over the perturbative breakdown of topological order. We show that the Heisenberg perturbation only renormalizes local operators at low orders, whereas mixing between topological sectors occurs only at a perturbative order proportional to the system size. At intermediate values of the Heisenberg interaction, the topological phase breaks down. We estimate the critical point through a combination of the fidelity susceptibility and the logarithmic susceptibility of non-contractible Wilson loops for various system sizes. Furthermore, we utilize the topological entanglement entropy to provide a comprehensive characterization of the phase transition. Beyond the transition, an antiferromagnetic $\pm X/\pm Z$ N\'eel phase emerges, characterized by a fourfold-degenerate symmetry-broken manifold, which is explicitly probed using staggered-magnetization-based diagnostics. Our results show how local two-spin interactions, which naturally arise in realistic implementations of the toric code, drive the breakdown of topological order. Moreover, we establish the SW approach as a systematic framework for analyzing such perturbations in combination with variational many-body methods.