# Ising order parameter and topological phase transitions: Toric code in a uniform magnetic field

## TL;DR

This paper introduces a string order parameter to characterize topological phase transitions in the Toric code model under a uniform magnetic field, linking local and non-local order parameters through a duality with the Ising model.

## Contribution

It proposes a novel string order parameter for topological phase transitions in the Toric code, bridging local and non-local order parameters via duality with the Ising model.

## Key findings

- String order parameter effectively characterizes topological phase transitions.
- Topological transitions can be analyzed using symmetry-breaking concepts via dual mappings.
- Results apply to various lattice geometries and dimensions.

## Abstract

Quantum Ising model in a transverse field is of the simplest quantum many body systems used for studying universal properties of quantum phase transitions. Interestingly, it is well-known that such phase transitions can be mapped to topological phase transitions in Toric code models. Therefore, one can expect that well known properties of the transverse Ising model are used for characterizing topological phase transition in Toric code model. In this paper, we consider the magnetization of Ising model and show while it is a local order parameter, it is mapped to a non local order parameter in the topological side. Consequently, we introduce a string order parameter for topological phase transitions in Toric code models defined on different lattices with different dimensions in presence of a uniform magnetic field. Since such string order parameter is dual of the Ising order parameter with well known properties, it leads to a topological (non local) characterization of phase transition in the Toric code model in the uniform field. Our results show that although topological phase transitions do not follow a symmetry breaking mechanism, it might be still possible to use concepts of symmetry breaking phase transitions for topological ones by finding suitable mappings.