Detecting a $Z_2$ topologically ordered phase from unbiased infinite projected entangled-pair state simulations

TL;DR

This paper introduces an unbiased iPEPS-based method to detect $Z_2$ topological order in quantum many-body systems, successfully identifying topological features without imposing symmetry constraints during tensor optimization.

Contribution

It presents a novel unbiased tensor network approach to identify topological order, including a new efficient method for computing topological invariants like the modular matrices.

Findings

Successfully identified $Z_2$ topological order in the toric code model

Generated stable topologically ordered states with correct entanglement entropy

Developed a more efficient corner-transfer matrix method for topological invariant computation

Abstract

We present an approach to identify topological order based on unbiased infinite projected entangled-pair states (iPEPS) simulations, i.e. where we do not impose a virtual symmetry on the tensors during the optimization of the tensor network ansatz. As an example we consider the ground state of the toric code model in a magnetic field exhibiting $Z_2$ topological order. The optimization is done by an efficient energy minimization approach based on a summation of tensor environments to compute the gradient. We show that the optimized tensors, when brought into the right gauge, are approximately $Z_2$ symmetric, and they can be fully symmetrized a posteriori to generate a stable topologically ordered state, yielding the correct topological entanglement entropy and modular S and U matrices. To compute the latter we develop a variant of the corner-transfer matrix method which is computationally more efficient than previous approaches based on the tensor renormalization group.