Toy model of boundary states with spurious topological entanglement entropy

Kohtaro Kato,Fernando G S L Brandão

doi:10.1103/physrevresearch.2.032005

Kohtaro Kato, Fernando G S L Brandão

Open Access

https://doi.org/10.1103/physrevresearch.2.032005

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Abstract

Topological entanglement entropy has been extensively used as an indicator of topologically ordered phases. We study the conditions needed for two-dimensional topologically trivial states to exhibit spurious contributions that contaminates topological entanglement entropy. We show that if the state at the boundary of a subregion is a stabilizer state, then it has a non-zero spurious contribution to the region if and only if, the state is in a non-trivial one-dimensional $G_1\times G_2$ symmetry-protected-topological (SPT) phase. However, we provide a candidate of a boundary state that has a non-zero spurious contribution but does not belong to any such SPT phase.

Highlights

Ordered phases are gapped quantum phases that cannot be detected by conventional local order parameters
We study the underlying mechanism behind spurious Topological entanglement entropy (TEE) in the trivial phase
We model the degrees of freedom at the boundaries of regions [Fig. 1(b)] by using matrixproduct states (MPSs) [11]

Summary

INTRODUCTION

Ordered phases are gapped quantum phases that cannot be detected by conventional local order parameters. To extract the TEE from a ground state, one can calculate suitable linear combinations of entropies for certain subsystems [see, e.g., Fig. 1(a)], known as conditional mutual information (CMI) in quantum information theory, such that the first leading terms cancel out [1,2]. In general, Eq (1) could contain an additional term, and the above argument does not always work This additional contribution, called spurious TEE [6,7], results in positive CMI for states in the trivial phase. We provide numerical evidence that, in general, there exist boundary states that have nonzero spurious TEE but do not belong to any such SPT phase. To the best of our knowledge, this is the first example of the mechanism of spurious TEE beyond the on-site G1 × G2 SPT phase at the boundary

MATRIX-PRODUCT-STATE MODEL OF BOUNDARY STATES

CHARACTERIZATION BY OPERATOR-ALGEBRA

RELATION TO SYMMETRY-PROTECTEDTOPOLOGICAL PHASES

A NONTRIVIAL EXAMPLE

FUTURE DIRECTIONS