Detection of symmetry-protected topological phases in one dimension with multiscale entanglement renormalization

TL;DR

This paper employs symmetric multiscale entanglement renormalization ansatz (MERA) tensor networks to identify and analyze symmetry-protected topological phases in a spin-1 Heisenberg chain, revealing geometric and RG flow insights.

Contribution

It introduces a geometrical brick-and-rope representation for inversion symmetric MERA tensors and demonstrates how symmetric entanglement renormalization captures RG fixed points of SPT phases.

Findings

Symmetric MERA effectively detects SPT phases.

The brick-and-rope model provides a geometric interpretation of inversion symmetry.

Entanglement renormalization with symmetric tensors yields correct RG fixed-point behavior.

Abstract

Symmetry-protected topological (SPT) phases are short-range entangled quantum phases with symmetry, which have gapped excitations in the bulk and gapless modes at the edge. In this paper, we study the SPT phases in the spin-1 Heisenberg chain with a single-ion anisotropy D, which has a quantum phase transition between a Haldane phase and a large-D phase. Using symmetric multiscale entanglement renormalization ansatz (MERA) tensor networks, we study the nonlocal order parameters for time-reversal and inversion symmetry. For the inversion symmetric MERA, we propose a brick-and-rope representation that gives a geometrical interpretation of inversion symmetric tensors. Finally, we study the symmetric renormalization group (RG) flow of the inversion symmetric string-order parameter, and show that entanglement renormalization with symmetric tensors produces proper behavior of the RG fixed-points.