Bulk-edge correspondence in the Haldane phase of the bilinear-biquadratic spin-1 Hamiltonian

TL;DR

This paper investigates the bulk-edge correspondence in the Haldane phase of spin-1 chains, revealing how entanglement spectra relate to edge states and exploring the connection to conformal field theory.

Contribution

It demonstrates a bulk-edge correspondence in the Haldane phase, linking entanglement spectra to physical edge states and identifying effective spin-1/2 descriptions.

Findings

Entanglement spectrum described by two effective spin-1/2s.

Entanglement Hamiltonian for non-contiguous partitions is a spin-1/2 Heisenberg model.

String order parameter relates to bulk-edge correspondence.

Abstract

The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum. Here we explore in depth this degeneracy for the spin-1 AKLT and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the entanglement spectrum can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.