Entanglement properties of the Haldane phases: A finite system-size approach

TL;DR

This paper investigates the entanglement properties of the Haldane phases in a bond-alternating Heisenberg model using DMRG and analytical methods, revealing boundary-condition-dependent topological invariants and phase diagrams.

Contribution

It introduces a finite-size DMRG approach to identify topological invariants via boundary conditions and entanglement spectrum, advancing numerical detection of SPT phases.

Findings

Parity quantum numbers correspond to topological invariants under APBC.

Two-fold degeneracy in entanglement spectrum signals topologically nontrivial phases.

Phase diagram determined with level spectroscopy method in DMRG.

Abstract

We study the bond-alternating Heisenberg model using the finite-size density-matrix renormalization group (DMRG) technique and analytical arguments based on the matrix product state, where we pay particular attention to the boundary-condition dependence on the entanglement spectrum of the system. We show that, in the antiperiodic boundary condition (APBC), the parity quantum numbers are equivalent to the topological invariants characterizing the topological phases protected by the bond-centered inversion and $\pi$ rotation about $z$ axis. We also show that the odd parity in the APBC, which characterizes topologically nontrivial phases, can be extracted as a two-fold degeneracy in the entanglement spectrum even with finite system size. We then determine the phase diagram of the model with the uniaxial single-ion anisotropy using the level spectroscopy method in the DMRG technique. These results not only suggest the detectability of the symmetry protected topological (SPT) phases via general twisted boundary conditions but also provide a useful and precise numerical tool for discussing the SPT phases in the exact diagonalization and DMRG techniques.