Symmetry fractionalization: Symmetry-protected topological phases of the bond-alternating spin-$1/2$ Heisenberg chain

TL;DR

This paper investigates the phases of a one-dimensional bond-alternating spin-1/2 Heisenberg chain using symmetry fractionalization, revealing trivial and symmetry-protected topological phases characterized by specific symmetry protections.

Contribution

It introduces a method to identify and classify phases in the bond-alternating spin chain via projective representations of symmetry groups, including a comprehensive analysis of the most general two-body model.

Findings

Identification of trivial and topological phases in the model

Classification of twelve distinct symmetry-protected topological phases

Use of infinite matrix-product states and symmetry fractionalization

Abstract

We study different phases of the one-dimensional bond-alternating spin-$1/2$ Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as $\sigma\rightarrow-\sigma$), parity (permutation of the chain about a specific bond), and dihedral ($\pi$-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-$1/2$ model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.