Quantum Entanglement and Topological Order in Hole-Doped Valence Bond Solid States

TL;DR

This paper investigates how doping valence bond solid states with fermionic holes can induce and stabilize topological order, revealing new symmetry-protected topological features through entanglement spectra and string-order parameters.

Contribution

It introduces supersymmetric extensions of VBS states and analyzes their topological properties, showing how doping recovers string order and affects entanglement spectra in these systems.

Findings

Doping induces string order in states where it was absent.

Fermionic sectors appear in entanglement spectra after doping.

Topological order relates to transformation properties of supermatrix-product states.

Abstract

We present a detailed analysis of topological properties of the valence bond solid (VBS) states doped with fermionic holes. As concrete examples, we consider the supersymmetric extension of the SU(2)- and the SO(5) VBS states, dubbed UOSp(1|2) and UOSp(1|4) supersymmetric VBS states, respectively. Specifically, we investigate the string-order parameters and the entanglement spectra of these states to find that, even when the parent states (bosonic VBS states) do not support the string order, they recover it when holes are doped and the fermionic sector appears in the entanglement spectrum. These peculiar properties are discussed in light of the symmetry-protected topological order. To this end, we characterize a few typical classes of symmetry-protected topological orders in terms of supermatrix-product states (SMPS). From this, we see that the topological order in the bulk manifests itself in the transformation properties of the SMPS in question and thereby affects the structure of the entanglement spectrum. Then, we explicitly relate the existence of the string order and the structure of the entanglement spectrum to explain the recovery and the stabilization of the string order in the supersymmetric systems.