What is the topological dual of the XXZ spin Chain?

TL;DR

This paper constructs a dual topological Hamiltonian for the $U(1)$ symmetric Heisenberg chain, revealing topological phases and their properties through non-local transformations and entanglement analysis.

Contribution

It introduces a novel dual SPT Hamiltonian for the anisotropic spin-$rac{1}{2}$ Heisenberg chain using a non-local unitary transformation, linking topological phases with the original model.

Findings

Identification of two topological phases separated by a Luttinger liquid.

Explicit construction of a local fermionic Hamiltonian with topological order.

Development of a non-local string order parameter and zero mode operator.

Abstract

We construct a dual symmetry-protected topological (SPT) Hamiltonian for the $U(1)$ symmetric anisotropic spin-$\frac{1}{2}$ Heisenberg chain-a model that has traditionally been used to study spontaneous symmetry breaking (SSB) in both ferromagnetic and antiferromagnetic phases, with an intervening extended Luttinger liquid phase. By performing a non-local unitary transformation, we explicitly construct a local fermionic Hamiltonian that exhibits two nontrivial topological phases separated by an extended Luttinger liquid regime. We demonstrate the topological nature of these phases by analyzing the entanglement structure, deriving a non-local string order parameter, and constructing an exact zero mode operator that connects states in different fermionic parity sectors.