Topology Changes and Quantum Phase Transition in Spin-Chain System

TL;DR

This paper proposes a topological framework using Chern numbers to characterize quantum phase transitions in the XY spin-chain model, providing a new way to identify phases without traditional order parameters.

Contribution

It introduces a topological description of quantum phase transitions via Chern numbers of U(1) bundles, linking topological invariants to quantum criticality.

Findings

Chern number singularity coincides with quantum phase transition

Ground states form a U(1) principal bundle on S^2

Topological order parameter effectively indicates phase change

Abstract

The standard Landau-Ginzburg scenario of phase transition is broken down for quantum phase transition. It is difficult to find an order parameter to indicate different phases for quantum fluctuations. Here, we suggest a topological description of the quantum phase transition for the XY model. The ground states are identified as a specialized U(1) principal bundle on the base manifold $S^2$. And then different first Chern numbers of U(1) principal bundle on the base manifold $S^2$ are associated to each phase of quantum fluctuations. The particle-hole picture is used to parameterized the ground states of the XY system. We show that a singularity of the Chern number of the ground states occurs simultaneously with a quantum phase transition. The Chern number is a suitable topological order of the quantum phase transition.