Abelian and Non-Abelian Quantum Geometric Tensor

TL;DR

This paper introduces a generalized quantum geometric tensor that unifies the description of topological and symmetry-breaking quantum phase transitions through local geometric measures, enhancing understanding of ground-state changes.

Contribution

It defines a new quantum geometric tensor incorporating non-Abelian Riemannian metric and Berry curvature to analyze phase transitions in many-body systems.

Findings

Identifies singular behavior of the tensor at phase transitions

Links symmetry-breaking and topological transitions to geometric singularities

Provides a geometric framework for understanding quantum criticality

Abstract

We propose a generalized quantum geometric tenor to understand topological quantum phase transitions, which can be defined on the parameter space with the adiabatic evolution of a quantum many-body system. The generalized quantum geometric tenor contains two different local measurements, the non-Abelian Riemannian metric and the non-Abelian Berry curvature, which are recognized as two natural geometric characterizations for the change of the ground-state properties when the parameter of the Hamiltonian varies. Our results show the symmetry-breaking and topological quantum phase transitions can be understood as the singular behavior of the local and topological properties of the quantum geometric tenor in the thermodynamic limit.