The Loschmidt Index

TL;DR

This paper introduces the Loschmidt index, a topological measure based on wavefunction overlap nodes, to analyze quantum phase transitions in multi-band systems, linking it to the wrapping number and applying it to the XY model.

Contribution

It defines the Loschmidt index and establishes its relationship with the wrapping number, providing a systematic way to classify quantum phase transitions.

Findings

Loschmidt index correlates with topological properties of quantum states.

Method applied successfully to classify phase transitions in the XY model.

Provides a unified framework for equilibrium and dynamical quantum phase transitions.

Abstract

We study the nodes of the wavefunction overlap between ground states of a parameter-dependent Hamiltonian. These nodes are topological, and we can use them to analyze in a unifying way both equilibrium and dynamical quantum phase transitions in multi-band systems. We define the Loschmidt index as the number of nodes in this overlap and discuss the relationship between this index and the wrapping number of a closed auxiliary hypersurface. This relationship allows us to compute this index systematically, using an integral representation of the wrapping number. We comment on the relationship between the Loschmidt index and other well-established topological numbers. As an example, we classify the equilibrium and dynamical quantum phase transitions of the XY model by counting the nodes in the wavefunction overlaps.