Quantum geometric tensor away from Equilibrium

TL;DR

This paper extends the concept of the quantum geometric tensor to non-equilibrium quantum states, analyzing its properties during time evolution and quantum quenches, with applications to the Cluster-XY model.

Contribution

It introduces a framework for studying the quantum geometric tensor away from equilibrium, including its phase diagram, equilibration, and relation to out-of-time-order correlators.

Findings

Quantum geometric tensor remains conserved during certain evolutions.

The tensor equilibrates after a quantum quench.

Time behavior of the tensor is governed by out-of-time-order correlators.

Abstract

The manifold of ground states of a family of quantum Hamiltonians can be endowed with a quantum geometric tensor whose singularities signal quantum phase transitions and give a general way to define quantum phases. In this paper, we show that the same information-theoretic and geometrical approach can be used to describe the geometry of quantum states away from equilibrium. We construct the quantum geometric tensor $Q_{\mu\nu}$ for ensembles of states that evolve in time and study its phase diagram and equilibration properties. If the initial ensemble is the manifold of ground states, we show that the phase diagram is conserved, that the geometric tensor equilibrates after a quantum quench, and that its time behavior is governed by out-of-time-order commutators (OTOCs). We finally demonstrate our results in the exactly solvable Cluster-XY model.