Topological Invariants for Quantum Quench Dynamics from Unitary Evolution

TL;DR

This paper introduces a new topological invariant based on loop unitaries that fully characterizes quantum quench dynamics, generalizing previous approaches and applicable to multiband systems and higher dimensions.

Contribution

The authors define a homotopy invariant from loop unitaries that captures the dynamical topology of quantum quenches, extending the classification beyond two-band models.

Findings

Invariant equals the change in Chern number for two-band systems.

Dynamical topology manifests as hedgehog defects and eigenvector linking.

Provides a systematic method to classify quantum quench dynamics.

Abstract

Recent experiments began to explore the topological properties of quench dynamics, i.e. the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. But it only works for topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator, and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary, and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.