Dynamical Topological Order Parameters far from Equilibrium

TL;DR

This paper introduces a new dynamical topological order parameter (DTOP) that captures topological changes during real-time evolution in quantum many-body systems, especially at dynamical quantum phase transitions.

Contribution

The paper defines the DTOP as a momentum space winding number based on the Pancharatnam geometric phase, extending topological invariants to dynamical settings.

Findings

DTOP changes at dynamical quantum phase transitions

DTOP detects topological changes after quantum quenches

DTOP relates to string order parameter dynamics

Abstract

We introduce a topological quantum number -- coined dynamical topological order parameter (DTOP) -- that is dynamically defined in the real-time evolution of a quantum many-body system and represented by a momentum space winding number of the Pancharatnam geometric phase. Our construction goes conceptually beyond the standard notion of topological invariants characterizing the wave-function of a system, which are constants of motion under coherent time evolution. In particular, we show that the DTOP can change its integer value at discrete times where so called dynamical quantum phase transitions occur, thus serving as a dynamical analog of an order parameter. Interestingly, studying quantum quenches in one-dimensional two-banded Bogoliubov de Gennes models, we find that the DTOP is capable of resolving if the topology of the system Hamiltonian has changed over the quench. Furthermore, we investigate the relation of the DTOP to the dynamics of the string order parameter that characterizes the topology of such systems in thermal equilibrium.