# Emergent topology and dynamical quantum phase transitions in two-dimensional closed quantum systems

## TL;DR

This paper introduces a dynamical topological order parameter (DTOP) for characterizing dynamical quantum phase transitions in two-dimensional closed quantum systems following a quench, extending previous one-dimensional concepts.

## Contribution

It generalizes the DTOP concept to two-dimensional systems and demonstrates its application to the Haldane model during quenches of the staggered mass.

## Key findings

- DTOP exhibits discontinuous jumps at critical times during DQPTs.
- Application to the Haldane model shows DTOP captures topological changes.
- The approach provides a gauge-invariant way to identify DQPTs.

## Abstract

We introduce the notion of a dynamical topological order parameter (DTOP) that characterises dynamical quantum phase transitions (DQPTs) occurring in the subsequent temporal evolution of "two dimensional" closed quantum systems, following a quench (or ramping) of a parameter of the Hamiltonian, {which generalizes the notion of DTOP introduced in Budich and Heyl, Phys. Rev. B 93, 085416 (2016) for one-dimensional situations}. This DTOP is obtained from the "gauge-invariant" Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur.