# Dynamical Topological Phase Transition in a Bose-Einstein Condensate

## TL;DR

This paper investigates dynamical topological phase transitions in a Bose-Einstein condensate, revealing how linear dispersion and mode displacement lead to phase changes, with implications for quantum simulation using 2D material membranes.

## Contribution

It introduces the concept of dynamical topological phase transitions in BECs and proposes a quantum simulation scheme using vibrational spectra of 2D membranes.

## Key findings

- Dynamical phase transition occurs at global periods with linear dispersion.
- Power law scaling with dynamical exponent of unity at critical times.
- Proposed quantum simulation scheme using membrane vibrational spectra.

## Abstract

We study a Bose-Einstein condensate (BEC) at low energy limit and show that their collective dynamics exhibit interesting topological behavior. The system undergoes dynamical topological phase transition at its global periods if its dispersion relation is strictly linear and when all of the bosonic mode are properly displaced from their equilibrium position. We corroborate the occurrence of dynamical phase transition by calculating Fisher zeros of the Loschmidt amplitude for sudden quench dynamics. A connection is established between the order of nonanalycity in the accumulated geometric phase and the spectral density of the system. Furthermore, it is shown that a power law scaling holds at all critical times for various displacement spectra, whose dynamical exponent equals unity. Eventually, a scheme for the quantum simulation of such dynamical phase transition is proposed. The scheme is based on the vibrational spectrum of a free-standing membrane of a two-dimensional material. To induce a displacement in the system and for tracking its collective geometrical dynamics, we propose to employ a spin that properly couples to the modes. When appropriate geometrical and boundary conditions are applied to the membrane, a spectrum with linear dispersion is attainable, then decoherence dynamics of the spin unveils occurrence of a dynamical topological phase transition.