Geometric quench in the fractional quantum Hall effect: exact solution in quantum Hall matrix models and comparison with bimetric theory

TL;DR

This paper provides an exact solution to the geometric quench in fractional quantum Hall states using matrix models, compares the dynamics with bimetric theory, and introduces higher-spin variables related to the $W_{}$ algebra.

Contribution

It formulates the geometric quench in quantum Hall matrix models, solves the post-quench dynamics exactly, and extends the bimetric theory to match these results for strong anisotropies.

Findings

Exact post-quench dynamics in matrix models

Agreement between matrix models and bimetric theory for weak quenches

Introduction of higher-spin variables related to $W_{}$ algebra

Abstract

We investigate the recently introduced geometric quench protocol for fractional quantum Hall (FQH) states within the framework of exactly solvable quantum Hall matrix models. In the geometric quench protocol a FQH state is subjected to a sudden change in the ambient geometry, which introduces anisotropy into the system. We formulate this quench in the matrix models and then we solve exactly for the post-quench dynamics of the system and the quantum fidelity (Loschmidt echo) of the post-quench state. Next, we explain how to define a spin-2 collective variable $\hat{g}_{ab}(t)$ in the matrix models, and we show that for a weak quench (small anisotropy) the dynamics of $\hat{g}_{ab}(t)$ agrees with the dynamics of the intrinsic metric governed by the recently discussed bimetric theory of FQH states. We also find a modification of the bimetric theory such that the predictions of the modified bimetric theory agree with those of the matrix model for arbitrarily strong quenches. Finally, we introduce a class of higher-spin collective variables for the matrix model, which are related to generators of the $W_{\infty}$ algebra, and we show that the geometric quench induces nontrivial dynamics for these variables.