Matrix Effective Theories of the Quantum Hall Effect

Andrea Cappelli A, Ivan Dario Jimenez Rodriguez

The present understanding of nonperturbative ground states in the fractional quantum Hall effect is based on effective theories of the Jain ‘composite fermion’ excitations. We review the approach based on matrix variables, i.e. D0 branes, originally introduced by Susskind and Polychronakos. We show that the Maxwell–Chern–Simons matrix gauge theory provides a matrix generalization of the quantum Hall effect, where the composite-fermion construction naturally follows from gauge invariance. The matrix ground states obtained by suitable projections of higher Landau levels are found to be in one-to-one correspondence with the Laughlin and Jain hierarchical states. The matrix theory possesses a physical limit for commuting matrices that could be reachable while staying in the same phase.

Allan H. MacDonald

Alexios P. Polychronakos

We propose a finite Chern-Simons matrix model on the plane as an effective description of fractional quantum Hall fluids of finite extent. The quantization of the inverse filling fraction and of the quasiparticle number is shown to arise quantum mechanically and to agree with Laughlin theory. We also point out the effective equivalence of this model, and therefore of the quantum Hall system, with the Calogero model.

R. Tao, Kazumi Maki

Matthew F. Lapa, Andrey Gromov, Taylor L. Hughes

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 postquench dynamics of the system and the quantum fidelity (Loschmidt echo) of the postquench state. Next, we explain how to define a spin-2 collective variable ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{g}}_{ab}(t)$ in the matrix models, and we show that for a weak quench (small anisotropy), the dynamics of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{g}}_{ab}(t)$ 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 ${W}_{\ensuremath{\infty}}$ algebra, and we show that the geometric quench induces nontrivial dynamics for these variables.