Hall Viscosity in the Chern-Simons Matrix Model of the Laughlin Fractional Quantum Hall States

We compute the Hall viscosity in the Chern-Simons matrix model (CSMM) of the Laughlin fractional quantum Hall (FQH) states. The CSMM is a matrix quantum mechanics model proposed by Polychronakos as a regularization of the noncommutative Chern-Simons theory of the Laughlin states proposed by Susskind. Both models can be understood as describing the electrons in a FQH state as forming a noncommutative fluid, i.e., a fluid occupying a noncommutative space. Here we revisit the CSMM in light of recent work on geometric response in the FQH effect, with the goal of determining whether the CSMM captures this aspect of the physics of the Laughlin states. To compute the Hall viscosity in the CSMM we first identify the quantum operators which generate area-preserving deformations of the noncommutative fluid coordinates, and then compute the Hall viscosity using standard methods (e.g., the Kubo formula) from previous works. We find that the Hall viscosity in the CSMM with level m is exactly equal to the guiding center Hall viscosity of the Laughlin state with filling fraction 1/m as computed by Park and Haldane. Thus, our result confirms that these noncommutative models accurately describe (at least some aspects of) the geometric response of the Laughlin states.