Probing the geometry of the Laughlin state

TL;DR

This paper investigates the quantum geometry of the Laughlin fractional quantum Hall state by numerically studying its response to metric deformations, revealing its intrinsic geometric properties and how they can be experimentally probed.

Contribution

It introduces numerical methods to analyze the geometric degree of freedom in the Laughlin state, including a pair amplitude operator to determine the intrinsic metric.

Findings

The Hall fluid response is proportional to the Gaussian curvature of the metric.

The intrinsic metric can be determined using a newly introduced pair amplitude operator.

The geometric probes work in experimentally relevant settings like mass anisotropy and electric field gradients.

Abstract

It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive numerical studies of the geometric degree of freedom for the simplest example of fractional quantum Hall states -- the filling $\nu=1/3$ Laughlin state. We perturb the system by a smooth, spatially dependent metric deformation and measure the response of the Hall fluid, finding it to be proportional to the Gaussian curvature of the metric. Further, we generalize the concept of coherent states to formulate the bulk off-diagonal long range order for the Laughlin state, and compute the deformations of the metric in the vicinity of the edge of the system. We introduce a "pair amplitude" operator and show that it can be used to numerically determine the intrinsic metric of the Laughlin state. These various probes are applied to several experimentally relevant settings that can expose the quantum geometry of the Laughlin state, in particular to systems with mass anisotropy and in the presence of an electric field gradient.