# Investigating anisotropic quantum Hall states with bi-metric geometry

## TL;DR

This paper develops a bi-metric effective theory for anisotropic fractional quantum Hall states, linking geometric responses to anisotropy and verifying predictions through numerical simulations, advancing understanding of nematic order in quantum Hall systems.

## Contribution

It introduces a novel bi-metric formalism for anisotropic FQH states and establishes a relationship between shift, Hall viscosity, and a new anisospin coupling.

## Key findings

- Derived a relationship between shift, Hall viscosity, and anisospin.
- Numerically verified Hall viscosity predictions using DMRG.
- Clarified the role of Berry phase coefficient in nematic order.

## Abstract

We construct a low energy effective theory of anisotropic fractional quantum Hall (FQH) states. We develop a formalism similar to that used in the bi-metric approach to massive gravity, and apply it to describe abelian anisotropic FQH states in the presence of external electromagnetic and geometric backgrounds. We derive a relationship between the shift, the Hall viscosity, and a new quantized coupling to anisotropy, which we term "anisospin". We verify this relationship by numerically computing the Hall viscosity for a variety of anisotropic quantum Hall states using the density matrix renormalization group (DMRG). Finally, we apply these techniques to the problem of nematic order and clarify certain disagreements that exist in the literature about the meaning of the coefficient of the Berry phase term in the nematic effective action.