# Numerical study of anisotropy in a composite Fermi liquid

## TL;DR

This study uses density-matrix renormalization group methods to analyze how anisotropy in electron dispersion affects composite fermions in a quantum Hall system, revealing a square root relationship for Coulomb interactions.

## Contribution

It provides the first numerical verification of the anisotropy relationship between electrons and composite fermions in a high magnetic field.

## Key findings

- For Coulomb interactions, $oldsymbol{ ext{alpha}_{CF}= extsqrt{ ext{alpha}_F}}$ within numerical accuracy.
- The anisotropy relationship varies with different electron-electron interactions.
- Results agree with recent experimental observations.

## Abstract

We perform density-matrix renormalization group studies of a two-dimensional electron gas in a high magnetic field and with an anisotropic band mass. At half-filling in the lowest Landau level, such a system is a Fermi liquid of composite fermions. By measuring the Fermi surface of these composite fermions, we determine a relationship between the anisotropy of composite fermion dispersion, $\alpha_{CF}$, and the original anisotropy $\alpha_F$ of the fermion dispersion at zero magnetic field. For systems where the electrons interact via a Coulomb interaction, we find $\alpha_{CF}=\sqrt{\alpha_F}$ within our numerical accuracy. The same result has been found concurrently in recent experiments. We also show results with other forms of the electron-electron interaction; this allows us (a) to benchmark our procedure against known exact results and (b) to show that the relationship between the anisotropies is dependent on the form of the interaction.