Unlocking new regimes in fractional quantum Hall effect with quaternions

TL;DR

This paper introduces a quaternion-based formulation of the composite-fermion theory for the fractional quantum Hall effect, enabling exploration of previously inaccessible phenomena like nematic and charge-density wave instabilities.

Contribution

The novel quaternion formulation significantly broadens the theoretical analysis capabilities for the FQH effect, inspired by mathematical tools from gravitational wave and cosmology studies.

Findings

Identified potential nematic and charge-density wave instabilities in FQH states.

Extended the theoretical framework of FQH to include quaternionic representations.

Linked mathematical techniques from cosmology to condensed matter physics.

Abstract

We demonstrate that formulating the composite-fermion theory of the fractional quantum Hall (FQH) effect in terms of quaternions greatly expands its reach and opens the door into many interesting issues that were previously beyond the reach of quantitative theoretical investigation. As an illustration, we investigate the possibility of a nematic or a charge-density wave instability of the composite-fermion Fermi sea at half-filled Landau level and of the nearby FQH states by looking for a magneto-roton instability. Our quaternion formulation of the FQH effect has been inspired by mathematical developments in the theoretical analyses of gravitational wave modes and cosmic microwave background radiation, where an important role is played by spin-weighted spherical harmonics which are nothing but monopole harmonics appearing in the spherical geometry for the FQH effect.