Geometry induced quantum Hall effect and Hall viscosity

TL;DR

This paper explores how geometry, specifically torsion in a 4D embedded surface, induces a quantum Hall effect and Hall viscosity in a 2D particle system, linking topological states to geometric responses.

Contribution

It introduces a novel geometric framework where torsion acts as a gauge potential, revealing a new mechanism for quantum Hall effects and Hall viscosity in 2D systems embedded in 4D space.

Findings

Torsion acts as a U(1) gauge potential affecting effective dynamics.

Quantum Hall effect arises from the response to torsion.

Hall viscosity is linked to deformation of torsion and topological states.

Abstract

For a particle confined to the two-dimensional helical surface embedded in four-dimensional (4D) Euclidean space, the effective Hamiltonian is deduced in the thin-layer quantization formalism. We find that the gauge structure of the effective dynamics is determined by torsion, which plays the role of U(1) gauge potential, and find that the topological structure of associated states is defined by orbital spin which originates from 4D space. Strikingly, the response to torsion contributes a quantum Hall effect, and the response to the deformation of torsion contributes Hall viscosity that is perfectly presented as a simultaneous occurrence of multiple channels for the quantum Hall effect. This result directly provides a way to probe Hall viscosity.