Lieb-Schultz-Mattis-Type and Laughlin-Type Argument for the Quantum Hall Effect in Lattice Fermions with Spiral Boundary Conditions

TL;DR

This paper derives a direct condition for the integer quantum Hall effect in lattice fermion systems with interactions using spiral boundary conditions, improving upon previous methods by reducing system-size dependence.

Contribution

It introduces a novel approach employing spiral boundary conditions to directly derive the quantum Hall condition, enhancing clarity and reducing redundancy compared to traditional methods.

Findings

Derived the condition $ heta u- ho ext{ in } extbf{Z}$ for quantum Hall effect.

Used spiral boundary conditions to treat the system as a 1D chain.

Provided a systematic way to control external force and response directions.

Abstract

We derive the condition for the occurrence of the integer quantum Hall effect in two-dimensional lattice systems with interactions, expressed as $\phi\nu-\rho\in\mathbb{Z}$, where $\phi$, $\nu$, and $\rho$ denote the magnetic flux, the Chern number, and the electron density, respectively. By employing spiral boundary conditions, which treat the system as an extended one-dimensional chain, this condition is obtained directly through a Lieb-Schultz-Mattis-type and Laughlin-type argument. This approach improves upon the preceding work based on conventional periodic boundary conditions, where the condition was derived indirectly with redundant system-size dependence. The key to this approach is that the spatial directions of the external force and the response can be systematically controlled by a factor of the system size.