Quantum Hall system in Tao-Thouless limit

TL;DR

This paper maps the quantum Hall problem onto a 1D lattice model in the Tao-Thouless limit, providing exact solutions at rational fillings and connecting these to bulk quantum Hall states, revealing phase transitions and the emergence of composite fermion states.

Contribution

It introduces an exact diagonalization approach in the Tao-Thouless limit for quantum Hall systems, linking 1D lattice solutions to 2D bulk states and analyzing phase transitions at various filling factors.

Findings

Exact solutions at rational fillings in the Tao-Thouless limit.

No phase transition for odd denominator fractions as system size increases.

Emergence of Luttinger liquid and composite fermion states at specific fillings.

Abstract

We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\pi/L_1$, where $L_1$ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit, $L_1\to 0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\nu=p/q\le 1$. For odd $q$, the ground state has the same qualitative properties as a bulk ($L_1 \to \infty$) quantum Hall hierarchy state and the lowest energy quasiparticle exitations have the same fractional charges as in the bulk. These states are the $L_1 \to 0$ limits of the Laughlin/Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd $q$, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states, {\it ie} that there is no phase transition as $L_1 \to \infty$ for filling factors where such states can be observed. For even denominator fractions, a phase transition occurs as $L_1$ increases. For $\nu=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite $L_1$, this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\nu=1/2$ system. The analysis generalizes to non-abelian quantum Hall states.