Link between the hierarchy of fractional quantum Hall states and Haldane's conjecture for quantum spin chains

TL;DR

This paper links the fractional quantum Hall state at filling factor 1/3 to Haldane's conjecture for spin chains by showing their adiabatic connection through an effective spin-1 Hamiltonian with string order.

Contribution

It introduces an effective spin-1 Hamiltonian derived from the fractional quantum Hall state, revealing a connection to Haldane and large-D phases and extending Haldane's conjecture.

Findings

The $ u=1/3$ state is connected to Haldane and large-D phases.

The state exhibits a dual string order parameter.

Quantum fluctuations induce an effective parity-breaking spin Hamiltonian.

Abstract

We study a strong coupling expansion of the $\nu=1/3$ fractional quantum Hall state away from the Tao-Thouless limit and show that the leading quantum fluctuations lead to an effective spin-1 Hamiltonian that lacks parity symmetry. By analyzing the energetics, discrete symmetries of low-lying excitations, and string order parameters, we demonstrate that the $\nu=1/3$ fractional quantum Hall state is adiabatically connected to both Haldane and large-$D$ phases, and is characterized by a string order parameter which is dual to the ordinary one. This result indicates a close relation between (a generalized form of) the Haldane conjecture for spin chains and the fractional quantum Hall effect.