Spectral Gaps and Incompressibility in a $${\varvec{\nu }}$$ = 1/3 Fractional Quantum Hall System

Abstract

AbstractWe study an effective Hamiltonian for the standard $$\nu =1/3$$ ν = 1 / 3 fractional quantum Hall system in the thin cylinder regime. We give a complete description of its ground state space in terms of what we call Fragmented Matrix Product States, which are labeled by a certain family of tilings of the one-dimensional lattice. We then prove that the model has a spectral gap above the ground states for a range of coupling constants that includes physical values. As a consequence of the gap we establish the incompressibility of the fractional quantum Hall states. We also show that all the ground states labeled by a tiling have a finite correlation length, for which we give an upper bound. We demonstrate by example, however, that not all superpositions of tiling states have exponential decay of correlations.