Spectral Gaps and Incompressibility in a ν=1/3 Fractional Quantum Hall System

We study an effective Hamiltonian for the standard nu=1/3 ractional 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.

Automated summary

The study explores an effective Hamiltonian for the standard nu=1/3 fractional quantum Hall system in the thin cylinder regime. It provides a complete description of the ground state space using Fragmented Matrix Product States and proves the existence of a spectral gap above the ground states for a range of coupling constants, establishing the incompressibility of the fractional quantum Hall states. Additionally, it shows that all ground states labeled by a tiling have a finite correlation length, with an upper bound given. However, it is demonstrated through an example that not all superpositions of tiling states exhibit exponential decay of correlations.