Spectral Gaps and Incompressibility in a {varvec{nu }}\xa0=\xa01/3 Fractional Quantum Hall System

Bruno Nachtergaele,Simone Warzel,Amanda Young

doi:10.1007/s00220-021-03997-0

Abstract

We study an effective Hamiltonian for the standard nu =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.

Highlights

The fractional quantum Hall effect (FQHE) is a result of the collective behavior of interacting charge degrees of freedom in a two-dimensional geometry with perpendicular magnetic field [50]
Two hallmarks characterize the remarkable properties of this quantum state of matter: the incompressibility of the liquid into which the charge carriers condense and the existence of an energy gap to excitations with fractional charge
An effective description of the observed features of excitations above the ground-state is based on Haldane pseudo-potentials, i.e. short-range repulsive interactions projected onto the lowest Landau level [16]

Summary

Introduction

The fractional quantum Hall effect (FQHE) is a result of the collective behavior of interacting charge degrees of freedom in a two-dimensional geometry with perpendicular magnetic field [50]. They are used in combination with the results of numerical simulations (e.g. in [27,32,42]) In their present form all these approaches produce lower bounds that contain as a factor the gap of the Hamiltonian with open boundary conditions H[a,b] for some finite interval. While the ground state of H on the whole Fock space F has zero energy, this may cease to be the case when restricting to a subspace with fixed particle number N , i.e. where N = x∈ nx is the number operator corresponding to which commutes with H. Note that the gap proven in Theorem 1.1 immediately implies a uniform lower bound on the N -particle groundstate energy for the second case in (1.16) From this we conclude that the compressibility vanishes in the thermodynamic limit. For the definition of the spin VMD states, see Sect. 2.2

Fragmented MPS

A A right right

Proof of the Spectral Gap

Exponential Clustering

A2 with