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

Abstract

We 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.

Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

“…There are a number of examples in the literature of such systems for which the question has been settled [1,11,16,43,44,[64][65][66]91,92]. For one-dimensional frustration-free systems arguments to prove a gap have been extended even further [2,23,39,59,61,62,70,77,84,98]. The stability results of Bravyi, Hastings, and Michalakis significantly amplify the class of models for which one can prove a spectral gap uniform in the system size [21,22,72].…”

Exactness of Linear Response in the Quantum Hall Effect

“…Concretely, the last two assumptions are satisfied for the family H s = H φ(s) introduced in Section 2 and Appendix 6.1, where φ ∈ C ∞ ([0, 1]; R) with φ(0) = 0, φ(1) = 2π and φ ≥ 0 is compactly supported in (0, 1). While Assumption 4.1 is believed to hold for (possibly fractional) quantum Hall systems, there is at the moment of writing no explicit microscopic model where this can be proved, except for perturbations of free systems (and therefore integer conductance), see [8,27,28]; see however [29] See [26] for details. We refer to operators with such finite |||•||| f as generalized extensive observables.…”

The spectral gap of a fractional quantum Hall system on a thin torus

“…Truncated Haldane potentials of the above form have been studied in [9,22,23,26,29] and are believed to capture, at least qualitatively, the main features of FQH systems such as their incompressibility (see also [3,4]). In the Hamiltonian description, the latter is explained by combining the maximal filling factor ν of their degenerate ground-state space with a spectral gap above these ground states, which is uniform in the particle number as well as the volume Λ L .…”

“…In the Hamiltonian description, the latter is explained by combining the maximal filling factor ν of their degenerate ground-state space with a spectral gap above these ground states, which is uniform in the particle number as well as the volume Λ L . For the case ν = 1/3 these properties were mathematically established in [22]. However, the estimates of the bulk gap were plagued by edge modes.…”