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

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.

References in corpus (31)

• Spectral Gap and Exponential Decay of Correlations
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Emergent SU(2) dynamics and perfect quantum many-body scars
• Statistical localization: from strong fragmentation to strong edge modes
• Stability of Frustration-Free Hamiltonians
• Incompressible Quantum Liquids and New Conservation Laws
• Half-Filled Lowest Landau Level on a Thin Torus
• The spectral gap for some spin chains with discrete symmetry breaking
• Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian
• Quantization of Hall Conductance For Interacting Electrons on a Torus
• An invariant of topologically ordered states under local unitary transformations
• A many-body index for quantum charge transport
• Exactly Solvable Fermion Chain Describing a $\nu =1/3$ Fractional Quantum Hall State
• Local gap threshold for frustration-free spin systems
• Persistence of exponential decay and spectral gaps for interacting fermions
• Spectral gaps of frustration-free spin systems with boundary
• Mott Insulators Without Symmetry Breaking
• Demonstrating the AKLT spectral gap on 2D degree-3 lattices
• Local Commuting Projector Hamiltonians and the Quantum Hall Effect
• Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice
• The origin of holomorphic states in Landau levels from non-commutative geometry, and a new formula for their overlaps on the torus
• A class of two-dimensional AKLT models with a gap
• Quantum Hall Systems: Braid groups, composite fermions, and fractional charge
• On the stability of charges in infinite quantum spin systems
• Local incompressibility estimates for the Laughlin phase
• Symmetry breaking in Laughlin's state on a cylinder
• Emergence of Haldane pseudo-potentials in systems with short-range interactions
• Spectral Gap and Edge Excitations of $d$-dimensional PVBS models on half-spaces
• Low-complexity eigenstates of a $\nu =1/3$ fractional quantum Hall system
• Fermionic and bosonic Laughlin state on thick cylinders
• Matrix-Product Ansatz for Excited States of Fractional Quantum Hall Systems

Cited by in corpus (13)

• Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results
• The spectral gap of a fractional quantum Hall system on a thin torus
• Bootstrapping the Quantum Hall problem
• Exactness of linear response in the quantum Hall effect
• Quantum statistics transmutation via magnetic flux attachment
• Quantitatively improved finite-size criteria for spectral gaps
• Random translation-invariant Hamiltonians and their spectral gaps
• Deformed Fredkin model for the $\nu =5/2$ Moore-Read state on thin cylinders
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Dynamical abelian anyons with bound states and scattering states
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• On a bulk gap strategy for quantum lattice models
• Excitation and tunneling spectra of a fractional quantum Hall system in the thin cylinder limit