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arXiv:2410.11645

The charge gap is greater than the neutral gap in fractional quantum Hall systems

Marius Lemm, Bruno Nachtergaele, Simone Warzel, Amanda Young

Published Oct 15, 2024• 0 citations
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Abstract

Past studies of fractional quantum Hall systems have found that the charge gap dominates the neutral gap for all relevant parameter choices. We report a wide-ranging proof that this domination is in fact a universal property of any Hamiltonian that satisfies a few simple structural properties: translation-invariance, charge conservation, dipole conservation, and a fractionally filled ground state. The result applies to both fermions and bosons. Our main tool is a new mathematical scheme, the gap comparison method, which provides a sequence of inequalities that relate the spectral gaps in successive particle number sectors. Our finding sheds new light on dipole conservation's profound effects on many-body physics.

DOI
10.48550/arxiv.2410.11645

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Past studies of fractional quantum Hall systems have found that the charge gap dominates the neutral gap for all relevant parameter choices. We report a wide-ranging proof that this domination is in fact a universal property of any Hamiltonian that satisfies a few simple structural properties: translation-invariance, charge conservation, dipole conservation, and a fractionally filled ground state. The result applies to both fermions and bosons. Our main tool is a new mathematical scheme, the gap comparison method, which provides a sequence of inequalities that relate the spectral gaps in successive particle number sectors. Our finding sheds new light on dipole conservation's profound effects on many-body physics.
We consider quantum lattice Hamiltonians and derive recursive spectral relations bridging successive particle number sectors. One relation gives conditions under which the charge gap dominates the neutral gap. We verify these conditions under a triad of symmetries (translation-invariance, charge and dipole conservation) that are present, e.g., in periodic fractional quantum Hall systems. Thus, this gap domination, previously observed numerically, is a universal feature imposed by symmetry. A second relation yields a new induction-on-particle-number method for deriving spectral gaps. The results cover both bosons and fermions.
We consider quantum lattice Hamiltonians and derive recursive spectral relations bridging successive particle number sectors. One relation gives conditions under which the charge gap dominates the neutral gap. We verify these conditions under a triad of symmetries (translation-invariance, charge and dipole conservation) that are present, e.g., in periodic fractional quantum Hall systems. Thus, this gap domination, previously observed numerically, is a universal feature imposed by symmetry. A second relation yields a new induction-on-particle-number method for deriving spectral gaps. The results cover both bosons and fermions.
The fractional quantum Hall (FQH) effect plays a prominent role in the study of topological phases of matter and of strongly correlated electron systems in general. FQH systems have been demonstrated to show many interesting novel properties such as fractional charges, and are believed to harbor even more intriguing phenomena such as fractional statistics. However, there remain many interesting questions to be addressed in this regime. The work reported in this thesis aims to push the envelope of our understanding of the low-energy properties of FQH states using microscopic principles. In the first part of the thesis, we present a systematic perturbative approach to study excitations in the thin cylinder/torus limit of the quantum Hall states. The approach is applied to the Haldane-Rezayi and Gaffnian quantum Hall states, which are both expected to have gapless excitations in the usual two-dimensional thermodynamic limit. For the Haldane-Rezayi state, we confirm that gapless excitations are present also in the ``one-dimensional'' thermodynamic limit of an infinite thin cylinder, in agreement with earlier considerations based on the wave functions alone. In contrast, we identify the lowest excitations of the Gaffnian state in the thin cylinder limit, and conclude that they are gapped, using a combination of perturbative and numerical means. We discuss possible scenarios for the cross-over between the two-dimensional and the one-dimensional thermodynamic limit in this case. In the second part of the thesis, we study the low energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of FQH states. Starting from the observation that positive many-body Hamiltonians must have ground state energies that increase monotonously in particle number, we explore what general additional constraints can be obtained for two-body interactions with "center-of-mass conservation" symmetry, both in the presence and absence of particle-hole symmetry. We find general bounds that constrain the evolution of the ground state energy with particle number, and in particular constrain the chemical potential at T=0. Special attention is given to Hamiltonians with zero modes, in which case similar bounds on the first excited state are also obtained, using a duality property. In this case, in particular an upper bound on the charge gap is also obtained. We further comment on center-of-mass and relative-decomposition in disk geometry within the framework of second quantization.
We study a fractional quantum Hall system with maximal filling ν = 1/3 in the thin torus limit. The corresponding Hamiltonian is a truncated version of Haldane’s pseudopotential, which upon a Jordan–Wigner transformation is equivalent to a one-dimensional quantum spin chain with periodic boundary conditions. Our main result is a lower bound on the spectral gap of this Hamiltonian, which is uniform in the system size and total particle number. The gap is also uniform with respect to small values of the coupling constant in the model. The proof adapts the strategy of individually estimating the gap in invariant subspaces used for the bosonic ν = 1/2 model to the present fermionic case.
We consider charge transport for interacting many-body systems with a gapped ground state subspace that is finitely degenerate and topologically ordered. To any locality-preserving, charge-conserving unitary that preserves the ground state space, we associate an index that is an integer multiple of 1/p, where p is the ground state degeneracy. We prove that the index is additive under composition of unitaries. This formalism gives rise to several applications: fractional quantum Hall conductance, a fractional Lieb–Schultz–Mattis (LSM) theorem that generalizes the standard LSM to systems where the translation-invariance is broken, and the interacting generalization of the Avron–Dana–Zak relation between the Hall conductance and the filling factor.
For the numerical simulation of the fractional quantum Hall (FQH) effects on a finite disk, the rotational symmetry is the only symmetry that is used in diagonalizing the Hamiltonian. In this work, we propose a method of using the weak translational symmetry for the center of mass of the many-body system. With this approach, the bulk properties, such as the energy gap and the magneto-roton excitation are consistent with those in the closed manifolds like the sphere and torus. As an application, we consider the FQH phase and its phase transition in the fast rotated dipolar fermions. We thus demonstrate the disk geometry having versatility in analyzing the bulk properties beside the usual edge physics.