Bounds for low-energy spectral properties of center-of-mass conserving positive two-body interactions

TL;DR

This paper derives bounds on the low-energy spectral properties of positive, center-of-mass conserving two-body Hamiltonians relevant to fractional quantum Hall models, focusing on ground-state energy evolution, zero modes, and charge gaps.

Contribution

It introduces general bounds on ground-state energies, chemical potentials, and charge gaps for such Hamiltonians, including cases with zero modes and duality properties.

Findings

Bounds on ground-state energy growth with particle number

Constraints on chemical potential at zero temperature

Upper bounds on charge gaps in zero-mode cases

Abstract

We study the low-energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of fractional quantum Hall 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.