Lieb's Theorem for Bose Hubbard Models

TL;DR

This paper proves the uniqueness and specific properties of the ground state in certain Bose Hubbard models using a cone-theoretical approach, applicable to finite-size lattices.

Contribution

It introduces a cone-theoretical method to establish ground state uniqueness and spin properties in two Bose Hubbard models, including a two-component variant.

Findings

Ground state is unique for both models under certain conditions.

Ground state has zero spin quantum number, i.e., is a singlet.

Method applies to any finite-size lattice.

Abstract

Using a cone-theoretical method, we prove the uniqueness of the ground state for two Bose Hubbard models. The first model is the usual Bose Hubbard model with real hopping coefficients and attractive interactions. The second model is a two-component Bose Hubbard model. Under certain conditions, we show that the ground state in the subspace with particle number $N=2n$($n$ is a positive integer) is unique for both models. For the second model, we show that the ground state has spin along the z-axis $S^{z}=0$. When the hopping coefficients are real, it has zero spin quantum number, i.e., it is a singlet. Our proofs work equally well for any arbitrary finite-size lattice.