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arXiv:2011.02296 · Physical Review Letters

Rigorous Results on the Ground State of the Attractive SU ( N ) Hubbard Model

Hironobu Yoshida, Hosho Katsura

Published Mar 12, 2021• 17 citations
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Abstract

We study the attractive SU(N) Hubbard model with particle-hole symmetry. The model is defined on a bipartite lattice with the number of sites N_{A} (N_{B}) in the A (B) sublattice. We prove three theorems that allow us to identify the basic ground-state properties: the degeneracy, the fermion number, and the SU(N) quantum number. We also show that the ground state exhibits charge density wave order when |N_{A}-N_{B}| is macroscopically large. The theorems hold for a bipartite lattice in any dimension, even without translation invariance.

DOI
10.1103/physrevlett.126.100201

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We study the attractive SU(N) Hubbard model with particle-hole symmetry. The model is defined on a bipartite lattice with the number of sites N_{A} (N_{B}) in the A (B) sublattice. We prove three theorems that allow us to identify the basic ground-state properties: the degeneracy, the fermion number, and the SU(N) quantum number. We also show that the ground state exhibits charge density wave order when |N_{A}-N_{B}| is macroscopically large. The theorems hold for a bipartite lattice in any dimension, even without translation invariance.
We study a one-dimensional SU(N) Hubbard model with an attractive on-site interaction and N>2N>2N>2 at half-filling on the bipartite lattice using density-matrix renormalization-group method and a perturbation theory. We find that the ground state of the SU(N) Hubbard model is a charge density wave state with two-fold degeneracy. All the excitations are found to be gapful, resulting in an insulating ground state, on contrary to that in the SU(2) case. Moreover, the charge gap is equal to the Cooperon gap, which behaves as −2Nt2/(N−1)U-2Nt^2/(N-1)U−2Nt2/(N−1)U in the strong coupling regime. However, the spin gap ΔsΔ_{s}Δs​ and the quasiparticle gap Δ1Δ_{1}Δ1​ as well open exponentially in the weak coupling region, while in the strong coupling region, they linearly depend on UUU such that Δs∼−U(N−1)Δ_{s}\sim -U(N-1)Δs​∼−U(N−1) and Δ1∼−U(N−1)/2Δ_{1}\sim -U(N-1)/2Δ1​∼−U(N−1)/2.
In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.
Abstract We study an attractive Hubbard model on bipartite lattices. In the grand canonical formalism, we prove the existence of superconducting long-range order in the ground state on Lieb lattices with the chemical potential corresponding to half filling. We also study the low-energy excitations above several ground states for the translationally invariant Hamiltonian. We prove the following: (i) The pairing excitations are gapless above the ground states when the number of fermions deviates from that of the half filling by the order of the volume. (ii) A certain class of single-fermion excitations shows a non-vanishing spectral gap above the ground state with an even number of fermions in a strong coupling and low-density regime.
We show that the exact ground state for a class of extended Hubbard models including bond-charge, exchange, and pair-hopping terms, is the Yang ``\ensuremath{\eta}-paired'' state for any nonvanishing positive value of the pair-hopping amplitude, at least when the on-site Coulomb interaction is attractive enough and the remaining physical parameters satisfy a single constraint. The ground state is thus rigorously superconducting. Our result holds on a bipartite lattice in any dimension, at any band filling, and for arbitrary electron hopping. \textcopyright{} 1996 The American Physical Society.
We present rigorous results for the SU(n)\mathrm{SU}(n)SU(n) Fermi-Hubbard model on the railroad-trestle lattice. We first study the model with a flat band at the bottom of the single-particle spectrum and prove that the ground states exhibit SU(n)\mathrm{SU}(n)SU(n) ferromagnetism when the total fermion number is the same as the number of unit cells. We then perturb the model by adding extra hopping terms and make the flat band dispersive. Under the same filling condition, it is proved that the ground states of the perturbed model remain SU(n)\mathrm{SU}(n)SU(n) ferromagnetic when the bottom band is nearly flat. This is the first rigorous example of the ferromagnetism in nonsingular SU(n)\mathrm{SU}(n)SU(n) Hubbard models in which both the single-particle density of states and the on-site repulsive interaction are finite.