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arXiv:2509.19780 · Physica Scripta

Superconductivity and low energy excitations in an attractive Hubbard model

Yukimi Goto, Tohru Koma, Hironobu Yoshida

Published Jul 21, 2026• 0 citations
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Abstract

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.

DOI
10.1088/1402-4896/ae8e4a

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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 investigate low-energy excitations of the one-dimensional half-filled SU(4) Hubbard model with an attractive on-site interaction U<0U<0U<0 using the density-matrix renormalization-group method as well as perturbation theory. We find that the ground state is a charge-density-wave state with long-range order. The ground state is completely incompressible since all the excitations are gapful. The charge gap, which is the same as the four-particle excitation gap, is a nonmonotonic function of UUU, while the spin gap and others increase with increasing $\ensuremath{\mid}U\ensuremath{\mid}$ and have linear asymptotic behaviors.
In 1989, Lieb proved two theorems about the Hubbard model. One showed that the ground state of the attractive model was a spin singlet state (S = 0) was unique and was positive definite. The other showed that the ground state of the repulsive model on a bipartite lattice at half-filling has a total spin given by [Formula: see text](N[Formula: see text]NB)/2[Formula: see text], corresponding to the difference of the number of lattice sites on the two sublattices divided by two. In the mid to late 1990s, Shen extended these proofs to show that the pseudospin of the attractive model was minimal until the electron number equaled 2NA where it became fixed at J = [Formula: see text](N[Formula: see text]NB)/2[Formula: see text] until the filling became 2NB, where it became minimal again. In addition, Shen showed that a spin tower exists for the spin eigenstates for the half-filled case on a bipartite lattice. The spin tower says the minimal energy state with spin S is higher in energy than the minimal energy state with spin S − 1 until we reach the ground-state spin given above. One long-standing conjecture about this model remains, namely, does the attractive model have such a spin tower for all fillings, which would then imply that the repulsive model has minimal pseudopsin in its ground state. While we do not prove this last conjecture, we provide a quick review of this previous work, provide a constructive proof of the pseudospin of the attractive model ground state, and describe the challenges by proving the remaining open conjecture.
In this paper, we show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half filling on some bipartite lattices. When NA{\mathit{N}}_{\mathit{A}}NA​>NB{\mathit{N}}_{\mathit{B}}NB​(NA{\mathit{N}}_{\mathit{A}}NA​ and NB{\mathit{N}}_{\mathit{B}}NB​ are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201 (1989)], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U>0 and is independent of the dimension of the lattices.
The low-temperature properties of the two-dimensional attractive Hubbard model are strongly influenced by the fermion density. Away from half-filling, there is a finite-temperature transition to a phase with sss-wave pairing order. However, Tc{T}_{\mathrm{c}}Tc​ is suppressed to zero at half-filling, where long-range charge-density-wave order also appears, degenerate with superconductivity. This paper presents determinant quantum Monte Carlo simulations of the attractive Hubbard model in the presence of a confining potential Vtrap{V}_{\mathrm{trap}}Vtrap​ which makes the fermion density $\ensuremath{\rho}$ inhomogeneous across the lattice. Pair correlations are shown to be large at low temperatures in regions of the trapped system with incommensurate filling, and to exhibit a minimum as the local density $\ensuremath{\rho}(\mathbit{i})$ passes through one fermion per site. In this ring of $\ensuremath{\rho}(\mathbit{i})=1$, charge order is enhanced. A comparison is made between treating Vtrap{V}_{\mathrm{trap}}Vtrap​ within the local-density approximation (LDA) and in an ab initio manner. It is argued that certain sharp features of the LDA result at integer filling do not survive the proximity of doped sites. The critical temperature of confined systems of fixed characteristic density is estimated.
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.