We consider states of BCS form in the 2D Hubbard model which, starting from some arbitrary point in state space in the neighborhood of a Hartree-Fock ground state, are relaxed within that BCS ansatz to local minima of the energy. As in the Hartree-Fock approximation there are a vast number of local minima, nearly degenerate in energy. What is new, and unlike the conventional Hartree-Fock states, is that there is a region in parameter space where these local minima are clearly associated with d-wave condensates of the form $d_{x^2-y^2}$ in the underdoped region. There are also indications of $d_{xy}$ condensation in the overdoped region, at least in this approximation to the 2D Hubbard model, as well as condensates over a range of parameters on triangular lattices.We consider states of BCS form in the 2D Hubbard model which, starting from some arbitrary point in state space in the neighborhood of a Hartree-Fock ground state, are relaxed within that BCS ansatz to local minima of the energy. As in the Hartree-Fock approximation there are a vast number of local minima, nearly degenerate in energy. What is new, and unlike the conventional Hartree-Fock states, is that there is a region in parameter space where these local minima are clearly associated with d-wave condensates of the form $d_{x^2-y^2}$ in the underdoped region. There are also indications of $d_{xy}$ condensation in the overdoped region, at least in this approximation to the 2D Hubbard model, as well as condensates over a range of parameters on triangular lattices.

arXiv:2507.01280Jul 2, 20250 citations

Variational cluster approach to quasi-two-dimensional Hubbard model

俊彦吉川

Jan 1, 20100 citations

On the mean-field antiferromagnetic gap for the half-filled 2D Hubbard model at zero temperature

Edwin Langmann A, Jonatan Lenells

Abstract We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree–Fock theory. It was conjectured by Hirsch in 1985 that this gap, $\Delta$ Δ , vanishes like $\exp (-2\pi \sqrt{t/U})$ exp ( - 2 π t / U ) in the weak-coupling limit $U/t\downarrow 0$ U / t ↓ 0 ( $U>0$ U > 0 and $t>0$ t > 0 are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.Abstract We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree–Fock theory. It was conjectured by Hirsch in 1985 that this gap, $$\Delta $$ Δ , vanishes like $$\exp (-2\pi \sqrt{t/U})$$ exp ( - 2 π t / U ) in the weak-coupling limit $$U/t\downarrow 0$$ U / t ↓ 0 ( $$U>0$$ U > 0 and $$t>0$$ t > 0 are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.

arXiv:2501.18141Letters in Mathematical PhysicsApr 20, 20260 citations

d_{x^2-y^2}

in the underdoped region. There are also indications of

d_{xy}

condensation in the overdoped region, at least in this approximation to the 2D Hubbard model, as well as condensates over a range of parameters on triangular lattices.

\Delta

Δ , vanishes like

\exp (-2\pi \sqrt{t/U})

exp ( - 2 π t / U ) in the weak-coupling limit

U/t\downarrow 0

U / t ↓ 0 (

U>0

U > 0 and

t>0

t > 0 are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.