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arXiv:2507.01280

BCS states and D-wave condensates in the 2D Hubbard model

Kazue Matsuyama, Jeff Greensite

Published Jul 2, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2507.01280

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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 dx2−y2d_{x^2-y^2}dx2−y2​ in the underdoped region. There are also indications of dxyd_{xy}dxy​ 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 explore certain properties of the Hartree–Fock approximation to the ground state of the two-dimensional Hubbard model, emphasizing the fact that in the Hartree approach there is an enormous multiplicity of self-consistent solutions which are nearly degenerate in energy, reminiscent of a spin glass, but which may differ substantially in other bulk properties. It is argued that this multiplicity is physically relevant at low temperatures. We study the localization properties of the one-particle wavefunctions comprising the Hartree–Fock states, and find that these are unlocalized at small and moderate values of U/t, in particular in the stripe region, but become highly localized at values corresponding to strong repulsion. We also find rectangular domains as well as stripes in the stripe region of the phase diagram, and study pair correlations in the neighborhood of half-filling.
[[abstract]]The ground-state energy, the concentration of double occupied sites and the density of bound electron pairs in low-dimensional attractive Hubbard model are calculated by means of the generalized self-consistent field (GSCF) approach. This approach gives an excellent agreement with the corresponding Bethe-ansatz solution and the quantum Monte-Carlo calculations in a wide range of interaction strength U < 0 and electron concentration n. A simple relationship is found between the order parameter and the density of bound electron pairs, valid for any dimension and in entire parameter space U, n and magnetic field h. The electron pairing and the crossover from the itinerant BCS regime into the Bose condensation regime of local pairs are studied in dependence of both U and n. The GSCF theory correctly displays the separation of the energy gap from the BCS order parameter at n not equal 1 as \U \ increases. (C) 2001 Elsevier Science B.V. All rights reserved.
We use the variational cluster approximation to study the superconducting ground state in the two-dimensional attractive Hubbard model, putting particular emphasis on the significance of quantum fluctuations of the system. We first show that the order parameter is suppressed in comparison with that obtained using the mean-field theory owing to the effects of spatial fluctuations in two-dimensional systems. We then show that the calculated Bogoliubov quasiparticle spectra and condensation amplitude clearly exhibit the character of Cooper pairs in momentum space and that the pair coherence length ξξξ evaluated from the condensation amplitude demonstrates a smooth crossover in real space from a weakly paired BCS state (ξ≫aξ\gg aξ≫a) to a BEC state of tightly bound pairs (ξ≪aξ\ll aξ≪a), where aaa is the lattice constant. The calculated kinetic and potential energies in the superconducting and normal ground states indicate that the superconducting state in the weak-coupling region is driven by the gain in potential energy, while that in the strongcoupling region is driven by the gain in kinetic energy.
Ground-state properties of the two-dimensional Hubbard model with point-defect disorder are investigated numerically in the Hartree-Fock approximation. The phase diagram in the p(point defect concentration)--\ensuremath{\delta}(deviation from half filling) plane exhibits antiferromagnetic, spin-density-wave, paramagnetic, and spin-glass-like phases. The disorder stabilizes the antiferromagnetic phase relative to the spin-density-wave phase. The presence of U strongly enhances the localization in the antiferromagnetic phase. The spin-density-wave and spin-glass-like phases are weakly localized.
The ground-state properties in one-dimensional Hubbard model with on-site attraction and repulsion of electrons in the presence of magnetic field h are calculated by means of the exact Bethe-ansatz formalism and the generalized self-consistent field (GSCF) approach for general electron concentrations n and arbitrary interaction strength. The ground-state properties, including the energy, the average spin (magnetization) and the kinetic energy are compared over a wide range of parameter space. The GSCF theory is in qualitative and in some cases in good quantitative agreement with the exact results. The GSCF theory at U≤0 (or U≥0) differentiates the spin (or charge) energy gap from the BCS (or antiferromagnetic) order parameter and suggests a smooth crossover from the phase with the itinerant BCS-like behavior to the Bose condensation regime of the local pairs.