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

TL;DR

This work investigates whether a BCS-like variational extension around Hartree-Fock states in the 2D Hubbard model can reveal a landscape of near-degenerate local minima with superconducting-like condensates. By relaxing a BCS ansatz near HF states on finite lattices, the authors find robust d_{x^2−y^2}–wave condensation in the underdoped regime and, in the overdoped regime, indications of d_{xy} condensation, without projection or symmetry imposition. The results show that many local minima exist with energies very close to the HF energy, suggesting a spin-glass–like multiplicity of near-ground states and revealing how pairing tendencies emerge from a nonprojected variational framework. These findings highlight a potential mechanism for d-wave pairing in correlated electron systems and motivate further exploration of relaxation criteria and finite-size effects in such variational schemes.

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, however, indications of $d_{xy}$ condensation in the overdoped region, at least in this approximation to the 2D Hubbard model.

Figures (7)

• Figure 1: Condensates of s and d-wave type vs. $U$ in the local minimum state at density $f=0.8$, $10\times 10$ lattice volume. The real part is shown.
• Figure 2: 3d view of the real part of (a) the d-wave order parameter; (c) the s-wave order parameter; and (d) the $s_{x^2+y^2}$ order parameter defined in the text, in the range of couplings $U$ and densities $f$ shown. Subfigure (b) is a "top down" display of the d-wave condensate with the height of the condensate indicated by color, rather than z-axis position, to help delineate the region of the d-wave peak.
• Figure 3: (a) Modulus of the momentum space condensate $P(k)$. (b) Spin density $D(x)$. (c) Charge (number density) $C(x)$. (d) The real part of the condensate spatial distribution $\Delta_1(x)$, which in this case is positive at all sites. All figures from single configurations at $U = 4, f = 0.8$ on a $20\times 20$ lattice with ${N=80}$. Other configurations display horizontal, rather than vertical stripes.
• Figure 4: Similar to Fig. \ref{['geometry']}, but at $U=3, f=0.85$. (a) Modulus of the momentum space condensate $P(k)$. (b) Spin density $D(x)$. (c) The real part of the condensate spatial distribution $\Delta_1(x)$. Stripes are less evident, and the condensate is not positive everywhere. All figures again taken from a single configuration on a $20\times 20$ lattice with $N=80$.
• Figure 5: $|P(k)|$ at a moderately strong coupling of $U=8,f=0.8$ on a $20\times 20$ lattice. A d-wave pattern is evident, although the amplitude is greatly reduced compared to the peak region.