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Superconductivity in the two-dimensional Hubbard model ?

D. Baeriswyl, Eichenberger, David, Gut, Bruno

Published Oct 24, 2007• 2 citations
OpenAlex

Abstract

A refined variational wave function for the two-dimensional repulsive Hubbard model is studied numerically, with the aim of approaching the difficult crossover regime of intermediate values of U. The issue of a superconducting ground state with d-wave symmetry is investigated for an average electron density n≈ 0.8 and for U = 8t. Due to finite-size effects a clear-cut answer to this fundamental question has not yet been reached.

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A refined variational wave function for the two-dimensional repulsive Hubbard model is studied numerically, with the aim of approaching the difficult crossover regime of intermediate values of U. The issue of a superconducting ground state with d-wave symmetry is investigated for an average electron density n≈ 0.8 and for U = 8t. Due to finite-size effects a clear-cut answer to this fundamental question has not yet been reached.
Abstract A refined variational wave function for the two‐dimensional repulsive Hubbard model is studied numerically, with the aim of approaching the difficult crossover regime of intermediate values of U. The issue of a superconducting ground state with d‐wave symmetry is investigated for an average electron density n ≈ 0.8 and for U = 8t. Due to finite‐size effects a clear‐cut answer to this fundamental question has not yet been reached. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength (U=8t)(U=8t)(U=8t) and for moderate sizes (from $6\ifmmode\times\else\texttimes\fi{}6$ to $10\ifmmode\times\else\texttimes\fi{}10$). Our ansatz is superior to other widely used variational wave functions. The results for order parameters and correlation functions provide insight into the antiferromagnetic state at half-filling as well as strong evidence for a superconducting phase away from half-filling.
The two-dimensional Hubbard model is studied for small values of the interaction strength (UUU of the order of the hopping amplitude ttt), using a variational ansatz well suited for this regime. The wave function, a refined Gutzwiller ansatz, has a BCS mean-field state with ddd-wave symmetry as its reference state. Superconducting order is found for densities n<1n<1n<1 (but not for n=1n=1n=1). This resolves a discrepancy between results obtained with the functional renormalization group, which do predict superconducting order for small values of UUU, and numerical simulations, which did not find superconductivity for $U\ensuremath{\lesssim}4t$. Both the gap parameter and the order parameter have a domelike shape as a function of nnn with a maximum for $n\ensuremath{\approx}0.8$. Expectation values for the energy, the particle number, and the superconducting order parameter are calculated using a linked-cluster expansion up to second order in UUU. In this way large systems (millions of sites) can be readily treated and well converged results are obtained. A big size is indeed required to see that the gap becomes very small at half filling and probably tends to zero in the thermodynamic limit, whereas away from half filling a finite asymptotic limit is reached. For a lattice of a given size the order parameter becomes finite only above a minimal coupling strength Uc{U}_{c}Uc​. This threshold value decreases steadily with increasing system size, which indicates that superconductivity exists for arbitrarily small UUU for an infinite system. For moderately large systems the size dependence is quite irregular, due to variations in level spacings at the Fermi energy. The fluctuations die out if the gap parameter spans several level spacings.
An elaborate variational wave function is used for studying superconductivity in the (repulsive) two-dimensional Hubbard model, including both nearest- and next-nearest-neighbor hoppings. A marked asymmetry is found between the ``localized'' hole-doped region and the more itinerant electron-doped region. Superconductivity with ddd-wave symmetry turns out to be restricted to densities where the Fermi surface crosses the magnetic zone boundary. A concomitant peak in the magnetic structure factor at $(\ensuremath{\pi},\ensuremath{\pi})$ clearly points to a magnetic mechanism.
We investigate the ground-state phase diagram of the two-dimensional Hubbard model based on the optimization variational Monte Carlo method. We use a wave function that is an off-diagonal type given as ψ=exp⁡(−λK)PGψ0ψ=\exp(-λK)P_Gψ_0ψ=exp(−λK)PG​ψ0​, where ψ0ψ_0ψ0​ is a one-particle state, PGP_GPG​ is the Gutzwiller operator, KKK is the kinetic operator, and λλλ is a variational parameter. The many-body effect plays an important role as an origin of spin correlation and superconductivity in correlated electron systems. We examine the competition between the antiferromagnetic state and superconducting state by varying the Coulomb repulsion UUU, the band parameter t′t't′ and the electron density nen_ene​. We show a phase diagram that includes superconducting and antiferromagnetic phases and that t′=0t'=0t′=0 is most favorable for superconductivity.
We use the C4vC_{4v}C4v​ symmetry group of the 4-site Hubbard model to construct a ground state variational wave function of two- and four interacting electrons. In the limit U→0U\rightarrow 0U→0, ground state energies of the two- and four interacting electrons system is of the order −4t-4t−4t. The variational wave function of the four interacting electrons obtained using the B1B_1B1​ irreducible representation is valid for on-site Coulomb repulsion, while the one obtained using the A1A_1A1​ representation is valid for negative values of the Coulomb interaction. The system exhibits antiferromagnetic correlations.