Abstract
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.
Highlights
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the twodimensional (2D) Hubbard model
The specific techniques used by the different groups to calculate effective vertices and susceptibilities differ in details, but the overall results are more or less consistent, with an antiferromagnetic instability at half filling and d-wave superconductivity away—but not too far away—from half filling
In the large-U limit, where double occupancy is suppressed, the Hubbard model can be replaced by the Heisenberg model at half filling [11] and by the t–J model close to half filling [12], defined by the Hamiltonian
Summary
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the twodimensional (2D) Hubbard model. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). There are several competing instabilities close to half filling, in particular, d-wave superconductivity and antiferromagnetism, or rather spin-density waves (SDWs). The problem can be solved in an elegant way using the functional renormalization group [7]–[10], which treats the competing density-wave and superconducting instabilities on the same footing. While the ground state of the 2D Heisenberg model is fairly well understood [13]—it is widely accepted that it exhibits long-range antiferromagnetic order with a reduced moment due to quantum fluctuations—the ground state of the t–J model remains an unsolved problem, despite an extensive use of sophisticated methods during the last two decades [14, 15].
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