A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength (U = 8t) and for moderate sizes (from 6 × 6 to 10 × 10). Our ansatz is clearly superior to other widely used variational wave functions. The results for order parameters and correlation functions provide new insight for the antiferromagnetic state at half filling as well as strong evidence for a superconducting phase away from half filling.PACS numbers: 71.10. Fd,74.20.Mn, The Hubbard model plays a key role in the analysis of correlated electron systems, and it is widely used for describing quantum antiferromagnetism, the Mott metalinsulator transition and, ever since Anderson's suggestion [1], superconductivity in the layered cuprates. Several approximate techniques have been developed to determine the various phases of the two-dimensional Hubbard model. For very weak coupling, the perturbative Renormalization Group extracts the dominant instabilities in an unbiased way, namely antiferromagnetism at half filling and d-wave superconductivity for moderate doping [2,3]. Quantum Monte Carlo simulations have been successful in extracting the antiferromagnetic correlations at half filling [4,5], but in the presence of holes the numerical procedure is plagued by the fermionic minus sign problem [6]. This problem appears to be less severe in dynamic cluster Monte Carlo simulations, which exhibit a clear tendency towards d-wave superconductivity for intermediate values of U [7].Variational techniques address directly the ground state and thus offer an alternative to quantum Monte Carlo simulations, which are limited to relatively high temperatures. Previous variational wave functions include mean-field trial states from which configurations with doubly occupied sites are either completely eliminated (full Gutzwiller projection) [8,9,10] or at least partially suppressed [11]. Recently, more sophisticated wave functions have been proposed, which include, besides the Gutzwiller projector, non-local operators related to charge and spin densities [12,13]. Our own variational wave function is based on the idea that for intermediate values of U the best ground state is a compromise between the conflicting requirements of low potential energy (small double occupancy) and low kinetic energy (delocalization). It is known that the addition of an operator involving the kinetic energy yields an order of magnitude improvement of the ground state energy with respect to a wave function with a Gutzwiller projector alone [14]. In this letter, we show that such an additional term allows us to draw an appealing picture of the ground state, both at half filling and as a function of doping (some preliminary results have been published [15,16]).In its most simple form, the 2D Hubbard model is composed of two terms,Ĥ = tT + UD , witĥHere c † iσ creates an electron at site i with spin σ, the summation is restricted to nearest-neighbor sites and n iσ = c † iσ c iσ . We consider a square lattice with peri...

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 ∆ (antiferromagnetic or superconducting) obtained with different variational wave functions. ∆ 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.

An elaborate variational wave function is used for studying superconductivity in the ͑repulsive͒ twodimensional 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 d-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 ͑ , ͒ clearly points to a magnetic mechanism.One of the central issues in the field of high-temperature superconductors has been-and for some researchers still is-the question whether pairing in the cuprates arises from purely repulsive interactions, as proposed by Anderson two decades ago. 1 This question has been studied extensively in the framework of the two-dimensional ͑2D͒ ͑repulsive͒ Hubbard model ͑and the related t-J model͒. Recent progress, both in dynamical mean-field theory 2,3 and in variational calculations, 4 has strengthened the case for the existence of a superconducting phase in the Hubbard model, with a ͑d-wave͒ gap parameter reasonably close to the experimental values for intermediate interaction strengths ͑U of the order of the bandwidth͒. This conclusion has been challenged on the basis of Monte Carlo simulations, 5 which we believe to be not conclusive, as discussed below.Most previous studies of the Hubbard model have been restricted to nearest-neighbor hopping, where electron doping does not differ from hole doping. Here we show that the addition of second-neighbor hopping ͑which breaks the electron-hole symmetry͒ changes this behavior substantially. While the hole-doped side is "localized" and shows kineticenergy-driven superconductivity with a large condensation energy, the electron-doped side is itinerant with a potentialenergy-driven superconductivity and a small condensation energy.The Hubbard Hamiltonian Ĥ = Ĥ 0 + UD consists of a hopping term ͑"kinetic energy"͒and an on-site repulsion UD , where D = ͚ i n i↑ n i↓ is the number of doubly occupied sites, n i = c i † c i , and the operator c i † ͑c i ͒ creates ͑annihilates͒ an electron at site i with spin . We use the trial ground statewhere g and h are ͑real͒ variational parameters and ͉⌿ 0 ͘ is a BCS state with d-wave symmetry. The first term in Eq. ͑2͒ promotes delocalization and yields a substantial improvement with respect to the Gutzwiller wave function ͑h =0͒. In fact, it has been demonstrated that ansatz ͑2͒ is very close to the exact ground state both for small 2D systems 6 and for the solvable 1 / r chain. 7In our previous study of the simple Hubbard model ͑t ij = t for nearest-neighbor sites and 0 otherwise͒, 8 we have found that wave function ͑2͒ exhibits d-wave pairing below a hole concentration of 0.18. At first sight this result seems to contradict recent Monte Carlo simulations, 5 where no signature for superconductivity was found, but a closer look shows that there is no discrepancy. In fact, three of the four hole densities consid...

A trial wave function is proposed for studying the instability of the two-dimensional Hubbard model with respect to d-wave superconductivity. Double occupancy is reduced in a similar way as in previous variational studies, but in addition our wave function both enhances the delocalization of holes and induces a kinetic exchange between the electron spins. These refinements lead to a large energy gain, while the pairing appears to be weakly affected by the additional term in the variational wave function.

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=0.8125 and for U=8t. Due to finite-size effects a clear-cut answer to this fundamental question has not yet been reached.Comment: 5 pages, 1 figure, Proc. 30th Int. Conf. of Theoretical Physics, Ustron, Poland, 2006, to be published in phys. stat. so