Numerical results for ground state and excited state properties (energies, double occupancies, and Matsubara-axis self energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods. arXiv:1505.02290v2 [cond-mat.str-el] 15

We introduce an efficient way to improve the accuracy of projected wave functions, widely used to study the two-dimensional Hubbard model. Taking the clue from the backflow contribution, whose relevance has been emphasized for various interacting systems on the continuum, we consider many-body correlations to construct a suitable approximation for the ground state at intermediate and strong couplings. In particular, we study the phase diagram of the frustrated t − tЈ Hubbard model on the square lattice and show that, thanks to backflow correlations, an insulating and nonmagnetic phase can be stabilized at strong coupling and sufficiently large frustrating ratio tЈ / t.

The two-dimensional Hubbard model on the anisotropic triangular lattice, with two different hopping amplitudes t and t ′ , is relevant to describe the low-energy physics of κ-(ET)2X, a family of organic salts. The ground-state properties of this model are studied by using Monte Carlo techniques, on the basis of a recent definition of backflow correlations for strongly-correlated lattice systems. The results show that there is no magnetic order for reasonably large values of the electron-electron interaction U and frustrating ratio t ′ /t = 0.85, suitable to describe the non-magnetic compound with X=Cu2(CN)3. On the contrary, Néel order takes place for weaker frustrations, i.e., t ′ /t ∼ 0.4 ÷ 0.6, suitable for materials with X=Cu2(SCN)2, Cu[N(CN)2]Cl, or Cu[N(CN)2]Br.

By using variational wave functions and quantum Monte Carlo techniques, we investigate the interplay between electron-electron and electron-phonon interactions in the two-dimensional HubbardHolstein model. Here, the ground-state phase diagram is triggered by several energy scales, i.e., the electron hopping t, the on-site electron-electron interaction U , the phonon energy ω0, and the electron-phonon coupling g. At half filling, the ground state is an antiferromagnetic insulator for U 2g 2 /ω0, while it is a charge-density-wave (or bi-polaronic) insulator for U 2g 2 /ω0. In addition to these phases, we find a superconducting phase that intrudes between them. For ω0/t = 1, superconductivity emerges when both U/t and 2g 2 /tω0 are small; then, by increasing the value of the phonon energy ω0, it extends along the transition line between antiferromagnetic and charge-density-wave insulators. Away from half filling, phase separation occurs when doping the charge-density-wave insulator, while a uniform (superconducting) ground state is found when doping the superconducting phase. In the analysis of finite-size effects, it is extremely important to average over twisted boundary conditions, especially in the weak-coupling limit and in the doped case.

We study the competition between magnetic and spin-liquid phases in the Hubbard model on the anisotropic triangular lattice, which is described by two hopping parameters t and t ′ in different spatial directions and is relevant for layered organic charge-transfer salts. By using a variational approach that includes spiral magnetic order, we provide solid evidence that a spin-liquid phase is stabilized in the strongly-correlated regime and close to the isotropic limit t ′ /t = 1. Otherwise, a magnetically ordered spiral state is found, connecting the (collinear) Néel and the (coplanar) 120• phases. The pitch vector of the spiral phase obtained from the unrestricted Hartree-Fock approximation is substantially renormalized in presence of electronic correlations, and the Néel phase is stabilized in a wide regime of the phase diagram, i.e., for t ′ /t < 0.75. We discuss these results in the context of organic charge-transfer salts.

We show that backflow correlations in the variational wave function for the Hubbard model greatly improve the previous results given by the Slater-Jastrow state, usually considered in this context. We provide evidence that, within this approach, it is possible to have a satisfactory connection with the strong-coupling regime. Moreover, we show that, for the Hubbard model on the lattice, backflow correlations are essentially short range, inducing an effective attraction between empty (holons) and doubly occupied sites (doublons). In presence of frustration, we report the evidence that the metal to Mott-insulator transition is marked by a discontinuity of the double occupancy, together with a similar discontinuity of the kinetic term that does not change the number of holons and doublons, while the other kinetic terms are continuous across the transition. Finally, we show the estimation of the charge gap, obtained by particle-hole excitationsà la Feynman over the ground-state wave function.

We consider the one-band Hubbard model on the square lattice by using variational and Green's function Monte Carlo methods, where the variational states contain Jastrow and backflow correlations on top of an uncorrelated wave function that includes BCS pairing and magnetic order. At half filling, where the ground state is antiferromagnetically ordered for any value of the on-site interaction U , we can identify a hidden critical point UMott, above which a finite BCS pairing is stabilized in the wave function. The existence of this point is reminiscent of the Mott transition in the paramagnetic sector and determines a separation between a Slater insulator (at small values of U ), where magnetism induces a potential energy gain, and a Mott insulator (at large values of U ), where magnetic correlations drive a kinetic energy gain. Most importantly, the existence of UMott has crucial consequences when doping the system: We observe a tendency to phase separation into a hole-rich and a hole-poor region only when doping the Slater insulator, while the system is uniform by doping the Mott insulator. Superconducting correlations are clearly observed above UMott, leading to the characteristic dome structure in doping. Furthermore, we show that the energy gain due to the presence of a finite BCS pairing above UMott shifts from the potential to the kinetic sector by increasing the value of the Coulomb repulsion.

We investigate the Hubbard model on the anisotropic triangular lattice with two hopping parameters t and t ′ in different spatial directions, interpolating between decoupled chains (t = 0) and the isotropic triangular lattice (t = t ′ ). Variational wave functions that include both Jastrow and backflow terms are used to compare spin-liquid and magnetic phases with different pitch vectors describing both collinear and coplanar (spiral) order. For relatively large values of the on-site interaction U/t ′ 10 and substantial frustration, i.e., 0.3 t/t ′ 0.8, the spin-liquid state is clearly favored over magnetic states. Spiral magnetic order is only stable in the vicinity of the isotropic point, while collinear order is obtained in a wide range of inter-chain hoppings from small to intermediate frustration.