Effective hamiltonian approach for strongly correlated lattice models

We review a recent approach for the simulation of many-body interacting systems based on an efficient generalization of the Lanczos method for Quantum Monte Carlo simulations. This technique allows to perform systematic corrections to a given variational wavefunction, that allow to estimate exact energies and correlation functions, whenever the starting variational wavefunction is a qualitatively correct description of the ground state. The stability of the variational wavefunction against possible phases, not described at the variational level can be tested by using the ''effective Hamiltonian'' approach. In fact Monte Carlo methods, such as the ''fixed node approximation'' and the ''generalized Lanczos technique'' (Phys. Rev. B 64,024512, 2001) allow to obtain exact ground state properties of an effective Hamiltonian, chosen to be as close as possible to the exact Hamiltonian, thus yielding the most reasonable estimates of correlation functions. We also describe a simplified one-parameter scheme that improve substantially the efficiency of the generalized Lanczos method. This is tested on the t-J model, with a special effort to obtain accurate pairing correlations, and provide a possible non-phonon mechanism for High temperature superconductivity.Comment: 19 pages, 7 colour figures, lecture notes for the Euro Winter School-Kerkrade-N

Wave function optimization in the variational Monte Carlo method

An appropriate iterative scheme for the minimization of the energy, based on the variational Monte Carlo (VMC) technique, is introduced and compared with existing stochastic schemes. We test the various methods for the 1D Heisenberg ring and the 2D t-J model and show that, with the present scheme, very accurate and efficient calculations are possible, even for several variational parameters. Indeed, by using a very efficient statistical evaluation of the first and the second energy derivatives, it is possible to define a very rapidly converging iterative scheme that, within VMC, is much more convenient than the standard Newton method. It is also shown how to optimize simultaneously both the Jastrow and the determinantal part of the wave function.Comment: 5 pages, 3 figures, to be published in Phys. Rev B (Rapid Comm.

Linearized Auxiliary fields Monte Carlo: efficient sampling of the fermion sign

We introduce a method that combines the power of both the lattice Green function Monte Carlo (LGFMC) with the auxiliary field techniques (AFQMC), and allows us to compute exact ground state properties of the Hubbard model for U<~ 4t on finite clusters. Thanks to LGFMC one obtains unbiased zero temperature results, not affected by the so called Trotter approximation of the imaginary time propagator exp(- H t). On the other hand the AFQMC formalism yields a remarkably fast convergence in t before the fermion sign problem becomes prohibitive. As a first application we report ground state energies in the Hubbard model at U/t=4 with up to one hundred sites.Comment: 5 pages, 3 figure

Effective hamiltonian approach and the lattice fixed node approximation

We define a numerical scheme that allows to approximate a given Hamiltonian by an effective one, by requiring several constraints determined by exact properties of generic ''short range'' Hamiltonians. In this way the standard lattice fixed node is also improved as far as the variational energy is concerned. The effective Hamiltonian is defined in terms of a guiding function $\psi_G$ and can be solved exactly by Quantum Monte Carlo methods. We argue that, for reasonable $\psi_G$ and away from phase transitions, the long distance, low energy properties are rather independent on the chosen guiding function, thus allowing to remove the well known problem of standard variational Monte Carlo schemes based only on total energy minimizations, and therefore insensitive to long distance low energy properties.Comment: 8 pages, for the proceedings of "The Monte Carlo Method in the Physical Sciences: Celebrating the 50th Anniversary of the Metropolis Algorithm", Los Alamos, June 9-11, 200

The New Resonating Valence Bond Method for Ab-Initio Electronic Simulations

The Resonating Valence Bond theory of the chemical bond was introduced soon after the discovery of quantum mechanics and has contributed to explain the role of electron correlation within a particularly simple and intuitive approach where the chemical bond between two nearby atoms is described by one or more singlet electron pairs. In this chapter Pauling's resonating valence bond theory of the chemical bond is revisited within a new formulation, introduced by P.W. Anderson after the discovery of High-Tc superconductivity. It is shown that this intuitive picture of electron correlation becomes now practical and efficient, since it allows us to faithfully exploit the locality of the electron correlation, and to describe several new phases of matter, such as Mott insulators, High-Tc superconductors, and spin liquid phases

Nagaoka ferromagnetism in the two-dimensional infinite-U Hubbard model

We present different numerical calculations based on variational quantum Monte Carlo simulations supporting a ferromagnetic ground-state for finite and small hole densities in the two-dimensional infinite-$U$ Hubbard model. Moreover, by studying the energies of different total spin sectors, these calculations strongly suggest that the paramagnetic phase is unstable against a phase with a partial polarization for large hole densities $\delta \sim 0.40$ with evidence for a second-order transition to the paramagnetic large doping phase.Comment: 4 page

Exact Jastrow-Slater wave function for the one-dimensional Luttinger model

We show that it is possible to describe the ground state of the Luttinger model in terms of a Jastrow-Slater wave function. Moreover, our findings reveal that one-particle excitations and their corresponding dynamics can be faithfully represented only when a Jastrow factor of a similar form is applied to a coherent superposition of many Slater determinants. We discuss the possible relevance of this approach for the theoretical description of photoemission spectra in higher dimensionality, where the present wave function can be straightforwardly generalized and can be used as a variational ansatz, that is exact for the 1D Luttinger model.Comment: 10 pages, one figure, to appear in Phys. Rev.