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arXiv:cond-mat/0201388

Effective hamiltonian approach for strongly correlated lattice models

Sandro Sorella

Published Jan 22, 2002• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0201388

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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.
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In Chapter 1, we introduce the physics of the HTSC, starting with an historical overview of the problem and describing some experimental results that characterize these materials. Subsequently, we introduce the t−J model, which allows a microscopic description of the HTSC and we introduce the Resonating Valence Bond (RVB) wave function. In Chapter 2, we will describe the numerical techniques used for obtaining the results of our thesis. We start from the Lanczos method, that enable us to obtain exact results for small cluster size and then we enter in the topic of the quantum Monte Carlo technique. We describe the variational Monte Carlo method, the optimization algorithm and we will introduce the Green’s function Monte Carlo and fixed-node approximation, that improve the variational results. In Chapter 3, we will introduce our new variational wave function which generalizes the RVB state we show our results on the charge fluctuations (phase separation problem) for the two-dimensional t−J model. The main results of this chapter has been published in Physical Review B [7]. In Chapter 4, we will study the magnetic and superconducting properties of the two-dimensional t−J and t−t′−J model, trying to understand the role of the next-nearest-neighbor hopping term on the magnetic and superconducting phases. We will show a phase diagram of the magnetic and superconducting correlations, which qualitatively reproduce the actual phase diagram of HTSC and gives some indication on the origin of the electronic pairing. The main results of this chapter were submitted to Physical Review B [8].
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We present, in this dissertation, a numerical simulation method to study interacting fermion systems. The general simulation procedures are discussed in connection with a description of a Quantum Monte Carlo simulation algorithm for interacting electrons on lattices. The algorithm presented here has been used to simulate interacting electrons on lattices, and it makes possible the study of substantially larger systems than can be studied by other numerical methods. As long as certain limits of applicability are respected, model Hamiltonians of interacting electrons can be studied without resort to uncontrolled approximations. The method then provides nearly exact solutions to model Hamiltonians of many-body systems, in the sense that the degree of error can be controlled. We also discuss some results obtained from simulations of the extended Hubbard model and and the periodic Anderson model. For the extended Hubbard model in two dimensions, different regions are identified where electron correlation produces antiferromagnetism, charge-density-waves, and singlet pairing superconducting behaviors. We also find regions where transitions from one type of correlation to another occur. For the symmetric periodic Anderson model, we observe the formation of local spin moments at high temperatures and their quenching at low temperatures, as in the single impurity Anderson model. In addition, we find antiferromagnetic interaction between f local moments at low temperatures which are not present in a single impurity problem.
Gutzwiller-projected fermionic states can be efficiently implemented within quantum Monte Carlo calculations to define extremely accurate variational wave functions for Heisenberg models on frustrated two-dimensional lattices, not only for the ground state but also for low-energy excitations. The application of few Lanczos steps on top of these states further improves their accuracy, allowing calculations on large clusters. In addition, by computing both the energy and its variance, it is possible to obtain reliable estimations of exact results. Here, we report the cases of the frustrated Heisenberg models on square and Kagome lattices.