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AIP conference proceedings

Algorithms and Applications of Path-Integral Renormalization Group Method

Masatoshi Imada

Published Jan 1, 2006• 0 citations
Publisher

Abstract

Path‐integral renormalization‐group (PIRG) method is a rapidly developing tool for computing ground state properties of interacting quantum systems on lattices, particularly models for strongly correlated electrons such as the Hubbard model. It has served in clarifying phase diagrams of the Hubbard model containing quantum spin liquid phase. PIRG has also been implemented as a low‐energy solver of the effective Hamiltonian for realistic systems. This makes it possible to construct a scheme of first‐principles calculation by the hybrid approach combined with the density functional theory.

DOI
10.1063/1.2178033

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Path‐integral renormalization‐group (PIRG) method is a rapidly developing tool for computing ground state properties of interacting quantum systems on lattices, particularly models for strongly correlated electrons such as the Hubbard model. It has served in clarifying phase diagrams of the Hubbard model containing quantum spin liquid phase. PIRG has also been implemented as a low‐energy solver of the effective Hamiltonian for realistic systems. This makes it possible to construct a scheme of first‐principles calculation by the hybrid approach combined with the density functional theory.
Path‐integral renormalization group (PIRG) method has been developed for studying strongly correlated electron systems. In a wide range of systems including Hubbard‐type models, this method overcomes a number of difficulties known in the Monte‐Carlo‐type methods such as the negative sign problem. This method has been combined with a procedure of quantum number projection and grand canonical ensemble method as well, which contribute to a wider applicability. The quantum‐number projected PIRG enables calculations of excited spectra with a specified momentum or spin. We review recent numerical results on strongly correlated electron systems studied by this method. By using the methods, we determine the phase diagram of the two‐dimensional Hubbard model in the parameter space of the onsite interaction U, strength of the geometrical frustration effects defined by the ratio between next‐nearest to nearest neighbor transfer integrals, t′/t, and the chemical potential. It reveals severe competitions of various phases. The phase diagram contains a remarkable nonmagnetic Mott insulator phase with gapless and dispersionless spin excitations, sandwitched by the first‐order Mott transition and the antiferromagnetic transition. The first‐order character becomes more continuous one with increasing the frustration effect.
Abstract We derive a practical algorithm to study many‐electron systems based on the path‐integral renormalization group (PIRG) method within the framework of the real‐space finite‐difference (RSFD) approach. The PIRG method, originally developed for investigating strongly correlated electron systems, has been successfully applied to some models such as Hubbard models. However, to apply this method to more realistic systems of electrons with long‐range Coulomb interactions in the RSFD approach, the one‐body Green's function, which requires large computational resources, must be removed. For the same reason, an efficient algorithm for computing the Fock matrix is needed. The newly proposed algorithm is free of the one‐body Green's function and enables us to compute the Fock matrix efficiently. Our result shows a significant reduction in CPU time and the possibility of using the present algorithm as a practical numerical tool. Copyright © 2008 John Wiley & Sons, Ltd.
We investigate the bandwidth-controlled Mott transition in the Hubbard model on the checkerboard lattice at half filling using the path-integral renormalization group (PIRG) method. It is demonstrated that the system undergoes a first-order phase transition to the plaquette-singlet insulating phase at a finite Hubbard interaction. This conclusion is drawn via a detailed analysis of the spin and charge correlations around the phase transition point by means of the PIRG method aided with a new iteration scheme introduced in this paper.
A numerical algorithm for studying strongly correlated electron systems is proposed. The groundstate wavefunction is projected out after numerical renormalization procedure in the path integral formalism. The wavefunction is expressed from the optimized linear combination of retained states in the truncated Hilbert space with numerically chosen basis. This algorithm does not suffer from the negative sign problem and can be applied to any type of Hamiltonian in any dimension. The efficiency is tested in examples of the Hubbard model where the basis of Slater determinants is numerically optimized. We show results on fast convergence and accuracy achieved with small number of retained states.
A new efficient numerical algorithm for interacting fermion systems is proposed and examined in detail. The ground state is expressed approximately by a linear combination of numerically chosen basis states in a truncated Hilbert space. Two procedures lead to a better approximation. The first is a numerical renormalization, which optimizes the chosen basis and projects onto the ground state within the fixed dimension, LLL, of the Hilbert space. The second is an increase of the dimension of the truncated Hilbert space, which enables the linear combination to converge to a better approximation. The extrapolation L→∞L\to\inftyL→∞ after the convergence removes the approximation error systematically. This algorithm does not suffer from the negative sign problem and can be applied to systems in any spatial dimension and arbitrary lattice structure. The efficiency is tested and the implementation explained for two-dimensional Hubbard models where Slater determinants are employed as chosen basis. Our results with less than 400 chosen basis indicate good accuracy within the errorbar of the best available results as those of the quantum Monte Carlo for energy and other physical quantities.