Projector Monte Carlo Method Research Articles

Observations on variational and projector Monte Carlo methods.

Variational Monte Carlo and various projector Monte Carlo (PMC) methods are presented in a unified manner. Similarities and differences between the methods and choices made in designing the methods are discussed. Both methods where the Monte Carlo walk is performed in a discrete space and methods where it is performed in a continuous space are considered. It is pointed out that the usual prescription for importance sampling may not be advantageous depending on the particular quantum Monte Carlo method used and the observables of interest, so alternate prescriptions are presented. The nature of the sign problem is discussed for various versions of PMC methods. A prescription for an exact PMC method in real space, i.e., a method that does not make a fixed-node or similar approximation and does not have a finite basis error, is presented. This method is likely to be practical for systems with a small number of electrons. Approximate PMC methods that are applicable to larger systems and go beyond the fixed-node approximation are also discussed.

Projector Monte Carlo method based on Slater determinants: a new sampling method for singlet state calculations

A new sampling method is proposed for projector Monte Carlo (PMC) calculations based on Slater determinants (SD) in singlet states. Using the symmetry of the α and β electron determinants, the number of configurations to be considered can be about one-half of the original sampling. We applied the new sampling to the PMC-SD calculations of the H2O molecule in the ground state. The results were always improved by the new sampling method both for the equilibrium and for bond-stretched structures.

Projector Monte Carlo method based on Slater determinants: Test application to singlet excited states of H 2O and LiF

A projector Monte Carlo method based on Slater determinants (PMC-SD) is expanded to excited state calculations. Target excited states are calculated state-by-state by eliminating the components of the lower states from the imaginary-time propagator. Test calculations are performed for the singlet excited states of H 2O and LiF. As the calculations of H 2O show, the accuracy of the PMC-SD method is improved systematically by increasing the number of walkers. The full-CI energies are obtainable as a limit for a given basis set. The avoided crossing of covalent and ionic states in the dissociation of LiF is well reproduced using the PMC-SD method.

Projector Monte Carlo method based on configuration state functions. Test applications to the H 4 system and dissociation of LiH

A projector Monte Carlo method based on configuration state functions (PMC-CSF) is developed, which does not require the nodes of trial wave functions. The accuracy of the PMC-CSF method is systematically improved by increasing the number of walkers. As demonstrated by test applications to the H 4 and LiH systems, full-CI energies are obtainable as a limit for a given basis set. In addition, the PMC-CSF method is suitable for parallel calculations.

Numerical solution of the spin-2 Heisenberg antiferromagnetic chains using a projector method.

An efficient projector Monte Carlo method has been used to calculate the energies of the ground state and the first excited state, and hence the gap between them, of the spin-2 Heisenberg antiferromagnetic chains with linear sizes up to N=32. By direct extrapolation, we found that a Haldane gap exists, \ensuremath{\Delta}E=0.05, and the value of the ground-state energy per site is ${\mathit{e}}_{0}$=-4.7545 as N\ensuremath{\rightarrow}\ensuremath{\infty}.

Monte Carlo calculations on the spin- [formula omitted] Heisenberg antiferromagnetic chains

An efficient projector Monte Carlo method has been used to calculate the energies of the ground state and the first excited state of the spin- 3 2 Heisenberg antiferromagnetic chains with linear sizes up to N = 32. We have found that the ground state energy per site converges to its thermodynamic limit value as fast as 1/ N 2 and the thermodynamic limit value is e 0 = -2.8248. All of our results show that the energy gap tends to zero as N→∞, which is consistent with the rigorous proof.

Numerical Solution of the Spin-1 Heisenberg Antiferromagnetic Chains by Using a Projector Method

An efficient projector Monte Carlo method has been used to calculate the energies of the ground state and the first excited state, and hence the gap between them, of the spin-1 Heisenberg antiferromagnetic chains with linear sizes up to . We have computed them on chains up to . We have found that the size-dependence of the ground state energy per site has the form , which implies that there is a gap between the ground state and the first excited state. The value of the ground state energy per site as is . The direct extrapolation of also shows that a gap of exists as .

A GEOMETRICAL VIEW OF THE MINUS-SIGN PROBLEM

It is shown that the sign of the fermionic determinant in the projector Monte Carlo method is directly related to a topological invariant. A key ingredient to obtain this result is the identification of the appropriate manifolds to describe the evolution of a fermionic trial wavefunction. They allow for a purely geometrical consideration of the minus-sign problem.

Stochastic truncation method for Hamiltonian lattice field theory.

A new Monte Carlo method is presented for estimating the dominant eigenvalue of a matrix Hamiltonian. It is a version of the power method, in which the basis-state amplitudes are stochastically rounded to integers. Its relation to the ensemble projector Monte Carlo method is discussed and some results are demonstrated for the example of the /ital Z//sub 2/ gauge model in 2+1 dimensions.

Monte Carlo calculation of elementary excitation of spin chains.

An efficient Monte Carlo method is proposed to calculate the elementary excitation spectrum of quantum systems. The lowest energy with arbitrary momentum is obtained by the projector Monte Carlo method. This is applied to the spin-(1/2 and -1 Heisenberg antiferromagnetic chain with length 32. For the S=(1/2 case, the spectrum coincides completely with the spectrum of des Cloiseaux and Pearson. For the S=1 case, the spectrum has a gap at momentum \ensuremath{\pi} as was predicted by Haldane. The value of the gap coincides with the calculation of Nightingale and Blote. The spectrum satisfies a variational relation with the structure factor.

Transverse Spin Correlation by Projector Monte Carlo Method with Large Cluster Decomposition

The cluster spin decomposition is applied to the projector Monte Carlo simulation. We investigate the ground state energy and transverse correlation functions of one dimensional X Y model. We enlarge the size of cluster up to nine. For fairly large τ, the agreement of Monte Carlo data with exact results is very good.

Quantum monte carlo studies of electron-electron interaction effects in conducting polymers

We discuss recent studies, using the quantum ensemble projector Monte Carlo (EPMC) method, of theoretical models of conducting polymers. Our focus is on the consequences of incorporating direct electron-electron interactions into the “standard” electron-phonon interaction models. Among the observables we examine one energetics of purely dimerized ground states, single solitons, soliton pairs, averaged spin and charge distributions, and local correlation functions.

Quantum Monte Carlo of coupled fermion chains

Computation involving fermions in more than one dimension presents a challenge to Monte Carlo simulations. One is either faced with calculation of the fermion determinant or the appearance of random walks that carry negative weights. The cancellations among positive and negative weights give rise to uncontrollable fluctuations. A new stochastic algorithm is proposed for treating quantum problems in which the matrix elements of the projection operator, U(..beta..) = e/sup -betaH/, carries both positive and negative signs. The algorithm is a variant of the population scheme of the Projector Monte Carlo method. We apply this algorithm to a double-chain fermion system where electron tunnelling across the chains gives rise to unavoidable negative matrix elements.

Ensemble projector Monte Carlo method, studying the lattice Schwinger model in the Hamiltonian formulation.

The ensemble projector Monte Carlo method is a promising method to study lattice gauge theories with fermions in the Hamiltonian formulation. We study the massive Schwinger model and show that consistent results are obtained in the presence of positive and negative matrix elements. The expectation values for the average energy calculated from matrix elements with negative and positive scores, and calculated from the average scores, are consistent with each other and with results obtained from the local Hamiltonian Monte Carlo method. In contrast with the latter method, the ensemble projector Monte Carlo method can be also applied to gauge field theories in 2+1 and 3+1 dimensions.

Application of the projector Monte Carlo method to the transfer matrix of the classical XY model

An application of the projector Monte Carlo method to the transfer-matrix eigenvalue problem is described, for the classical planar spin model. The free energy of the system is computed by the growth estimate. Importance sampling is implemented, using a mean-field approximation of the eigenfunction as a trial wave function. The mixed expectation values of spin correlations are measured by their averages in the Monte Carlo ensemble.

A lattice calculation in (1 + 1)-dimensional QCD by the hamiltonian ensemble projector Monte Carlo method

The SU(2) gauge theory coupled to one unflavored fermion in two dimensions is studied on the lattice by a local hamiltonian Monte Carlo method. The vacuum energy, the quark condensate and the correlation function of the lightest meson are measured. For massless quarks the chiral symmetry is unbroken as expected in the continuum. A vanishing condensate is found and the meson mass is determined.

Numerical simulation of quantum spin models.

The general application of the projector Monte Carlo method to spin systems is discussed. The purpose of the paper is to present several variants of the method that improve its accuracy and convergence in specific problems and to give numerical examples of the improvements. More detailed application to specific spin models is left for subsequent publications.

Monte Carlo simulation of the ‘‘Kondo necklace’’

The projector Monte Carlo method and finite-size scaling are applied to the solution of the ground-state properties of the ``Kondo necklace'' Hamiltonian. The energy and various spin-spin correlation functions are calculated for chains up to 16 sites in length. Correlations between the spins on different sites are evaluated, and the nature of the ground-state phase as a function of the ratio of the exchange coupling J to the bandwidth W is examined. Evaluation of the correlation function 〈S\ensuremath{\rightarrow}(i)\ensuremath{\cdot}\ensuremath{\tau}\ensuremath{\rightarrow}(i)〉 shows that as the coupling between the localized S spins and the \ensuremath{\tau} spin chain increases, the system becomes composed of independent singlets on each site. Contrary to both mean-field and various approximate renormalization-group calculations, we find that our results are consistent with an absence of magnetic order as soon as J is different from 0.

Projector Monte Carlo study of confinement and roughening in (2+1)-dimensional U(1) lattice gauge theory

Using the projector Monte Carlo method of Blankenbecler and Sugar to compute Hamiltonian eigenvalues, we report string-tension evaluation down to $g=0.75$ in (2+1)-dimensional U(1) lattice gauge theory. Moreover, we present some evidence of string roughening via the study of the string width.