Quantum Monte Carlo Methods in Condensed Matter Physics

The following sections are included:

• PREFACE

• CONTENTS

A general decomposition theory of exponential operators is reviewed, and its possible applications to quantum Monte Carlo simulations are discussed. The generalized Trotter formula, the improved Trotter-like formula, and higher-order decomposition formulas are presented, and the convergence of these formulas is also discussed.

Quantum Monte Carlo methods are reviewed with emphasis on their methodological aspects. Methods based on the Suzuki-Trotter approximation, namely the world-line approach and the auxiliary-field approach, are mainly described. The transfer-matrix method, which is closely related to the quantum Monte Carlo methods, is also reviewed. Recently developed arguments on the negative-sign problem are presented.

We report current progress on the synthesis of methods to alleviate two major difficulties in implementing a Monte Carlo Renormalization Group (MCRG) for quantum systems. In particular, we have utilized the loop-algorithm to reduce critical slowing down, and we have implemented an MCRG method in which the symmetries of the classical equivalent model need not be fully understood, since the Renormalization Group is given by the Monte Carlo simulation. We report preliminary results obtained when the resulting MCRG method is applied to the d=1 XXZ model. Our results are encouraging. However, since this model has a Kosterlitz-Thouless transition, it does not provide a full test of our MCRG method.

The classical d+1-dimensional spin systems used for the simulation of quantum spin systems in d dimensions are, quite generally, vertex models. Standard simulation methods for such models strongly suffer from critical slowing down. Recently, we developed the loop algorithm, a new type of cluster algorithm that to a large extent overcomes critical slowing down for vertex models. We present the basic ideas on the example of the F model, a special case of the 6-vertex model. Numerical results clearly demonstrate the effectiveness of the loop algorithm. Then, using the framework for cluster algorithms developed by Kandel and Domany, we explain how to adapt our algorithm to the cases of the 6-vertex model and the 8-vertex model, which are relevant for spin systems. The techniqes presented here can be applied without modification to 2-dimensional spin systems, provided that in the Suzuki-Trotter formula the Hamiltonian is broken up into 4 sums of link terms. Generalizations to more complicated situations (higher spins, different uses of the Suzuki-Trotter formula) are, at least in principle, straightforward.

The spin-1 antiferromagnetic and ferromagnetic chain in the presence of two-fold random anisotropy have been investigated by Trotter-Suzuki decomposition and transfer matrix methods. Results for the energy and the effect of randomness on die Haldane gap are presented for the antiferromagnetic chain. For the ferromagnetic chain results for the magnetization and susceptibility and given which indicate the irrelevance of the anisotropy.

The spin-½ chain frustrated by random bonds and fields has also been investigated. Results for the entropy and magnetization are given.

Inhomogeneity effects in quantum spin systems are studied by quantum Monte Carlo methods. In particular, the ground-state and thermodynamic properties of diluted S = 1/2 Heisenberg antiferromagnets on the square lattice and S = 1 antiferromagnetic Heisenberg chains with open boundaries are investigated. Methodological problems which are inherent in inhomogeneous systems are are also described.

The quantum transfer matrix method based on the checkerboard decomposition and its application to one-dimensional spin systems are briefly reviewed. It is demonstrated that the extrapolation in the quantum limit is quite effective so that the properties at fairly low temperatures can be described by using approximations with a modest number of Trotter slicings.

We review the quantum transfer matrix method based on the path integral idea, and summarize useful related theorems. The eigenvalues of the transfer matrix in a quantum many-body system lead to the free energy and the correlation lengths in the thermodynamic limit. For calculating these eigenvalues efficiently, the thermal Bethe ansatz method and the Monte Carlo power method are presented. We treat also transfer matrices in quantum many-body systems at zero temperature.

We give an efficient Monte Carlo method of calculating the elementary excitation spectrum of quantum systems. The lowest energy with an arbitrary momentum is obtained by the projector Monte Carlo method. This method is applied to the spin-1/2 and spin-1 Heisenberg antiferromagnetic chains of length 32. For the S=1/2 case, the spectrum coincides completely with des Cloiseaux and Pearson’s spectrum. For the S=1 case, the spectrum has a gap at the momentum π as was predicted by Haldane. The value of the gap coincides with Nightingale and Blöte’s calculation. The spectrum satisfies the variational relation to the structure factor. The properties of S = 1 Heisenberg antiferromagnetic chains are completely different from those of S = 1/2 Heisenberg antiferromagnetic chains. This method is also applied to the anisotropic S = 1 Heisenberg chain. The results are in excellent agreement with those of the diagonalization method and experiment.

The decoupled cell method(DCM) and modified decoupled cell method(mDCM) are introduced as new methods of quantum Monte Carlo calculations. Both methods are applied to the one-dimensional XY-model. The results are compared with the exact ones.

Decoupled Cell Monte Carlo calculations of the critical properties of the spin-1/2 Heisenberg model on cubic lattices are presented. Critical exponents for the ferromagnetic model are determined, establishing the utility of the Decoupled Cell Method for critical models in three dimensions. Our results are not in agreement with the ɛ-expansion, or Monte Carlo calculations for the classical Heisenberg model, but do obey the usual scaling laws.

A review is given on some of recent applications of the variational Monte Carlo method to correlated electrons. The systems discussed in this article include the Hubbard model, the t – J model and the Kondo lattice, which represent various aspects of correlated electrons.

Application of the quantum Monte Carlo method to multiband fermion systems is described. The result for our recent model for superconductivity in a class of multiband systems comprising a metallic band interacting repulsively with an insulating band has unraveled the superconducting ground state as hallmarked by the pairing correlation. function similar to that for the attractive Hubbard model. The similarity persists, surprisingly, well into the non-perturbative regime. This confirms the generic mechanism for superconductivity proposed for the system having a metallic band on top of another, which may be either weakly or strongly correlated charge-transfer insulator, Mott-Hubbard system, etc, and provides an example of the remarkable strength of the quantum Monte Carlo method in a non-perturbative regime that is otherwise difficult to probe.

A new Quantum Monte Carlo algorithm is used to solve models of strongly correlated systems in the limits of infinite spatial dimensions and lattice size. A detailed description of this algorithm is given, concentrating on the methods used to reduce systematic errors and to measure the two-particle properties. The algorithm is then applied to the Hubbard model, which is shown to retain the qualitative features expected in the limit of finite dimensions. These include a Mott gap and antiferromagnetism for the half-filled model. Away from half filling the Mott gap disappears, replaced by a Kondo-like peak in the density of states associated with screening of the local spin. At low temperatures and low doping, a transition to an incommensurate magnetic state is also observed. In the paramagnetic state of the lightly doped model, we find anomalies in the resistivity and inverse NMR rate which are strikingly similar to those characteristic for the cuprate high temperature superconductors (ρ ~ T and 1/T₁ ~ T at high temperatures).

The interpretation of the sign problem is discussed with reference to auxiliary field quantum Monte Carlo calculations on the Hubbard and Heisenberg models. In many cases one must use a vector auxiliary field, in which case the sign problem can be interpreted geometrically in terms of a Berry phase. The classical effective Hamiltonian for the auxiliary field, obtained by integrating out finite-frequencies, is complex as a result of the negative-weight Paths. The static approximation is the leading term in a 1/N expansion of this Hamiltonian.

We summarize results of quantum Monte Carlo simulations of the degenerate, single-impurity, Anderson model. Using Maximum Entropy methods, we performed the analytic continuation of the imaginary-time Green’s functions produced by these simulations to obtain their real-frequency, single-particle, spectral densities for degeneracies of N = 2, 4, and 6. In several cases, we split the degeneracies to mimic crystal electric field and external magnetic field effects. All simulations were done on a cluster of workstations using a simple message-passing model. This model is also briefly described.

The ground state properties of a many body hamiltonian H can be calculated by a recently developed technique, known as “the Projection Quantum Monte Carlo method”. Instead of simulating the many body system at finite temperature, one attempts to reach the zero temperature limit by applying to a trial wavefunction, suitably chosen, a many body propagator e^−Ht. For large enough t -t playing a role of an effective inverse temperature- the ground state component of the trial state is filtered out usually much faster than the corresponding finite temperature method. The Hubbard Stratonovich transformation allows to formulate the many body propagation of the trial state as that of sampling a distribution. This distribution is constructed by propagating the trial wavefunction under the influence of a one-body time dependent external field. However this distribution may be not positive definite, especially for large t, and serious problems occur when the average sign of the distribution is too small. We introduce a symmetric Hubbard Stratonovich transformation that in principle may solve this problem with large enough computer time. We also discuss how to apply the “Projection Quantum Monte Carlo” technique to the infinite U Hubbard model in a simple way.

This paper discusses the method of projecting out the ground-state of an interacting fermion system using propagation of a trial state in imaginary-time, with particle interactions represented exactly by the auxiliary-field (Hubbard-Stratonovitch) approach. The sampling of fields using the hybrid molecular-dynamics-Monte-Carlo method is discussed in this context. An understanding of sign problems which occur in the auxiliary-field ground-state projection method in terms of a diffusion-like equation is reviewed and an approximate ground-state projection method which arises naturally from this understanding is presented.

Recently developed simulation algorithms for strongly correlated lattice fermions are reviewed. The simulation methods which we describe are based on the path-integral formalism. Basic algorithms for auxiliary-field methods as well as prescriptions for reducing the minus-sign problem are presented. As an important application, ground state properties of the Hubbard model are discussed in the light of recent simulation results. Clarified so far, important features of antiferromagnetic transitions, metal-insulator transitions and pairing mechanisms in strongly correlated systems are summarized. Recent applications of the fermion Monte Carlo method to other fermion models with degenerate orbitals such as the d-p model are also reviewed. Heavy fermion systems, exhibiting a variety of crossovers such as the temperature crossover between the intersite antiferromagnetic correlation and the local singlet formation, are also discussed. Pairing mechanisms examined numerically in a variety of extended lattice models are briefly sketched.

The phase transition in an interacting boson system in 2D from a superfluid to an insulator with increasing amounts of disorder is emerging as a paradigm for a large class of problems. Among them are ⁴He adsorbed in porous media, superconductor-insulator transition in homogeneous and granular films, Josephson junction arrays, vortices in bulk type II superconductors, the quantum Hall transition from a Hall liquid to a Hall insulator, and random magnets. We review our work on the the application of quantum Monte Carlo methods (checker board and path integral algorithms) to the dirty boson problem. We present first qualitative results for a model of bosons on a 2D square lattice with a random potential of strength V and on-site repulsion U. By calculating the superfluid density and the excitation spectrum we show the existence of three phases at T = 0- (i) a superfluid phase, (ii) a localized or ‘Bose glass’ phase with gapless excitations, and (iii) a Matt phase with a gap to excitations (found only at commensurate densities). We discuss unusual effects arising from the interplay between interaction and disorder effects. We next investigate the critical properties of the superfluid to Bose glass transition for a hard core bose system as a function of disorder. Using finite size scaling on 64 × 64 lattices we find the dynamical exponent z = 0.5 ± 0.1, the correlation length exponent ν = 2.2±0.2, and a non-zero compressibility κ, at the transition. Our conclusions differ from the existing scaling theory as well as from simulations on simplified models argued to be in the same universality class. Our results are suggestive of new low lying collective excitations in the disordered system that are modified from usual phonons.

A review is given of recent efforts at using the path-integral formulation of the quantum Monte Carlo to study the static and dynamic properties of crystals. The computation of the static properties of energy, specific heat and lattice constant as functions of temperature for systems of atoms interacting by Lennard-Jones (12-6) potentials is first considered. Results for fcc systems appropriate to neon and argon are presented and these are compared with analytical results from the effective potential and improved self-consistent theories and with results from the harmonic approximation. In addition, some earlier investigations on the static properties of a one-dimensional chain of atoms are also considered and compared with the exact solutions for these properties in the corresponding classical system. A discussion of the time-dependent thermodynamic properties of a one-dimensional quantum chain of atoms is made in terms of the response functions which describe the inelastic neutron scattering from the chain. A continued fraction representation, based on the work of Mori [1], is presented for these response functions and the coefficients in the continued fraction are computed using the static path-integral quantum Monte Carlo method. Improvements on Mori’s method applied to the quantum chain are made by using molecular dynamics results for the corresponding classical chain of atoms to guide in terminating the continued fraction representation of the quantum response functions.

We discuss numerical methods for calculating time-correlation functions for quantum systems. We review several methods including analytical continuation methods, stationary phase methods and exact enumeration. We present some results for the relaxation of a quantum spin in a fluctuating medium obtained by exact enumeration.