A new Monte Carlo power method for the eigenvalue problem of transfer matrices

M. P. Nightingale, Robert G. Caflisch

James E. Gubernatis, Naoki Kawashima, Philipp Werner

Quantum Monte Carlo versions of the power method for finding ground states come with different names, including the projector method, diffusion Monte Carlo, and Green's function Monte Carlo. These quantum Monte Carlo methods are primarily adoptions of methods developed for classical problems. We summarize the basics of deterministic power methods, detail key features of Monte Carlo power methods, and put these concepts into the context of quantum ground state calculations. We conclude the chapter by outlining power methods that allow the computation of a few excited states. In the computation of excited states, we encounter sign problems, which are discussed inmore detail in Chapter 11. Themethods and techniques of the present chapter are very general and most usefully applied to systems of Bosons and to systems of Fermions and quantum spins that do not suffer from a sign problem. Deterministic direct and inverse power methods The power method is over a century old. As a deterministic method, it originally was combined with the deflation technique (Meyer, 2000; Stewart, 2001b) as a method to compute all the eigenvalues and eigenvectors of small matrices. Today, its deterministic use is limited primarily to special applications involving very large, sparse matrices. Many of these applications are in the study of quantum lattice models as the corresponding Hamiltonian matrices are typically very sparse. In these applications, we can perhaps initially store in computer memory the relatively small number of nonzero matrix elements and all the vectors. Soon, because of what is undoubtedly the now familiar exponentially increasing number of basis states that determines the order of the Hamiltonian matrix and hence the size of our vectors, we reach the point where computer memory restrictions allow us to store only a few vectors and we need to compute the matrix elements on the fly. When we have problems for which we cannot even store all the components of one vector, the power method's expression as a Monte Carlo procedure is our only option. With it, we can compute a few of the extremal eigenvalues, that is, a few of the largest or smallest eigenvalues. The mathematical basis for the method is simple.

Yutaka Okabe, Yusuke Tomita, Chiaki Yamaguchi

Brenda M. Rubenstein

Quantum Monte Carlo methods are a powerful suite of techniques for solving the quantum many-body problem. By using random numbers to stochastically sample quantum properties, QMC methods are capable of studying low-temperature quantum systems well beyond the reach of conventional deterministic techniques. QMC techniques have likewise been indispensible tools for augmenting our current knowledge of superfluidity and superconductivity. In this thesis, I present two new quantum Monte Carlo techniques, the Monte Carlo Power Method and Bose-Fermi Auxiliary-Field Quantum Monte Carlo, and apply previously developed Path Integral Monte Carlo methods to explore two new phases of quantum hard spheres and hydrogen. I lay the foundation for a subsequent description of my research by first reviewing the physics of quantum liquids in Chapter One and the mathematics behind Quantum Monte Carlo algorithms in Chapter Two. I then discuss the Monte Carlo Power Method, a stochastic way of computing the first several extremal eigenvalues of a matrix too memory-intensive to be stored and therefore diagonalized. As an illustration of the technique, I demonstrate how it can be used to determine the second eigenvalues of the transition matrices of several popular Monte Carlo algorithms. This information may be used to quantify how rapidly a Monte Carlo algorithm is converging to the equilibrium probability distribution it is sampling. I next present the Bose-Fermi Auxiliary-Field Quantum Monte Carlo algorithm. This algorithm generalizes the well-known Auxiliary-Field Quantum Monte Carlo algorithm for fermions to bosons and Bose-Fermi mixtures. Despite some shortcomings, the Bose-Fermi Auxiliary-Field Quantum Monte Carlo algorithm represents the first exact technique capable of studying Bose-Fermi mixtures of any size in any dimension. In Chapter Six, I describe a new Constant Stress Path Integral Monte Carlo algorithm for the study of quantum mechanical systems under high pressures. While the eventual hope is to apply this algorithm to the exploration of yet unidentified high-pressure, low-temperature phases of hydrogen, I employ this algorithm to determine whether or not quantum hard spheres can form a low-temperature bcc solid if exchange is not taken into account. In the final chapter of this thesis, I use Path Integral Monte Carlo once again to explore whether glassy para-hydrogen exhibits superfluidity. Physicists have long searched for ways to coax hydrogen into becoming a superfluid. I present evidence that, while glassy hydrogen does not crystallize at the temperatures at which hydrogen might become a superfluid, it nevertheless does not exhibit superfluidity. This is because the average binding energy per p-H2 molecule poses a severe barrier to exchange regardless of whether the system is crystalline. All in all, this work extends the reach of Quantum Monte Carlo methods to new systems and brings the power of existing methods to bear on new problems.

Nicolas J. Cerf, Olivier Martin

Projection methods such as Green's function and diffusion Monte Carlo are commonly used to calculate the leading eigenvalue and eigenvector of operators or large matrices. They thereby give access to ground state properties of quantum systems, and finite temperature properties of classical statistical mechanical systems having a transfer matrix. The basis of these approaches is a stochastic application of the power method in which a "projection" operator is applied iteratively. For the systematic errors to be small, the number of iterations must be large; however, in that limit, the statistical errors grow tremendously. We present an analytical study of the main variance reduction methods used for dealing with this problem. In particular, we discuss the consequences of guiding, replication, and population control on statistical and systematic errors.

Ralf Gamillscheg, Gundolf Haase, Wolfgang von der Linden