This thesis work deals with Monte Carlo (MC) simulations with the focus on quantum mechanical systems. I start with an introduction about Monte-Carlo methods in general and their application to the classical Ising model. The second, most important part deals with the problem in quantum mechanics, and presents a possible simulation method, the Stochastic Series Expansion (SSE). I give an introduction about the idea behind that method and give some results obtained for simulations for finite spin chains. All programming was performed in FORTRAN.
This thesis work deals with Monte Carlo (MC) simulations with the focus on quantum mechanical systems. I start with an introduction about Monte-Carlo methods in general and their application to the classical Ising model. The second, most important part deals with the problem in quantum mechanics, and presents a possible simulation method, the Stochastic Series Expansion (SSE). I give an introduction about the idea behind that method and give some results obtained for simulations for finite spin chains. All programming was performed in FORTRAN.
The Stochastic Series Expansion (SSE) technique is a quantum Monte Carlo method that is especially efficient for many quantum spin systems and boson models. It was the first generic method free from the discretization errors affecting previous path integral based approaches. These lecture notes give a brief overview of the SSE method and its applications. In the introductory section, the representation of quantum statistical mechanics by the power series expansion of e−βH will be compared with path integrals in discrete and continuous imaginary time. Extensions of the SSE approach to ground state projection and quantum annealing in imaginary time will also be briefly discussed. The later sections introduce efficient sampling schemes (loop and cluster updates) that have been developed for many classes of models. A summary of generic forms of estimators for important observables are also given. Applications are discussed in the last section.
A quantum implementation of the Stochastic Series Expansion (SSE) Monte Carlo method is proposed, and it is shown that quantum SSE offers significant advantages over classical implementations of SSE. In particular, for problems where classical SSE encounters the sign problem, the cost of implementing a Monte Carlo iteration scales only linearly with system size in quantum SSE, while it may scale exponentially with system size in classical SSE. In cases where classical SSE can be efficiently implemented, quantum SSE still offers an advantage by allowing for more general observables to be measured.
… a general method, suitable for electronic computing machines, of calculating the properties of any substance which may be considered as composed of interacting individual molecules. N. Metropolis et al. , 1953 Summary Quantum Monte Carlo methods have been very useful in providing exact results, or, at least, exact constraints on properties of electronic systems, in particular for the homogeneous electron gas. The results are, in many cases, more accurate than those from other quantum many-body methods, and provide unique capabilities and insights. In this chapter we introduce the general properties of stochastic methods and motivate their use on the quantum many-body problem. In particular, we discuss Markov chains and the computation of error estimates. The methods that we introduce in the next four chapters are quite different from those in Parts II and III: stochastic or quantum Monte Carlo methods. In stochastic methods, instead of solving deterministically for properties of the quantum many-body system, one sets up a random walk that samples for the properties. Historically the most important role of QMC for the electronic structure field has been to provide input into the other methods, most notably the QMC calculation of the HEG [109], used for the exchange– correlation functional in DFT. A second important role has been as benchmarks for other methods such asGW. There are systems for which QMC is uniquely suited, for example the Wigner transition in the low-density electron gas, see Sec. 3.1. In this chapter we introduce general properties of simulations, in particular Markov chains, and error estimates. In the following chapters we will apply this theory to three general classes of quantum Monte Carlo algorithms, namely variational (Ch. 23), projector (Ch. 24), and path-integral Monte Carlo (Ch. 25); Ch. 18 already introduced the QMC calculation of the impurity Green's function used in the dynamical mean-field method.We note that there are a variety of other QMC methods not covered in this book. Simulations First let us define what we mean by a simulation, since the word has other meanings in applied science. The dimensionality of phase space (i.e., the Hilbert space for a quantum system) is large or infinite. Even a classical system requires the positions and momenta of all particles, and, hence, the phase space for N classical particles has dimensionality 6 N .
Many strongly correlated quantum systems are difficult to study numerically because of the exponential growth of the Hilbert space combined with the failure of numerical methods in important parameter regimes. This includes the long-standing problems of high-temperature superconductivity and frustrated magnetism. A promising solution to this dilemma are quantum simulators, well-controlled (synthetic) quantum systems, emulating model Hamiltonians not only of condensed matter systems, where parameters can be tuned at will and model complexity can be added step by step. This dissertation presents numerical simulation studies that are motivated by quantum simulators. Using large-scale quantum Monte Carlo computer simulations and mean-field theory the semiquantitative ground state phase diagram of a Penning ion trap quantum simulator, realizing a two-dimensional model of quantum magnetism with long-range interactions, is obtained. We study also thermal and quantum phase transitions in one-dimensional transverse-field Ising models with long-range interactions, which exhibit a Kosterlitz-Thouless transition and universality classes with continuously varying critical exponents. The second part of this thesis is concerned with the Fermi-Hubbard model, which has been realized recently with ultracold atoms in optical lattices, both for repulsive and attractive interactions. Equipped with a quantum gas microscope, which allows single-site single-atom detection, these experimental setups can collect histograms of the particle number from repeated projective measurements. This dissertation presents a method to compute such full counting statistics for quadratic operators such as particle number or magnetization in the framework of determinantal quantum Monte Carlo. From an analysis of the full counting statistics of the particle number for different subsystem sizes the size of preformed pairs or Cooper pairs in the attractive Hubbard model is inferred. Furthermore, excellent agreement of the numerically computed probability distribution of the staggered magnetization with a recent quantum gas experiment demonstrates that current experiments are capable of resolving the differences between the Hubbard model and its limiting Heisenberg model. An extensive method section reviews the stochastic series expansion quantum Monte Carlo method for Ising models in a transverse field, which is free of the sign problem for arbitrary interactions, with special emphasis on the efficient simulation of long-range interactions. Also a self-contained introduction to the determinantal quantum Monte Carlo method is provided.
A quantum Monte Carlo method is simply a Monte Carlo method applied to a quantum problem. What distinguishes a quantum Monte Carlo method from a classical one is the initial effort necessary to represent the quantum problem in a form that is suitable for Monte Carlo simulation. It is in making this transformation that the quantum nature of the problem asserts itself not only through such obvious issues as the noncommutivity of the physical variables and the need to symmetrize or antisymmetrize the wave function, but also through less obvious issues such as the sign problem. Almost always, the transformation replaces the quantum degrees of freedom by classical ones, and it is to these classical degrees of freedom that the Monte Carlo method is actually applied. Succeeding chapters present and explain many of the quantum Monte Carlo methods being successfully used on a variety of quantum problems. In Chapters 1 and 2 we focus on discussing what the Monte Carlo method is and why it is useful. The Monte Carlo method The Monte Carlo method is not a specific technique but a general strategy for solving problems too complex to solve analytically or too intensive numerically to solve deterministically. Often a specific strategy incorporates several different Monte Carlo techniques. In what is likely the first journal article to use the phrase “Monte Carlo,” Metropolis and Ulam (1949) discuss this strategy. To paraphrase them, The Monte Carlo method is an iterative stochastic procedure, consistent with a defining relation for some function, which allows an estimate of the function without completely determining it. This is quite different from the colloquialism, “a method that uses random numbers.” Let us examine the definition piece by piece. A key point will emerge. Ulam and Metropolis were presenting the motivation and a general description of a statistical approach to the study of differential and integro-differential equations. These equations were their “defining relation for some function.” The “function” was the solution of these equations. This function is of course unknown a priori. Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953) a few years later would propose a statistical approach to the study of equilibrium statistical mechanics. The defining relation there was a thermodynamic average of a physical quantity over the Boltzmann distribution. The function was the physical quantity, and the unknown its average.
We have provided a concise introduction to the Ising model as one of the most important models in statistical mechanics and in studying the phenomenon of phase transition. The required theoretical background and derivation of the Hamiltonian of the model have also been presented. We finally have discussed the computational method and details to numerically solve the two- and three-dimensional Ising problems using Monte Carlo simulations. The related computer codes in both Python and Fortran, as well as a simulation trick to visualize the spin lattice, have also been provided.