Quantum Monte Carlo methods. Algorithms for lattice models (Q2814192)

The book presents detailed explanations of the major algorithms used in quantum Monte Carlo simulations It is composed in twenty-thirteen chapters divided into four parts and fifteen appendices.NEWLINENEWLINEThe first part deals with Monte Carlo basics and is composed in five chapters.NEWLINENEWLINEChapter one is the introduction about quantum Monte Carlo. A quantum Monte Carlo method is a simply a Monte Carlo method applied to a quantum problem in which the quantum problem is represented in a form that is suitable for a Monte Carlo simulation. Monte Carlo method is a general strategy for solving problems too complex to solve analytically or to intensive numerically to solve deterministically. Chapter two deals with Monte Carlo Basics. Some probability concepts are introduced, such a the sample space or a random variable that maps an outcome to real number. Joint probability, law of total probability or the conditional probability are introduced. The next section deals with random sampling. To sample a probability distribution means that we generate events with a frequency proportional to it. Discrete and continuous distributions are described. The next section deals with Markov chain Monte Carlo followed by the Metropolis algorithm. The Metropolis algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. The purpose of the Metropolis-Hastings algorithm is to generate a collection of states according to a desired distribution \(P(x)\). The algorithm uses a Markov process which asymptotically reaches a unique stationary distribution. A Markov process is uniquely defined by its transition probabilities, the probability of transitioning from any given state to any other given state. It has a unique stationary distribution when it is unique and a detailed balance condition is present. Detailed balance is sufficient but not necessary condition. It requires that each transition is reversible: The Metropolis algorithm thus consists in the following, initialisation: pick an initial state \(x\) at random; randomly pick a state \(x'\) according to \(g(x'|x)\), accept the state according to an acceptance matrix. If not accepted, transition doesn't take place, and so there is no need to update anything. Else, the system transits to \(x'\). Continue the procedure until \(N\) states were generated. The value of \(N\) must be chosen according to different factors such as the proposal distribution. In the next subsection, the generalised Metropolis algorithm is introduced, The matrix describing the trial transitions must not be symmetric anymore. Then, the Heat-bath algorithm is described in which the value of the local variable is independent of its current value and is always accepted. The variable is sampled from a univariate density that is conditional on the value of a small number of variables. Then, Rosenbluth's theorem is introduced that indicates special character of the convergence and the validity of the Metropolis algorithm. Chapter three deals with data analysis, with calculating averages and estimating errors and blocking analysis and data sufficiency, for example it is assumed that 32 uncorrelated values are enough to perform a Monte Carlo simulation. The chapter end with a structure of a Monte Carlo program. The following Chapter four describes classical examples of many-body problems. By the example of the Ising model three different Monte Carlo algorithms are presented. Finite-temperature properties are simulates by the Metropolis algorithm using local updates by describing a single spin-flip update. The second algorithm describes the cluster updates by the Metropolis algorithm that was proposed by Swedsen-Wang by the example of the Ising model. It preforms cluster updates on lattices. The third algorithms preforms worm updates in which the phase space is the set of all closed graphs. The task is to construct a Markov process for generating closed graphs with a certain frequency. To preform this one decomposes the local Boltzmann factor resulting in a partition function. A new state cannot be proposed by choosing a pair of nearest neighbour sites at random and then removing or adding an edge to connect them, because doing so results in a state with two odd vertices and such a state does not contribute to the partial function, Instead one generates ``artificial'' states that do not contribute to the partition function. For at most two odd vertices in the artificial state, there is a high chance of observing closed graphs with sufficient frequency.NEWLINENEWLINEChapter five is titled ``The quantum Monte Carlo primer''. The main idea is to map a Quantum system in dimension \(d\) to a classical system in dimension \(d+1\). In quantum statistical mechanics, this additional dimension is an imaginary-time axis. After the mapping \(d\) classical Monte Carlo on the equivalent problem is done. We need to map the quantum partition function to a classical problem. The quantum version of the classical one-dimensional Ising model is presented by the continuous-time cluster algorithm for transverse-field Ising model. Then, the simulation with loops and worms is presented. The negative-sign problem is explainedNEWLINENEWLINEThe second part ``Finite temperature'' is composed in three chapters.NEWLINENEWLINEChapter six deals with finite-temperature quantum spin algorithms. The Feynman's path integral is introduced with imaginary time. Differences in how the worm exactly moves lead to different algorithms that are presented. Chapter seven presents the determinant method for simulating fermions, power methods for computing ground and excited states, and the variational Monte Carlo method. Chapter eight deals with continuous-time impurity solvers heir use within dynamical mean-field theory.NEWLINENEWLINEThe third part ``Zero temperature'' is composed in three chapters.NEWLINENEWLINEChapter nine describes the variational Monte-Carlo. It is quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system. Then, the quantum Monte Carlo version of the power method for finding the ground is introduced followed by Chapter eleven that presents Fermion ground state methods.NEWLINENEWLINEThe fourth part is called ``Other topics'' and is composed in three chapters.NEWLINENEWLINEChapter twelve deals with analytic continuation. Numerical methods for extracting information for finite-temperature quantum Monte Carlo methods are discussed, that used numerical methods that are usually used in Bayesian statistical interference. Central to a Bayesian method is the Bayes' theorem that is composed of the prior probability multiplied with the likelihood function and divided by the evidence resulting in the posterior probability. The principle of the maximum entropy maximises a constrained entropy. The principle of maximum entropy says that to assign probabilities on the basis of partial information, we maximise the entropy, constrained by whatever information we know about the probability. What is produced is the least informative probability consistent with the constraints. In general, the principle of maximum entropy is useful for suggesting the functional forms of probabilities. The approaches were used in several Monte Carlo methods for finding ground state gaps and excited state. Chapter thirteen deals with parallelisation of Monte Carlo simulations.NEWLINENEWLINEThe four parts are followed by fifteen useful appendices, indicating the alias method, rejection method to the Bryan algorithm and many more.NEWLINENEWLINEThe book is the first of its own, its is an essential source of information for everyone interested in quantum Monte Carlo techniques.