Abstract This chapter introduces a few typical numerical methods used in modern studies of phase transitions and critical phenomena in spin systems. The first section describes the stochastic dynamics of a generic system with discrete degrees of freedom following the master equation. This section serves as a theoretical basis for the Monte Carlo method that includes the heat bath and Metropolis algorithms of configuration updates. Another useful numerical technique is the transfer matrix method, described in the last section, and which is applied for numerically exact evaluation of the free energy and related physical quantities.
Abstract This chapter introduces a few typical numerical methods used in modern studies of phase transitions and critical phenomena in spin systems. The first section describes the stochastic dynamics of a generic system with discrete degrees of freedom following the master equation. This section serves as a theoretical basis for the Monte Carlo method that includes the heat bath and Metropolis algorithms of configuration updates. Another useful numerical technique is the transfer matrix method, described in the last section, and which is applied for numerically exact evaluation of the free energy and related physical quantities.
A new field opened when modern computers offered the possibility of performing extensive simulations of large systems. This allows known behaviours to be checked and provides an exploratory guide in circumstances where theoretical tools are absent. Measurements of observables can be compared both to existing theoretical expectations – providing a crucial test for their validity – and to experiments – checking the modelling of a physical system –. This chapter presents the background material needed to design simulations on a (relatively) large scale. Some numerical examples have already been presented in previous chapters, and we only give a few further illustrations, pertaining mainly to lattice gauge theory, the usefulness of which relies extensively on this method. We also describe a practical implementation of real space renormalization, known as the Monte Carlo Renormalization Group method (Ma, Swendsen, Wilson). Finally, we discuss specific issues relevant to the extension of the simulations to fermionic systems. Algorithms Generalities Systems with up to 10 6 to 10 7 variables can be handled by computers, and these numbers may soon be significantly increased. The measurements can be sufficiently numerous to allow statistical accuracy. Although simulated systems still have a modest size as compared to macroscopic systems, collective effects already clearly appear and accurate results about critical phenomena emerge from the numerical simulations. It turns out, when investigating more closely the available numerical methods, that one gets a better insight into the foundations of equilibrium statistical physics, the ergodicity problems, the meaning of probabilities and, last but not least, ways to approach equilibrium.
Introduction In Chapters 8 and 10 we studied methods for simulating classical systems consisting of many interacting degrees of freedom. In these methods, a sequence of system configurations is generated, and from this sequence averages of physical quantities, given as functions of the degrees of freedom (positions and momenta, or spins), can be determined. These quantities are called mechanical quantities, and the expressions of their expectation values are called mechanical averages . There exist, however, quantities that cannot be determined straightforwardly using these methods. These quantities include free energies and chemical potentials. The point is that these quantities are not given as a normalised average, which for mechanical quantities is replaced by an average over the configurations generated in the simulation. In the previous chapters we have seen that it is not straightforward to find free energies and chemical potentials using MC and MD methods (see Section 10.5). In this chapter we discuss a method which enables us to find free energies for lattice spin models with very high accuracy; this is the transfer matrix method . This method calculates the free energy of a model defined on a strip of finite width and infinite length directly in terms of the largest eigenvalue of a large matrix, the transfer matrix . This matrix contains essentially the Boltzmann weights for adding an extra row of spins to the strip. In Chapter 12 we shall see that the transfer matrix is the analogue of the time evolution operator in quantum mechanics.
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.
Abstract In the last chapter we saw that the Metropolis algorithm with single-spin-flip dynamics is an excellent Monte Carlo algorithm for simulating the using model when we are interested in temperatures well away from the critical temperature Tc However, as we approach the critical temperature, the combination of large critical fluctuations and long correlation time makes the errors on measured quantities grow enormously. As we pointed out, there is little to be done about the critical fluctuations, since these are an intrinsic property of the model near its phase transition (and are, what’s more, precisely the kind of interesting physical effect that we want to study with our simulations). On the other hand, the increase in the correlation time close to the phase transition is a function of the particular algorithm we are using-the Metropolis algorithm in this case-and it turns out that by using different algorithms we can greatly reduce this undesirable effect. In the first part of this chapter we will study one of the most widely used and successful such algorithms, the Wolff algorithm. Before introducing the algorithm however, we need to define a few terms.
This discussion serves as an introduction to the use of Monte Carlo simulations as a useful way to evaluate the observables of a ferromagnet. Key background is given about the relevance and effectiveness of this stochastic approach and in particular the applicability of the Metropolis-Hastings algorithm. Importantly the potentially devastating effects of spontaneous magnetization are highlighted and a means to avert this is examined. An Ising model is introduced and used to investigate the properties of a two dimensional ferromagnet with respect to its magnetization and energy at varying temperatures. The observables are calculated and a phase transition at a critical temperature is also illustrated and evaluated. Lastly a finite size scaling analysis is undertaken to determine the critical exponents and the Curie temperature is calculated using a ratio of cumulants with differing lattice sizes. The results obtained from the simulation are compared to exact calculations to endorse the validity of this numerical process. A copy of the code used, written in C++, is enclosed and is freely available for use and modification under the General Public License.
As we have seen on several occasions, a great many of the physical phenomena of interest concern collective behaviour or strongly correlated particles. While it is sometimes possible to obtain useful insight using approximation methods, such as perturbation theory, reliable controlled approximations do not exist. In such situations, numerical simulations have become an indispensable tool. We shall not attempt in this chapter to discuss computer programming nor shall we discuss many of the powerful methods, both algorithmic and in data analysis, that have been developed since numerical simulations have become an important research tool. Instead, we shall assume that the reader already has enough familiarity with computer programming to apply the notions which we discuss here. Our goal is to explain enough of the basics of the Monte Carlo method (classical and quantum) to allow the reader to apply it readily to interesting equilibrium statistical physics problems of the type discussed in this book. The problems in this chapter have been tailored with this in mind. The programs needed are relatively simple and the physics illustrates many of the phenomena already seen: phase transitions with and without spontaneous symmetry breaking, critical exponents, diverging susceptibilities, scaling, vortices, superfluids etc. Consequently, some of the problems are somewhat long and may be better considered as mini-projects. Markov chains, convergence and detailed balance Even relatively innocent looking models are impossible to solve exactly. The exact solution of the two-dimensional Ising model is known but not in the presence of an external magnetic field. In three dimensions, even in the absence of a magnetic field, the exact solution is not known. In fact, relatively few models can be solved exactly; see Reference [14] for an excellent review of the subject.