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Cambridge University Press eBooks

Transfer matrix and diagonalisation of spin chains

Joseph Marie Thijssen

Published Mar 22, 2007• 0 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9781139171397.013

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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.
We have thus far looked at the classical statistical mechanics of a collection of Ising spins in terms of their partition function. We have seen how to extract the free energy per site and correlation functions of the d = 1 Ising model. We will now consider a second way to do this. While the results are, of course, going to be the same, the method exposes a deep connection between classical statistical mechanics in d = 1 and the quantum mechanics of a single spin-12 particle. This is the simplest way to learn about a connection that has a much wider range of applicability: the classical statistical mechanics of a problem in d dimensions may be mapped on to a quantum problem in d – 1 dimensions. The nature of the quantum variable will depend on the nature of the classical variable. In general, the allowed values of the classical variable ( s =±1 in our example) will correspond to the maximal set of simultaneous eigenvalues of the operators in the quantum problem ( σz in our example). Subsequently we will study a problem where the classical variable called x lies in the range –∞ < x < ∞ and will correspond to the eigenvalues of the familiar position operator X that appears in the quantum problem. The correlation functions will become the expectation values in the ground state of a certain transfer matrix . The different sites in the d = 1 lattice will correspond to different, discrete, times in the life of the quantum degree of freedom. By the same token, a quantum problem may be mapped into a statistical mechanics problem in one-higher dimensions, essentially by running the derivation backwards. In this case the resulting partition function is also called the path integral for the quantum problem. There is, however, no guarantee that the classical partition function generated by a legitimate quantum problem (with a Hermitian Hamiltonian) will always make physical sense: the corresponding Boltzmann weights may be negative, or even complex! In the quantum mechanics we first learn, time t is a real parameter. For our purposes we also need to get familiar with quantum mechanics in which time takes on purely imaginary values t = –iτ , where τ is real. It turns out that it is possible to define such a Euclidean quantum mechanics .
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 The aim of this chapter is to describe how transfer matrices can be used to solve one-dimensional classical spin models. The idea is to write down the partition function in terms of a matrix, the transfer matrix. The thermodynamic properties of the model are then wholly described by the eigenspectrum of the matrix. In particular the free energy per spin in the thermodynamic limit depends only on the largest eigenvalue and the correlation length only on the two largest eigenvalues through simple formulae.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds nnn lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve only interactions between nearest neighbours on the lattice. For such models, correlation functions can be calculated by a transfer matrix formalism. The chapter first describes some general properties of transfer matrices in one-dimensional models. This formalism is used to establish various properties of correlation functions, like the thermodynamic or infinite volume limit, the large-distance behaviour of the two-point correlation function, and introduces the very important concept of correlation length. Connected correlation functions, cumulants of the distribution, play a particularly important role. Indeed, these functions satisfy the cluster property, which characterizes their decay at large distance. The transfer matrix formalism is applied to the example of a Gaussian Boltzmann weight, which is studied in detail. The chapter calculates the partition function and correlation functions explicitly, and observes that , the correlation length diverges, making it possible to define a continuum limit. It shows that results of the continuum limit can be reproduced directly by solving a partial differential equation in which all traces of the initial lattice structure have disappeared. Finally, it exhibits a slightly more general class of models which share the same properties: divergent correlation length and continuum limit. Exercises are provided at the end of the chapter.
Abstract This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point can be identified by means of the lowest eigenvalue. The chapter covers Baxter's approach, the Yang–Baxter equation and its relation to the Boltzmann weights, the R-matrix, and discusses activity away from the critical point, the six-vertex model, as well as functional equations and symmetries.