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Duality

Hidetoshi Nishimori, Gerardo Ortíz

Published Dec 2, 2010• 0 citations
Publisher

Abstract

Abstract Exact solutions of model systems are the most reliable source of information. However, there are not many model systems that can be exactly solved and, consequently, many approximate schemes such as series expansions have been developed. It is sometimes possible, though, to extract exact information without directly solving those model systems. Arguments using duality transformations make it possible to derive the exact location of the phase transition point and the exact value of the energy at the transition point when the model is self dual. Duality not only determines the exact location of the transition point of the two-dimensional Ising model and related models but also is useful to rewrite the $XY$ model into a different form, the Villain model and its dual, which reveals new physical aspects of the system. Classical and quantum dualities can be related by certain mappings and Fourier transformation is at the root of such deep results.

DOI
10.1093/acprof:oso/9780199577224.003.0010

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Abstract The Ising model is the simplest model for describing many-body effects in classical statistical mechanics. A duality analysis leads to its critical point under several assumptions. The Ising model has Z2Z_2Z2​-symmetry. The basis of duality analysis is a nontrivial relationship between low- and high-temperature expansions. However, discrete Fourier transformation automatically determines the hidden relationship. The duality analysis can naturally extend to systems with various degrees of freedom, ZqZ_qZq​ symmetry, and random spin systems. Furthermore, we obtained the duality relation in a series of permutation models in the present study by considering the symmetric group SqS_qSq​ and its Fourier transformation. The permutation model in a symmetric group is closely related to random quantum circuits and random tensor network models, which are frequently discussed in quantum computing. It also relates to the holographic principle, a property of string theories and quantum gravity. We provide a systematic approach using duality analysis to examine the phase transition in these models.
Abstract We have now seen how powerful the tools of statistical mechanics can be in the study of complex systems in thermal equilibrium and how they can be used to determine exactly the behaviour of some simplified, but nevertheless interesting, models of phase transitions. However, these exact solutions are sometimes lengthy and difficult and there is no guarantee that any particular model will yield exact results. For example, we were able to solve the two-dimensional, but not the three-dimensional, Ising model, and that only in zero magnetic field. So, it is natural to consider whether we can use a computer to simulate these systems. In this chapter we shall see that this is indeed possible; a great deal of our understanding of critical phenomena is based wholly or partly on the results of such numerical work. However, as we shall also see, such simulations are not entirely straightforward and a number of tricks are needed to obtain accurate results. This is partly a consequence of the very physics that makes the phase transitions interesting to start with. One should probably think of numerical simulation as a whole subject, almost an art, of its own; we shall have time only to scratch at the surface and discuss a few of the most important ideas.
The Ising model is the simplest to describe many-body effects in classical statistical mechanics. Duality analysis leads to a critical point under several assumptions. The Ising model itself has Z(2)Z(2)Z(2) symmetry. The basis of the duality analysis is a nontrivial relationship between low and high-temperature expansions. However, the discrete Fourier transformation finds the hidden relationship automatically. The duality analysis can be naturally generalized into the case with the degrees of freedom with Z(q)Z(q)Z(q) symmetry and random spin systems. We further obtain the duality in a series of permutation models in the present study by considering the symmetric group SqS_qSq​ and its Fourier transformation. The permutation model in the symmetric group is closely related to the random quantum circuits and random tensor network model, often discussed in the context of quantum computing and the holographic principle, a property of string theories and quantum gravity. We provide a systematic way by our duality analysis to analyze the phase transition in these models.
Abstract Chapter 4 begins by discussing the Peierls argument, which allows us to prove the existence of a phase transition in the two-dimensional Ising model. The remaining sections of the chapter deal with duality transformations (duality in square, hexagonal and triangular lattices) that link the low- and high-temperature phases of several statistical models. Particularly important is the proof of the so-called star-triangle identity. This identity will be crucial in the later discussion of the transfer matrix of the Ising model. Finally, it covers the aspect of duality in two dimensions. An appendix provides information about the Poisson sum formula.
Abstract Only a limited number of models of phase transitions and critical phenomena can be solved exactly. These examples nevertheless play important roles in many aspects including the verification of the accuracy of approximation theories such as the mean-field theory and renormalization group. Mathematical methods to solve such examples are interesting in their own right and constitute an important subfield of mathematical physics. In particular the exact solution of the two-dimensional Ising model occupies an outstanding status as one of the founding studies of the modern theory of phase transitions and critical phenomena. The present chapter shows simple but typical examples of exact solutions of classical spin systems such as the one-dimensional Ising model with various boundary conditions, the n-vector model, the spherical model, the one-dimensional quantum XYXYXY model, and the two-dimensional Ising model. An account on the Yang-Lee theory will also be given as a set of basic rigorous results on phase transitions.