Geoffrey Grimmett
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.003.0008
- Subject:
- Mathematics, Probability / Statistics
The Tutte polynomial and its relatives play important roles in matroid theory, computational complexity, and models of statistical physics. They provide the natural way to count and relate a variety ...
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The Tutte polynomial and its relatives play important roles in matroid theory, computational complexity, and models of statistical physics. They provide the natural way to count and relate a variety of objects defined on graphs. This chapter shows that they permit a representation of the two-point correlation function of a ferromagnetic Potts model on a graph G in terms of the flow polynomials of certain related random graphs. This representation extends to general Potts models the so-called random-current expansion for Ising models, and it amplifies the links between the Potts partition function and the Tutte polynomial.Less
The Tutte polynomial and its relatives play important roles in matroid theory, computational complexity, and models of statistical physics. They provide the natural way to count and relate a variety of objects defined on graphs. This chapter shows that they permit a representation of the two-point correlation function of a ferromagnetic Potts model on a graph G in terms of the flow polynomials of certain related random graphs. This representation extends to general Potts models the so-called random-current expansion for Ising models, and it amplifies the links between the Potts partition function and the Tutte polynomial.
Barry M. McCoy
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199556632
- eISBN:
- 9780191723278
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556632.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter presents the exact results for the grand partition function of the hard hexagon model in both low and high density regions are presented. In the low density region, these results are ...
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This chapter presents the exact results for the grand partition function of the hard hexagon model in both low and high density regions are presented. In the low density region, these results are used to derive the first 25 virial coefficients of the virial expansion. The analyticity of the pressure in the density plane in both the low and high density regions are then presented. The general theory of the chiral Potts model as a two-dimensional statistical model is presented and the eigenvalues of the three-component superintegrable case are computed in detail. The order parameter is discussed and the phase diagram of the general three-component integrable chiral Potts spin chain is given. Open questions are discussed about Q operators, eight-vertex model degeneracies, and conjectures for correlations functions of the superintegrable chiral Potts model.Less
This chapter presents the exact results for the grand partition function of the hard hexagon model in both low and high density regions are presented. In the low density region, these results are used to derive the first 25 virial coefficients of the virial expansion. The analyticity of the pressure in the density plane in both the low and high density regions are then presented. The general theory of the chiral Potts model as a two-dimensional statistical model is presented and the eigenvalues of the three-component superintegrable case are computed in detail. The order parameter is discussed and the phase diagram of the general three-component integrable chiral Potts spin chain is given. Open questions are discussed about Q operators, eight-vertex model degeneracies, and conjectures for correlations functions of the superintegrable chiral Potts model.
Barry M. McCoy
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199556632
- eISBN:
- 9780191723278
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556632.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) ...
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This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.Less
This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.
Barry M McCoy
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199556632
- eISBN:
- 9780191723278
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556632.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on ...
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This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on general theory which includes a summary of the basic principles of statistical mechanics; a presentation of the physical phenomena covered and the models used to discuss them; theorems on the existence and uniqueness of partition functions; theorems on order; and critical phenomena and scaling theory. The second part is on series and numerical methods which includes derivations of the Mayer and Ree–Hoover expansions of the low density virial equation of state; Groeneveld's theorems; the application to hard spheres and discs; a summary of numerical studies of systems at high density; and the use of high temperature series expansions to estimate critical exponents for magnets. The third part covers exactly solvable models which includes a detailed presentation of the Pfaffian methods of computing the Ising partition function, magnetization, correlation functions, and susceptibility; the star-triangle (Yang–Baxter equation); functional equations and the free energy for the eight-vertex model; and the hard hexagon and chiral Potts models. All needed mathematics is developed in detail and many open questions are discussed. The goal is to guide the reader to the current forefront of research.Less
This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on general theory which includes a summary of the basic principles of statistical mechanics; a presentation of the physical phenomena covered and the models used to discuss them; theorems on the existence and uniqueness of partition functions; theorems on order; and critical phenomena and scaling theory. The second part is on series and numerical methods which includes derivations of the Mayer and Ree–Hoover expansions of the low density virial equation of state; Groeneveld's theorems; the application to hard spheres and discs; a summary of numerical studies of systems at high density; and the use of high temperature series expansions to estimate critical exponents for magnets. The third part covers exactly solvable models which includes a detailed presentation of the Pfaffian methods of computing the Ising partition function, magnetization, correlation functions, and susceptibility; the star-triangle (Yang–Baxter equation); functional equations and the free energy for the eight-vertex model; and the hard hexagon and chiral Potts models. All needed mathematics is developed in detail and many open questions are discussed. The goal is to guide the reader to the current forefront of research.
Hidetoshi Nishimori and Gerardo Ortiz
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and ...
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Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and critical phenomena. Although randomness may seem to obscure singular behaviour such as divergence of physical quantities at the critical temperature, it is established that well-defined phase transitions exist as long as randomness is not too strong, but the critical behaviour may get modified with respect to the pure sample. After the introduction of basic concepts and methods such as self-averaging and replica method, it is elucidated what type of phase transitions exist in the random-field Ising model and the SK model of spin glasses. Also explained are the percolation transitions using the fractal structure and the Potts model.Less
Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and critical phenomena. Although randomness may seem to obscure singular behaviour such as divergence of physical quantities at the critical temperature, it is established that well-defined phase transitions exist as long as randomness is not too strong, but the critical behaviour may get modified with respect to the pure sample. After the introduction of basic concepts and methods such as self-averaging and replica method, it is elucidated what type of phase transitions exist in the random-field Ising model and the SK model of spin glasses. Also explained are the percolation transitions using the fractal structure and the Potts model.
Jesper Lykke Jacobsen
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780198828150
- eISBN:
- 9780191866937
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198828150.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter illustrates basic concepts of quantum integrable systems on two important models of statistical physics: the Q-state Potts model and the O(n) model. Both models are transformed into loop ...
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This chapter illustrates basic concepts of quantum integrable systems on two important models of statistical physics: the Q-state Potts model and the O(n) model. Both models are transformed into loop and vertex models that provide representations of the dense and dilute Temperley–Lieb algebras. The identification of the corresponding integrable R-matrices leads to the solution of both models by the algebraic Bethe Ansatz technique. Elementary excitations are discussed in the critical case and the link to conformal field theory in the thermodynamic limit is established. The concluding sections outline the solution of a specific model of the theta point of collapsing polymers, leading to a continuum limit with a non-compact target space.Less
This chapter illustrates basic concepts of quantum integrable systems on two important models of statistical physics: the Q-state Potts model and the O(n) model. Both models are transformed into loop and vertex models that provide representations of the dense and dilute Temperley–Lieb algebras. The identification of the corresponding integrable R-matrices leads to the solution of both models by the algebraic Bethe Ansatz technique. Elementary excitations are discussed in the critical case and the link to conformal field theory in the thermodynamic limit is established. The concluding sections outline the solution of a specific model of the theta point of collapsing polymers, leading to a continuum limit with a non-compact target space.
