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International Journal of Modern Physics B

A STEP-BY-STEP APPROACH TO INTEGRABILITY

David Hansel, J.-M. Maillard, Pál Ruján

Published Oct 1, 1989• 5 citations
Publisher

Abstract

We try to elucidate the role played by different symmetries in simple models of statistical mechanics. Starting with obvious symmetries for the partition function and combining them with duality relations we obtain a set of constraints on the possible algebraic varieties relevant for the integrable manifolds and the phase diagram of the lattice model. When imposing in addition the inversion relation special polynomials are obtained, which are close and sometimes identical to the set of equations defining the parameter subspace of the integrable models. Our procedure is detailed on the q-state chiral Potts models on a square lattice, in particular for q=3 and 4.

DOI
10.1142/s0217979289000993

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Abstract The integrable quantum field theories analyzed in the previous chapters provide the exact solution of many statistical models away from the critical point. Despite the elegance and the undeniable success of these methods, the generic situation that occurs in statistical physics is that of non-integrable dynamics: many interesting statistical models fall within this class and therefore it would be highly desirable to develop an appropriate formalism to deal with the lack of integrability. This task is notoriously difficult for the rich phenomenology that arises when the dynamics are not integrable: there are decays and production scattering processes, confinement phenomena and nucleation of false vacua, resonance peaks in the cross-sections, etc.

•Aug 27, 2009•0 citations
We try to elucidate the role played by different symmetries in simple models of statistical mechanics. Starting with obvious symmetries for the partition function and combining them with duality relations we obtain a set of constraints on the possible algebraic varieties relevant for the integrable manifolds and the phase diagram of the lattice model. When imposing in addition the inversion relation special polynomials are obtained, which are close and sometimes identical to the set of equations defining the parameter subspace of the integrable models. Our procedure is detailed on the q-state chiral Potts models on a square lattice, in particular for q=3 and 4.
Exactly solvable models have played a key role in our understanding of low-dimensional statistical systems and condensed matter physics. Underlying these models is the famous Yang-Baxter equation, and the authors present here a large class of new integrable quantum many-body models following a rational solution to this equation. Moreover, a duality symmetry connecting different integrable models is uncovered. This is illustrated using an interacting topological system, where the topological phase transition is related to the modular subgroup of the duality symmetry.
The Ising and QQQ-state Potts models are statistical mechanical models of spins interaction on crystal lattices. \nWe study the partition functions on a range of lattices, particularly two- and three-dimensional cases. The study aims to investigate cooperative phenomena −-− how higher level structure is affected by the detailed activity of a very large number of lower level structures. \nWe investigate the analytic properties of the partition functions and their relationship to physical observables in equilibrium near phase transition. \nOur study is focussed on describing the partition function and the distribution of zeros of the partition function in the complex-temperature plane close to phase transitions. \nHere we first consider the solved case of the Ising model on square lattice as a benchmark for checking our method of computation and analysis. \nThe partition function is computed using a transfer matrix approach and the zeros are found numerically by Newton-Raphson method. \nWe extend the study of QQQ-state Potts models to a more general case called the ZQZ_QZQ​-symmetric model. \nWe evidence the existence of multiple phase transitions for this model in case Q=5,6,Q=5,6,Q=5,6, and discuss the possible connection to different stages of disordered state. \nGiven sufficient and efficient coding and computing resources, we extend many previously studied cases to larger lattice sizes. \nOur analysis of zeros distribution close to phase transition point is based on a certain power law relation which leads to critical exponent of physical observable. \nWe evidence for example, that our method can be used to give numerical estimates of the specific heat critical exponent alpha\\alphaalpha.
This thesis deals with two two-dimensional lattice models in statistical mechanics. The first of these is the zero-field 20-vertex model on a triangular lattice. This model is the analogue to the ice-type, zero-field six-vertex model on a square lattice. We solve this problem exactly for a restricted set of vertex weights. The other model is the q-component Potts model on a square lattice. This model is equivalent to a staggered ice-type model on a square lattice. Using a variational method we estimate the critical exponent ß for several values of q. In part I we outline the fundamental statistical mechanical ideas needed, including the definition of the partition function and its relation to the thermodynamic quantities. Mention is also made of the existence and uniqueness of the thermodynamic limit for these models. The 20-vertex model is discussed in part II. The transfer matrix approach is adopted and a Bethe ansatz is used for the eigenvector. Unlike the square lattice ice-type model, the triangular lattice 20-vertex model is only soluble by this method for a restricted set of vertex weights. In particular, the ten parameters determining the vertex weights must be expressed in terms of four independent parameters. This results in only one type of non-trivial temperature independent solution. This solution exhibits critical behaviour similar to the square lattice KDP model. The general solution has three states: disordered, antiferroelectric and ferroelectric. Which state the model is in is determined by one of the above four parameters. The transition from the disordered to the antiferroelectric state is of infinite-order (i.e the free energy has an essential singularity), and from the disordered to the ferroelectric state is of finite-order. We also compare our results to the free-fermion solution of the same model and note where the two methods overlap. In part III the Potts model is considered. The model is defined and shown to be equivalent to a staggered ice-type model. A variational principle for the eigenvalues of the transfer matrix of the staggered ice model is determined. Using the zero-temperature solution as a ^uide a simple form for a sequence of approximations to the eigenvector, below T , is developed. By using the variational principle we arrive at a set of matrix equations determining the eigenvector at a given level of approximation. By noting some of the symmetries of the transfer matrix we are able to separate some of the dependences of the matrix elements and consequently simplify the original set of equations. We are also able to express the percolation probability ( the analogue of the spontaneous magnetization for general values of q ) of the Potts model as a simple function of these matrix elements. At each level of approximation there is a set of algebraic equations to solve. We have used an electronic computer to solve these equations for the first five levels of approximation, and for several values of q between zero and four. With our results we estimate the critical exponent 3 for the Potts model. These estimates agree reasonably well with previous work and indicate 3=0.145 ±.017 for q=l.
Abstract In this paper we investigate certain fusion relations associated to an integrable vertex model on the square lattice which is invariant under Sp (4) symmetry. We establish a set of functional relations which include a transfer matrix inversion identity. The solution of these relations in the thermodynamic limit allows us to compute the partition function per site of the fundamental Sp (4) representation of the vertex model. As a byproduct we also obtain the partition function per site of a vertex model mixing the four and five dimensional representations of the Sp (4) symmetry.
Abstract The integrable quantum field theories analyzed in the previous chapters provide the exact solution of many statistical models away from the critical point. Despite the elegance and the undeniable success of these methods, the generic situation that occurs in statistical physics is that of non-integrable dynamics: many interesting statistical models fall within this class and therefore it would be highly desirable to develop an appropriate formalism to deal with the lack of integrability. This task is notoriously difficult for the rich phenomenology that arises when the dynamics are not integrable: there are decays and production scattering processes, confinement phenomena and nucleation of false vacua, resonance peaks in the cross-sections, etc.