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Introduction to statistical physics I Silvio R.A. Salinas. p. cm. -(Graduate texts in contemporary physics) IncJudes bibliographical references and index.

We study a model for anisotropic ferromagnetic quantum domain walls. The large degeneracy of the ground state in the extreme anisotropic (Ising) limit, associated with the translational invariance of the "kink center, " is lifted in the quantum system in a peculiar way. The critical point, at which the Hamiltonian is invariant under the quantum group U~[SU(2)j, is exactly determined by a cluster method. We also find the ground state wave function at the critical point. Some generalizations of these results for arbitrary spin and dimension are obtained.A simple model of a spin-S quantum Heisenberg ferromagnet with a domain wall is given by the spin Hamiltonian = -J P(S"'S"+s + S~S"+s) r, 6 ags"'S"'"h g S"'g S"', (1) r, cst t r EF rEFs.where 1 ) 0 and 6~J are exchange parameters, and r is a lattice vector, with a neighbor r + 6, on a d-dimensional cubic lattice of side L. The effective field h~0 represents the interactions of the spins with the boundary surfaces F+ (F ) with positive (negative) normal vectors. In one dimension, a fully isotropic (J = 6) spin-2 quantum

We formulate the Ising model with competing interactions on a Cayley tree, in the infinitecoordination limit, as a two-dimensional nonlinear mapping. The phase diagram displays a Lifshitz point and many modulated phases. We perform calculations to show the existence of a complete devil's staircase at low temperatures. Also, we give strong numerical evidence for the existence of chaotic phases associated with strange attractors.

We obtain the phase diagram of a ferromagnetic mixed Ising system, consisting of spin--, and spin-5 variables, on a Bethe lattice of coordination number z, with nearest-neighbor exchange interactions and single-ion terms. The problem is formulated as a discrete nonlinear map. There is a tricritical point for S integer and z~5. In the infinite-coordination-number limit, we regain the results of an exact calculation for a Curie-Weiss version of the model.

Fourth-order cumulants of physical quantities have been used to characterize the nature of a phase transition. In this paper we report some Monte Carlo simulations to illustrate the behavior of fourth-order cumulants of magnetization and energy across second and rstorder transitions in the phase diagram of a well known spin-1 Ising model. Simple ideas from the theory of thermodynamic uctuations are used to account for the behavior of these cumulants. I IntroductionThere are many attempts to characterize the order of a phase transition on the basis of the analysis of numerical data obtained from simulations of nite spin systems. One of the approaches to this problem consists in the analysis of the behavior of fourth-order cumulants of physical quantities as the order parameter and the energy associated with the systems under consideration 1, 2 . Properties of the fourth-order cumulants of magnetization and energy have been investigated in the context of nite-size e ects in magnetically and thermally driven rst-order transitions in Ising and Potts models 3, 4 , 5 , 6 as well as in the case of some other systems 7, 8 .In this paper, we perform Monte Carlo simulations for the well known Blume-Capel model 9, 10 to illustrate the behavior of the fourth-order cumulants of magnetization and energy across rst and second-order transitions in the phase diagram of this system. We show that it is possible to draw some conclusions from the study of relatively small lattices. The general features of the cumulants can be accounted for by simple arguments from the theory of thermodynamic uctuations. In particular, we emphasize the di erences between the two t ypes of cumulants, and the alternative de nitions of the cumulant of energy which has not been fully appreciated in previous investigations.The layout of this paper is as follows. In Section 2 we de ne the cumulants of a physical quantity. In Section 3 we i n troduce some further de nitions, and discuss some properties of the Blume-Capel model. Simulations for the fourth-order cumulants of magnetization and energy are reported in Sections 4 and 5, respectively. W e h o p e t o h a ve provided another example of the use of these cumulants to characterize the order of a phase transition. II De nition of the cumulantsThe cumulants of a quantity x can be obtained from an expansion of the form

We consider a simple Maier-Saupe statistical model with the inclusion of disorder degrees of freedom to mimic the phase diagram of a mixture of rod-like and disc-like molecules. A quenched distribution of shapes leads to the existence of a stable biaxial nematic phase, in qualitative agreement with experimental findings for some ternary lyotropic liquid mixtures. An annealed distribution, however, which is more adequate to liquid mixtures, precludes the stability of this biaxial phase. We then use a two-temperature formalism, and assume a separation of relaxation times, to show that a partial degree of annealing is already sufficient to stabilize a biaxial nematic structure.Quenched and annealed degrees of freedom of statistical systems are known to produce phase diagrams with a number of distinct features [1]. The ferromagnetic site-diluted Ising model provides an example of a continuous transition, in the quenched case, which turns into a first-order boundary beyond a certain tricritical point, if we consider thermalized site dilution [2]. Disordered degrees of freedom in solid compounds, as random magnets and spin-glasses, are examples of quenched disorder, which lead to well-known problems related to averages of sets of disordered free energies. In liquid systems, however, relaxation times are shorter, and the simpler problems of annealed disorder are more relevant from the physical perspective. In this paper, we show that distinctions between quenched and annealed degrees 1

Using a weak-disorder scheme and real-space renormalization-group techniques, we obtain analytical results for the critical behavior of various q-state Potts models with correlated disordered exchange interactions along d1 of d spatial dimensions on hierarchical (Migdal-Kadanoff) lattices. Our results indicate qualitative differences between the cases d − d1 = 1 (for which we find nonphysical random fixed points, suggesting the existence of nonperturbative fixed distributions) and d − d1 > 1 (for which we do find acceptable perturbartive random fixed points), in agreement with previous numerical calculations by Andelman and Aharony. We also rederive a criterion for relevance of correlated disorder, which generalizes the usual Harris criterion.

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