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arXiv:1310.3458 · Journal of Statistical Mechanics Theory and Experiment

Phase transitions in systems with non-additive long-range interactions

Takashi Mori

Published Oct 11, 2013• 43 citations
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Abstract

We consider spin systems with long-range interactions in a non-additive regime. When the non-additive scaling limit is employed, the energy and the entropy compete and the system exhibits some phase transitions. Such systems do not satisfy the additivity, which results in some unfamiliar properties related to phase transitions. In this paper, the concept of additivity and its consequences are explained, and recent progress on the statistical mechanics of long-range interacting systems is reviewed. It is shown that parameter space is clearly decomposed into three regions according to the stability of the uniform state predicted by the mean-field (MF) theory. Based on this parameter space decomposition, recent results on the exactness of the MF theory are explained. When the interaction is non-negative (ferromagnetic), the analysis of the MF theory is exact and a typical spin configuration is always uniform in the canonical ensemble. However, in the restricted canonical ensemble, i.e., the canonical ensemble with a restriction of the value of the magnetization, it is shown that the MF theory does not necessarily give the exact description of the system and phase transitions between the MF uniform states (MF phase) and the inhomogeneous states (non-MF phase) occur. A new finding is that when the interaction potential changes its sign depending on the distance, the non-MF phase appears even in the canonical ensemble.

DOI
10.1088/1742-5468/2013/10/p10003

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We consider spin systems with long-range interactions in a non-additive regime. When the non-additive scaling limit is employed, the energy and the entropy compete and the system exhibits some phase transitions. Such systems do not satisfy the additivity, which results in some unfamiliar properties related to phase transitions. In this paper, the concept of additivity and its consequences are explained, and recent progress on the statistical mechanics of long-range interacting systems is reviewed. It is shown that parameter space is clearly decomposed into three regions according to the stability of the uniform state predicted by the mean-field (MF) theory. Based on this parameter space decomposition, recent results on the exactness of the MF theory are explained. When the interaction is non-negative (ferromagnetic), the analysis of the MF theory is exact and a typical spin configuration is always uniform in the canonical ensemble. However, in the restricted canonical ensemble, i.e., the canonical ensemble with a restriction of the value of the magnetization, it is shown that the MF theory does not necessarily give the exact description of the system and phase transitions between the MF uniform states (MF phase) and the inhomogeneous states (non-MF phase) occur. A new finding is that when the interaction potential changes its sign depending on the distance, the non-MF phase appears even in the canonical ensemble.
Equilibrium properties of long-range interacting systems on lattices are investigated. There was a conjecture by Cannas et al. [Phys. Rev. B 61, 11521 (2000)] that the mean-field theory is exact for spin systems with nonadditive long-range interactions. This is called "exactness of the mean-field theory." We show that the exactness of the mean-field theory holds for systems on a lattice with nonadditive two-body long-range interactions in the canonical ensemble with unfixed order parameters. We also show that in a canonical ensemble with fixed order parameters, exactness of the mean-field theory does not hold in one parameter region, which we call the "non-mean-field region." In the non-mean-field region, an inhomogeneous configuration appears, in contrast to the uniform configuration in the region where the mean-field theory holds. This inhomogeneous configuration is not the one given by the standard phase separation. Therefore, the mean-field picture is not adequate to describe these states. We discuss phase transitions between the mean-field region and the non-mean-field region. Exactness of the mean-field theory in spin glasses is also discussed.
ENGLISH ABSTRACT:In this thesis we discuss some of the anomalies present in systems with long-range interactions, for instance negative specifc heat and negative magnetic susceptibility, and show how they can be related to the convexity properties of the thermodynamic potentials and nonequivalence of ensembles. We also discuss the possibility of engineering long-range quantum spin systems with cold atoms in optical lattices to experimentally verify the existence of nonequivalence of ensembles. We then formulate an expression for the density of states when the energy and magnetisation correspond to a pair of non-commuting operators. Finally we analytically compute the entropy s(;m) as a function of energy, and magnetisation, m, for the anisotropic Heisenberg model with Curie-Weiss type interactions. The results show that the entropy is non-concave in terms of magnetisation under certain circumstances which in turn indicates that the microcanonical and canonical ensembles are not equivalent and that the magnetic susceptibility is negative. After making an appropriate change of variables we show that a second-order phase transition can be present at negative temperatures in the microcanonical ensemble which cannot be represented in the canonical ensemble.
Abstract Methods of statistical mechanics have been enormously successful in clarifying the macroscopic properties of many-body systems. Typical examples are found in magnetic systems, which have been a test bed for a variety of techniques. This chapter introduces the Ising model of magnetic systems and explains its mean-field treatment, a very useful technique of analysis of many-body systems by statistical mechanics. Mean-field theory explained here forms the basis of the methods used repeatedly throughout this book. The arguments in the present chapter represent a general mean-field theory of phase transitions in the Ising model with uniform ferromagnetic interactions. Special features of spin glasses and related disordered systems are taken into account in subsequent chapters.
We study equilibrium states of quantum spin systems with nonadditive long-range interactions by adopting an appropriate scaling of the interaction strength, i.e., the so-called Kac prescription. In classical spin systems, it is known that the equilibrium free energy is obtained by minimizing the free-energy functional over the coarse-grained magnetization. Here we show that it is also true for quantum spin systems. From this observation, it is found that when the canonical ensemble and the microcanonical ensemble are not equivalent in some parameter region, it is not necessarily justified to replace the actual long-range interaction by the infinite-range interaction (Curie-Weiss-type interaction). On the other hand, in the parameter region where the two ensembles are equivalent, this replacement is always justified. We examine the Heisenberg XXZ model as an illustrative example and discuss the relation to experiments.
Engineering long-range interactions in experimental platforms has been achieved with great success in a large variety of quantum systems in recent years. Inspired by this progress, we propose a generalization of the classical Hamiltonian mean-field model to fermionic particles. We study the phase diagram and thermodynamic properties of the model in the canonical ensemble for ferromagnetic interactions as a function of temperature and hopping. At zero temperature, small charge fluctuations drive the many-body system through a first-order quantum phase transition from an ordered to a disordered phase. At higher temperatures, the fluctuation-induced phase transition remains first order initially and switches to second-order only at a tricritical point. Our results offer an intriguing example of tricriticality in a quantum system with long-range couplings, which bears direct experimental relevance. The analysis is performed by exact diagonalization and mean-field theory.
Mixed order phase transitions (MOT), which display discontinuous order parameter and diverging correlation length, appear in several seemingly unrelated settings ranging from equilibrium models with long-range interactions to models far from thermal equilibrium. In a recent paper [ 1 ], an exactly soluble spin model with long-range interactions that exhibits MOT was introduced and analyzed both by a grand canonical calculation and a renormalization group analysis. The model was shown to form a bridge between two classes of 1D models exhibiting MOT, namely between spin models with inverse distance square interactions and surface depinning models. In this paper, we elaborate on the calculations performed in [ 1 ]. We also analyze the model in the canonical ensemble, which yields a better insight into the mechanism of MOT. In addition, we generalize the model to include Potts and general Ising spins and also consider a broader class of interactions that decay with distance using a power law different from 2.