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arXiv:1106.4920 · Physical Review E

Instability of the mean-field states and generalization of phase separation in long-range interacting systems

Takashi Mori

Published Sep 27, 2011• 35 citations
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Abstract

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.

DOI
10.1103/physreve.84.031128

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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.
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.
Relationships between general long-range interacting classical systems on a lattice and the corresponding mean-field models (infinitely long-range interacting models) are investigated. We study systems in arbitrary dimension d for periodic boundary conditions and focus on the free energy for fixed value of the total magnetization. As a result, it is shown that the equilibrium free energy of the long-range interacting systems are exactly the same as that of the corresponding mean-field models (exactness of the mean-field theory). Moreover, the mean-field metastable states can be also preserved in general long-range interacting systems. It is found that in the case that the magnetization is conserved, the mean-field theory does not give correct property in some parameter region.
We study the random-field Ising model with long-range interactions and show the exactness of the mean-field theory under certain mild conditions. This is a generalization of the result of Mori for the non-random and spin-glass cases. To treat random fields, we evoke the self-averaging property of a function of random fields, without recourse to the replica method. The result is that the mean-field theory gives the exact expression of the canonical free energy for systems with power-decaying interactions if the power is smaller than or equal to the spatial dimension.
Renormalization in pure systems In Section 2.2 we reviewed the basic facts about broken symmetry and phase transitions in pure systems, within mean field theory. This helped set the stage for the mean field description of spin glasses that we studied in Chapters 3–6. We will now want to study the phase transition and the nature of the low-temperature phase for spin glasses with fairly short-ranged interactions in low dimensionality (2 or 3). This situation is far from the region where mean field theory is a good guide, so we will have to learn new methods. Again, we start by learning about them in pure systems. Renormalization is a very general approach to problems with many strongly interacting degrees of freedom. In a statistical mechanical problem, the basic idea is to carry out the trace in the partition function over some of the variables, leaving a new problem with fewer degrees of freedom. One follows how the parameters in the Hamiltonian of the system change as this procedure is iterated many times. Generally, the partial trace corresponds to removing the short-distance degrees of freedom, so each successive effective Hamiltonian has a larger lattice constant (or equivalent microscopic length) than its predecessor: each renormalization step corresponds to a change of scale. Different thermodynamic phases are identified with flows to different fixed points in the parameter space of the Hamiltonian under this semigroup.
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.
The mean field theory discussed in the preceding chapters revealed a surprisingly rich structure and in particular a very complex ordered phase with many ‘pure’ states. However, we know from periodic systems that the mean field theory does not always give the right answer. In this and the next two chapters, we will discuss alternative approaches to the study of spin glasses with short-range interactions. The most important one will be the renormalization group which led to a deep understanding of phase transitions (and in particular critical behaviour) of nonrandom systems and which has also turned out to be extremely useful for the study of spin glasses. Together with Monte Carlo simulations and experimental data, this will give us a fairly complete picture, at least of static spin glass properties below T f . Critical behaviour near T f and scaling arguments for T < T f have also been considered for the spin glass dynamics but here the situation seems to be less clear. Depending on the system, the remanent magnetization decays with quite different decay laws (see Chapter 1), which indicates that not all dynamic processes are universal. However, all spin glasses have a huge range of characteristic relaxation times, ranging from 10 −13 s (the Korringa relaxation in a metallic spin glass) to 10 −6 – 10 −8 s. This long-time limit is not inherent to the system but is simply determined by the patience of an experimentalist (or the average time of a student's thesis).