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arXiv:1410.1279 · Journal of Physics A Mathematical and Theoretical

Existence of shape-dependent thermodynamic limit in spin systems with short- and long-range interactions

Takashi Mori

Published Mar 16, 2015• 1 citation
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Abstract

The existence of the thermodynamic limit in spin systems with short- and long-range interactions is established. We consider the infinite-volume limit with a fixed shape of the system. The variational expressions of the entropy density and the free energy density are obtained, which explicitly depend on the shape of the system. This shape dependence of thermodynamic functions implies the nonadditivity, which is one of the most important characteristics of long-range interacting systems.

DOI
10.1088/1751-8113/48/14/145001

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The existence of the thermodynamic limit in spin systems with short- and long-range interactions is established. We consider the infinite-volume limit with a fixed shape of the system. The variational expressions of the entropy density and the free energy density are obtained, which explicitly depend on the shape of the system. This shape dependence of thermodynamic functions implies the nonadditivity, which is one of the most important characteristics of long-range interacting systems.
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.
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.
Quantum thermalization of closed systems -- i.e. the relaxation of local properties to their thermodynamic value along the Hamiltonian evolution -- is generically expected in non-integrable systems at finite energy density. In this work we show that quantum spin systems with at least one spin component coupled through long-range ferromagnetic interactions, and a second spin component coupled to a field, cannot in general thermalize to a paramagnetic state whenever they have a finite, negative energy per spin. The long-range interactions we consider decay as a power law of the distance, ∼r−α\sim r^{-α}∼r−α with α<dα< dα<d, where ddd is the number of spatial dimensions of the system. The lack of thermalization is generic to an infinite family of Hamiltonians, and it is not associated to any form of integrability. It can instead be understood by the anomalous features of the thermodynamics of the system in the paramagnetic phase, which cannot be reflected in the properties of the eigenstates. This leads to a breakdown of eigenstate typicality and of the eigenstate thermalization hypothesis. Interestingly, the non-thermal states emerging at long times in the dynamics display long-range multipartite entanglement, elevating the breakdown of thermalization to a resource.
This paper is a contribution to the general theory of quantum spin systems. We deal with a general theory in the sense that no concrete interaction shows up but an arbitrary relatively short-range interaction is chosen. It is well known that the mean entropy plays an important role in the thermodynamics of quantum spin systems: it is one of the ingredients of the Lanford–Ruelle–Robinson variational principle. We show that in the background of the existence of the mean entropy, there is an operator convergence which resembles the McMillan theorem from information theory. Asymptotic equipartition property and several entropy densities are investigated, in particular, the relation of the Gibbs condition to relative entropies. Although this paper is not intended to be a review, we try to give an overview of the theory of quantum Gibbs states in the mathematical sense. The paper is organized as follows. Section 1 provides an introduction and further sections contain our new results. Each section is closed with a short discussion on sources and references.
The infinite-volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamic entropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.
Abstract We propose a topological approach suitable to establish a connection between thermodynamics and topology in the microcanonical ensemble. Indeed, we report on results that point to the possibility of describing interacting classical spin systems in the thermodynamic limit, including the occurrence of a phase transition, using topological arguments only. Our approach relies on Morse theory, through the determination of the critical points of the potential energy, which is the proper Morse function. Our main finding is that, in the context of the classical models studied, the Euler characteristic χ ( E ) embeds the necessary features for a correct description of several magnetic thermodynamic quantities of the systems, such as the magnetization, correlation function, susceptibility, and critical temperature. Despite the classical nature of the models, such quantities are those that do not violate the laws of thermodynamics (with the proviso that van der Waals loop states are mean field (MF) artifacts). We also discuss the subtle connection between our approach using the Euler entropy, defined by the logarithm of the modulus of χ ( E ) per site, and that using the Boltzmann microcanonical entropy. Moreover, the results suggest that the loss of regularity in the Morse function is associated with the occurrence of unstable and metastable thermodynamic solutions in the MF case. The reliability of our approach is tested in two exactly soluble systems: the infinite-range and the one-dimensional short-range XY models in the presence of a magnetic field. In particular, we confirm that the topological hypothesis holds for both the infinite-range ( T c ≠ 0 ) and the short-range ( T c = 0) XY models. Further studies are very desirable in order to clarify the extension of the validity of our proposal.