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arXiv:0909.5638 · Journal of Statistical Mechanics Theory and Experiment

Microcanonical entropy of the spherical model with nearest-neighbour interactions

Michael Kästner

Published Dec 10, 2009• 6 citations
arXivHTMLPDF

Abstract

For the spherical model with nearest-neighbour interactions, the microcanonical entropy s (ε, m ) is computed analytically in the thermodynamic limit for all accessible values of the energy ε and the magnetization m per spin. The entropy function is found to be concave (albeit not strictly concave), implying that the microcanonical and the canonical ensembles are equivalent, despite the long-range nature of the spherical constraint that the spins have to obey. Two transition lines are identified in the (ε, m )-plane, separating a paramagnetic phase from a ferromagnetic and an antiferromagnetic one. The resulting microcanonical phase diagram is compared to the more familiar canonical one.

DOI
10.1088/1742-5468/2009/12/p12007

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For the spherical model with nearest-neighbour interactions, the microcanonical entropy s (ε, m ) is computed analytically in the thermodynamic limit for all accessible values of the energy ε and the magnetization m per spin. The entropy function is found to be concave (albeit not strictly concave), implying that the microcanonical and the canonical ensembles are equivalent, despite the long-range nature of the spherical constraint that the spins have to obey. Two transition lines are identified in the (ε, m )-plane, separating a paramagnetic phase from a ferromagnetic and an antiferromagnetic one. The resulting microcanonical phase diagram is compared to the more familiar canonical one.
We study the problem of ensemble equivalence in spin systems with short-range interactions under the existence of a first-order phase transition. The spherical model with nonlinear nearest-neighbour interactions is solved exactly for both canonical and microcanonical ensembles. The result reveals apparent ensemble inequivalence at the first-order transition point in the sense that the microcanonical entropy is non-concave as a function of the energy and consequently the specific heat is negative. In order to resolve the paradox, we show that an unconventional saddle point should be chosen in the microcanonical calculation that represents a phase separation. The XY model with nonlinear interactions is also studied by microcanonical Monte Carlo simulations in two dimensions to see how this model behaves in comparison with the spherical model.
Due to the equivalence of the statistical ensembles, thermostatic properties of physical systems with short-range interactions can be calculated in different ensembles leading to the same physics. In particular, the ensemble equivalence holds for systems that undergo a continuous phase transition in the infinite volume limit with the result that the properties of the transition can also be investigated in the microcanonical approach. Considering as an example the spherical model, the ensemble equivalence is explicitly demonstrated by calculating the critical properties in the microcanonical ensemble and comparing them to the well known canonical results.
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.
The microcanonical entropy s ( e , m ) as a function of the energy e and the magnetization m is computed analytically for the anisotropic quantum Heisenberg model with Curie–Weiss-type interactions. The result shows a number of interesting properties which are peculiar to long-range interacting systems, including nonequivalence of ensembles and partial equivalence. Furthermore, from the shape of the entropy it follows that the Curie–Weiss Heisenberg model is indistinguishable from the Curie–Weiss Ising model in canonical thermodynamics, although they do in general differ in microcanonical thermodynamics. The possibility of experimentally realizing quantum spin models with long-range interactions in a microcanonical setting by means of cold dipolar gases in optical lattices is discussed.
A phase diagram is a graph in parameter space showing the phase boundaries of a many-particle system. Commonly, the control parameters are chosen to be those of the (generalized) canonical ensemble, such as temperature and magnetic field. However, depending on the physical situation of interest, the (generalized) microcanonical ensemble may be more appropriate, with the corresponding control parameters being energy and magnetization. We show that the phase diagram on this parameter space looks remarkably different from the canonical one. The general features of such a microcanonical phase diagram are investigated by studying two models of ferromagnets with short-range interactions. The physical consequences of the findings are discussed, including possible applications to nuclear fragmentation, adatoms on surfaces, and cold atoms in optical lattices.
The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where ddd is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and ddd, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and ddd. In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature Tc{T}_{c}Tc​. At T=TcT={T}_{c}T=Tc​, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large rrr as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as 1r2\frac{1}{{r}^{2}}r21​ in certain directions only are also investigated.