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arXiv:cond-mat/9906340 · Physical review. B, Condensed matter

Evidence of exactness of the mean-field theory in the nonextensive regime of long-range classical spin models

Sergio Alejandro Cannas, A C N de Magalhães, Francisco A. Tamarit

Published May 1, 2000• 46 citations
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Abstract

The q-state Potts model with long-range interactions that decay as ${1/r}^{\ensuremath{\alpha}}$ subjected to a uniform magnetic field on d-dimensional lattices is analyzed for different values of q in the nonextensive regime $0<~\ensuremath{\alpha}<~d.$ We calculate the mean-field solution of the model for all q and performed, for some values of q, Monte Carlo simulations for the spontaneous magnetization in the one-dimensional case. We show that, using a derived scaling which properly describes the nonextensive thermodynamic behavior, both types of calculations present an excellent agreement for $0<~\ensuremath{\alpha}<d.$ We also consider the two-dimensional antiferromagnetic Ising model with competing antiferromagnetic long-range interactions and ferromagnetic first neighbor ones in the presence of a uniform magnetic field. We calculate the mean-field magnetization for this case and compare it with Monte Carlo numerical data from Sampaio et al. They also show a very good agreement for $\ensuremath{\alpha}<d.$ These results, together with some previous ones, led us to conjecture that the mean-field theory is exact for nonextensive classical spin models with $0<~\ensuremath{\alpha}<d.$

DOI
10.1103/physrevb.61.11521

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The q-state Potts model with long-range interactions that decay as 1/r\ensuremathα{1/r}^{\ensuremath{\alpha}}1/r\ensuremathα subjected to a uniform magnetic field on d-dimensional lattices is analyzed for different values of q in the nonextensive regime 0< \ensuremathα< d.0<~\ensuremath{\alpha}<~d.0< \ensuremathα< d. We calculate the mean-field solution of the model for all q and performed, for some values of q, Monte Carlo simulations for the spontaneous magnetization in the one-dimensional case. We show that, using a derived scaling which properly describes the nonextensive thermodynamic behavior, both types of calculations present an excellent agreement for 0< \ensuremathα<d.0<~\ensuremath{\alpha}<d.0< \ensuremathα<d. We also consider the two-dimensional antiferromagnetic Ising model with competing antiferromagnetic long-range interactions and ferromagnetic first neighbor ones in the presence of a uniform magnetic field. We calculate the mean-field magnetization for this case and compare it with Monte Carlo numerical data from Sampaio et al. They also show a very good agreement for \ensuremathα<d.\ensuremath{\alpha}<d.\ensuremathα<d. These results, together with some previous ones, led us to conjecture that the mean-field theory is exact for nonextensive classical spin models with 0< \ensuremathα<d.0<~\ensuremath{\alpha}<d.0< \ensuremathα<d.
The Ising model with ferromagnetic interactions that decay as $\frac{1}{{r}^{\ensuremath{\alpha}}}$ is analyzed in the nonextensive regime $0<~\ensuremath{\alpha}<~d$, where the thermodynamic limit is not defined. In order to study the asymptotic properties of the model in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit (NNN being the number of spins) we propose a generalization of the Curie-Weiss model, for which the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit is well defined for all $\ensuremath{\alpha}>~0$. We conjecture that mean-field theory is exact in the last model for all $0<~\ensuremath{\alpha}<~d$. This conjecture is supported by Monte Carlo heat-bath simulations in the d=1d=1d=1 case. Moreover, we confirm a recently conjectured scaling by Tsallis that allows for a unification of extensive ($\ensuremath{\alpha}>d$) and nonextensive ($0<~\ensuremath{\alpha}<~d$) regimes.
Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\ensuremath{\alpha}$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\ensuremath{\alpha}$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,\ensuremath{-}\frac{M}{2},\ensuremath{-}\frac{M}{2})$ with unsaturated M<1M<1M<1 in a process known as ``order by disorder'' similar to the nearest-neighbor antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest-neighbor and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.
We study the nature of the phase transition of the qqq-state Potts model with long-range ferromagnetic interactions decaying as $1/{r}^{d+\ensuremath{\sigma}}$, in dimension d=1d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1d=1, using a histogram Monte Carlo (MC) technique. The model can exhibit a first-order transition or a second-order phase transition with nonstandard critical exponents. The critical value of qqq above which a first-order transition occurs decreases with decreasing $\ensuremath{\sigma}$, from qc=8{q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8qc​=8 for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ to qc=2{q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2qc​=2 for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.3$. Detailed results for various $\ensuremath{\sigma}$ will be shown and discussed. Mean-field calculation confirms the tendency of our MC results.
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We present a possible way to study long-range interacting particles in finite-infinite systems with periodic boundary conditions. A symmetrical lattice and its contributions overall space are being used. In this context, we study the Ising model with ferromagnetic interaction that decays as a 1/r a law. We verify by Monte Carlo heat-bath simulations in the D (dimension) = 1 case that the thermodynamics quantities scale in a way proposed by Tsallis, and we confirm that mean field theory is exact in the last model for all 0 < a < D suggested by Cannas and Tamarit.
There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance rrr as $1∕{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. Here, we review old nonextensive scaling. In addition, we propose a Hamiltonian scaled by $2\frac{{(N∕2)}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1}{1\ensuremath{-}\ensuremath{\alpha}}$ that explicitly includes symmetry of the lattice and dependence on the size NNN of the system. The approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.