REVISTA MEXICANA DE F´ISICA 48 SUPLEMENTO 3, 21–23
DICIEMBRE 2002
Long-range interactions and periodic boundary conditions in ferromagnetic spin models S. Curilef Departamento de F´ısica, Universidad Cat´olica del Norte, Casilla 1280, Antofagasta, Chile e-mail:
[email protected] Recibido el 18 de enero de 2001; aceptado el 3 de agosto del 2001
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/rα 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 ≤ α ≤ D suggested by Cannas and Tamarit. Keywords: Ising lattice, spin models, ferromagnetism. Pesentamos una manera de estudiar part´ıculas en sistemas finitos e infinitos con condiciones de borde peri´odicas. Se usa una red sim´etrica y consideramos sus contribuciones sobre todo el espacio. Estudiamos el modelo de Ising con interacciones ferromagn´eticas que decaen lentamente como 1/rα . Verificamos por simulaciones de Monte Carlo con ba˜no t´ermico en sistemas con D (dimensi´on) = 1 que las cantidades temodin´amicas escalan de la forma propuesta por Tsallis, y confirmamos que la teor´ıa del campo medio es exacta en este modelo para 0 ≤ α ≤ D como fue sugerido previamente por Cannas y Tamarit. Descriptores: Redes de ising, modelos de spin, ferromagnetismo. PACS: 05.10.Lm, 0.5.90+m,75.10Hk
It is known since several decades [1] that when the potential attractive tail behaves as 1/rα , the thermodynamical extensivity imposes α > D (short-range interactions) in classical systems like the present one. However, in recent years, much attention has been paid to physical systems with microscopic long-range interactions (see [2] and references therein), in regard to their relationship with neural systems modeling, where far away localized neurons interact through an action potential that decays slowly along the axon. One related problem is the spin system with Ruderman-KittelKasuya-Yosida (RKKY) like interactions, which are present in spin glasses [3]. The most simple case, the ferromagnetic model, presents nontrivial nonextensive behaviors and therefore it represents a good starting point for studying of more complex models. On one hand, standard mean field (or van der Waals) theory for an integrable (i.e., α > D) potential uses a cutoff rc with corrections Z ∞ ∝ v(r)dD r. rc
However, no corrections are considered here. We propose to discuss the trend of potentials as a function of N (the size of the system). When the interactions are short-ranged (α > D), the size is not very important, systems with N ≥ 102 have behaviors very similar. This is not true when interactions are long-ranged (α ≤ D). On the other hand, let us go to use of a nonextensive scaling proposed recently by Tsallis [4] and partially revised by
several authors [5, 6]. That way begins by evaluating the internal energy associated to systems which include potentials with an attractive tail that decays as 1/rα . It gives Z N 1/D 1 (1) UN ∝ dD rg(r) α , r 1 where N is evaluated according to Eq.(4). If we take into account that g(r) ≈ 1 for r >> 1 where g(r) is the pair distribution function, it follows from precedent equation that UN ∝ N ∗ /D; where, N ∗ ≡ (N 1−α/D − 1)/(1 − α/D) and we can expect that quantities like internal energy, free energy, etc., per particle scale with N ∗ . In general, variables include scaling with N ∗ ; more explicitly, variables known as intensive (i.e., T, P, etc.) scale with N ∗ as it is conjectured [4]. Due to the absence of exact and analytical solutions, much effort has been devoted to handling long-range interactions in computational systems by molecular dynamics and Monte Carlo simulations. We consider particles interacting with Ising ferromagnets with two-particle potential, that means, a system described by the Hamiltonian X H=− J(rij )Si Sj (Si = ±1 ∀i), (2) (i,j)
where rij is the distance between two sites. Periodic boundary conditions are taken into account by repetitions of a central cell to infinity in (1D) one dimmension. So k=M X 1 J(r) = J , (3) |r + k|α k=−M
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ENGLISHS. CURILEF
where the sum in k represents the contributions of replications overall space and M is a positive integer for finite systems that represents the cutoff for the sum and interactions. Therefore, the size of the systems depends on M as N = (2M + 1)N ,
(4)
where N is the number of particles in every cell. So, N → ∞ is reached when M → ∞ and/or N → ∞. Furthermore, the variable given by |r + k| is the distance between two particles of different cells. Non vanishing corrections on results [7] are introduced when this way of application of periodic boundary conditions is used in systems with long-range interactions [5]. So, the main aim of this work is to compare the previous results [7] to corrected results with the appropriated application of the periodic boundary conditions. We consider the model that generalizes the Curie-Weiss one. Such model is described by the Hamiltonian X H0 = − J 0 (rij )Si Sj , (5) (i,j) 0
0
where H y J are normalized in the way as it is suggested in Ref. [7] k=M X J 1 J (r) = α ∗ , 2 N (α) |r + k|α 0
(6)
k=−M
We performed a Monte Carlo simulation on a chain of N spins with Hamiltonian using heat-bath dynamics. Simulations were done with the following size for the central cell N = 201, for several values of the parameter M ; namely, M = 1, 10, 100. We calculate root mean square of the magnetization of the system m(M, T ) as a function of the temperature T . Results were averaged over 20 samples with different random number sequences. In Fig. 1, it is depicted curves through the following relation 1 − m/mM F as a function of the scaled temperature T 0 , where T 0 = T /2α N ∗ , m is the root mean square of the magnetization obtained from the simulation compared to mM F , the result from the mean field theory, this is mM F = tanh(mM F /T 0 ). In this case, it is observed that curves fall into a single one when M ≥ 10, where 2M + 1 represents the total number of cells overall space. Non vanishing and important corrections appear on previous results [7] when the present way is applied. Thermodynamical limit is reached when M → ∞ and N → ∞. This manner for scaling thermal quantities provides a novel point of view for defining the thermodynamical limit. Summarizing, the thermodynamical limit for systems where the range of interactions is lesser than the size of the system (e.g., finite range with infinite system size) is very
F IGURE 1. Simulated results for a chain of N = 201 spins, we depicted 1 − m/mM F as a function of T 0 = T /(2α N ∗ ). The root mean square of the magnetization from simulation m is compared to the result from the mean field theory (solid line) mM F . The upper curve corresponds to the previous result as in Ref. [7], the intermediate one to M = 1 the two lower curves correspond to M = 10 and M = 100 and these curves practically coincide.
well defined and mean values of quantity per particle is not depend on the size of system. When the range of interactions is greater than the size of the system (e.g., infinite range with finite system size) quantities per particle depend strongly on the size of the system [8]. In the present point of view, we have given a picture where the scaled quantities per particle does not depend on the size of the system. We hope to have provided enough arguments to show that thermodynamics and the concept of thermodynamical equilibrium can very well accommodate throughout a nonextensive scaling. Finally, it is important to understand the microscopic behavior of the particles, because computation of the mesoscopic and macroscopic thermodynamical properties comes from the averages of quantities per particle when the number of particles is infinity. However, computer simulations (i.e., molecular dynamics, Monte Carlo procedure, etc.,) produce configurations of some finite numbers of particles within a central cell. To minimize wall effects, periodic boundary conditions repeat the central cell to infinite. Certainly, the detailed study of more such systems will provide a view for thermodynamical limit for this class of systems.
Acknowledgments This work has received partial support by FONDECYT 1010776 and 3980014, C´atedra Presidencial en Ciencias (F. Claro) and Proyecto Milenio ICM P99-135.
Rev. Mex. F´ıs. 48 S3 (2002) 21–23
ENGLISHLONG-RANGE INTERACTIONS AND PERIODIC BOUNDARY CONDITIONS IN FERROMAGNETIC SPIN MODELS
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6. S.A. Cannas, A.C.N. de Magalh˜aes, F.A. Tamarit, Phys. Rev. B 61 (2000) 11521; F.A. Tamarit and C. Anteneodo, Phys. Rev. Lett., 84 (2000) 208; S. Curilef and C. Tsallis, Phys. Lett. A 264 (1999) 270; S.A. Cannas, Physica A 358 (1998) 32; L.C. Sampaio, M.P. Alburquerque and F.S. Menezes, Phys. Rev. B 55 (1997) 5611; J.R. Grigera, Phys. Lett. A 217 (1996) 47. 7. S.A. Cannas and F. A. Tamarit, Phys. Rev. B 54 (1996) 12661. 8. B.J. Hiley, G.S. Joyce, Proc.Phys.Soc., 493 (1965).
Rev. Mex. F´ıs. 48 S3 (2002) 21–23
