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Register a new userPossibility of a new universality class: The anisotropic Heisenberg model with dipolar interactions
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Added by
Lucas Alvares da Silva M\\'ol
Publication date
2008
fields
Physics
and research's language is
English
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In this work we have used extensive Monte Carlo calculations to study the planar to paramagnetic phase transition in the two-dimensional anisotropic Heisenberg model with dipolar interactions (AHd) considering the true long-range character of the dipolar interactions by means of the Ewald summation. Our results are consistent with an order-disorder phase transition with unusual critical exponents in agreement with our previous results for the Planar Rotator model with dipolar interactions. Nevertheless, our results disagrees with the Renormalization Group results of Maier and Schwabl [PRB, 70, 134430 (2004)] and the results of Rapini et. al. [PRB, 75, 014425 (2007)], where the AHd was studied using a cut-off in the evaluation of the dipolar interactions. We argue that besides the long-range character of dipolar interactions their anisotropic character may have a deeper effect in the system than previously believed. Besides, our results shows that the use of a cut-off radius in the evaluation of dipolar interactions must be avoided when analyzing the critical behavior of magnetic systems, since it may lead to erroneous results.
The two-dimensional ($2d$) fully frustrated Planar Rotator model on a square lattice has been the subject of a long controversy due to the simultaneous $Z_2$ and $O(2)$ symmetry existing in the model. The $O(2)$ symmetry being responsible for the Berezinskii - Kosterlitz - Thouless transition ($BKT$) while the $Z_2$ drives an Ising-like transition. There are arguments supporting two possible scenarios, one advocating that the loss of $Ising$ and $BKT$ order take place at the same temperature $T_{t}$ and the other that the $Z_2$ transition occurs at a higher temperature than the $BKT$ one. In the first case an immediate consequence is that this model is in a new universality class. Most of the studies take hand of some order parameter like the stiffness, Binders cumulant or magnetization to obtain the transition temperature. Considering that the transition temperatures are obtained, in general, as an average over the estimates taken about several of those quantities, it is difficult to decide if they are describing the same or slightly separate transitions. In this paper we describe an iterative method based on the knowledge of the complex zeros of the energy probability distribution to study the critical behavior of the system. The method is general with advantages over most conventional techniques since it does not need to identify any order parameter emph{a priori}. The critical temperature and exponents can be obtained with good precision. We apply the method to study the Fully Frustrated Planar Rotator ($PR$) and the Anisotropic Heisenberg ($XY$) models in two dimensions. We show that both models are in a new universality class with $T_{PR}=0.45286(32)$ and $T_{XY}=0.36916(16)$ and the transition exponent $ u=0.824(30)$ ($frac{1}{ u}=1.22(4)$).
A two state sandpile model with preferential sand distribution is developed and studied numerically on scale free networks with power-law degree ($k$) distribution, {em i.e.}: $P_ksim k^{-alpha}$. In this model, upon toppling of a critical node sand grains are given one to each of the neighbouring nodes with highest and lowest degrees instead of two randomly selected neighbouring nodes as in a stochastic sandpile model. The critical behaviour of the model is determined by characterizing various avalanche properties at the steady state varying the network structure from scale free to random, tuning $alpha$ from $2$ to $5$. The model exhibits mean field scaling on the random networks, $alpha>4$. However, in the scale free regime, $2<alpha<4$, the scaling behaviour of the model not only deviates from the mean-field scaling but also the exponents describing the scaling behaviour are found to decrease continuously as $alpha$ decreases. In this regime, the critical exponents of the present model are found to be different from those of the two state stochastic sandpile model on similar networks. The preferential sand distribution thus has non-trivial effects on the sandpile dynamics which leads the model to a new universality class.
Ultrathin magnetic films can be modeled as an anisotropic Heisenberg model with long-range dipolar interactions. It is believed that the phase diagram presents three phases: An ordered ferromagnetic phase I, a phase characterized by a change from out-of-plane to in-plane in the magnetization II, and a high-temperature paramagnetic phase III. It is claimed that the border lines from phase I to III and II to III are of second order and from I to II is first order. In the present work we have performed a very careful Monte Carlo simulation of the model. Our results strongly support that the line separating phases II and III is of the BKT type.
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 their microcanonical thermodynamics in general differs. 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.
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