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Physical review. E

Phase transitions and multicritical behavior in the Ising model with dipolar interactions

M. A. Bab, C. M. Horowitz, M. Leticia Rubio Puzzo, Gustavo P. Saracco

Published Oct 4, 2016• 10 citations
Publisher

Abstract

In this work, the phase diagram of the ferromagnetic Ising model with dipole interactions is revisited with the aim of determining the nature of the phase transition between stripe-ordered phases with width $n$ $({h}_{n})$ and tetragonal liquid (TL) phases. Extensive Monte Carlo simulations are performed in order to study the short-time dynamic behavior of the observables for selected values of the ratio between the ferromagnetic exchange and dipolar constants $\ensuremath{\delta}$. The obtained results indicate that the ${h}_{1}$-TL phase transition line is continuous up to $\ensuremath{\delta}=1.2585$, while for the ${h}_{2}$-TL line a weak first-order character is found in the interval $1.2585\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1.36$ and becomes continuous for $1.37\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1.9$. This result suggests the existence of a tricritical point close to $\ensuremath{\delta}=1.37$. When it is appropriate, the complete set of critical exponents is obtained, and in all the studied cases they depend on $\ensuremath{\delta}$ but do not belong to the Ising universality class. Furthermore, short-time dynamic studies reveal that at the point where the mentioned lines meet the ${h}_{1}\ensuremath{-}{h}_{2}$ line, i.e., at $\ensuremath{\delta}=1.2585$, the critical phase corresponding to the ${h}_{1}\ensuremath{-}\mathrm{TL}$ transition coexists with the ${h}_{2}$ phase.

DOI
10.1103/physreve.94.042104

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In this work, the phase diagram of the ferromagnetic Ising model with dipole interactions is revisited with the aim of determining the nature of the phase transition between stripe-ordered phases with width nnn (hn)({h}_{n})(hn​) and tetragonal liquid (TL) phases. Extensive Monte Carlo simulations are performed in order to study the short-time dynamic behavior of the observables for selected values of the ratio between the ferromagnetic exchange and dipolar constants \ensuremathδ\ensuremath{\delta}\ensuremathδ. The obtained results indicate that the h1{h}_{1}h1​-TL phase transition line is continuous up to \ensuremathδ=1.2585\ensuremath{\delta}=1.2585\ensuremathδ=1.2585, while for the h2{h}_{2}h2​-TL line a weak first-order character is found in the interval 1.2585\ensuremath≤\ensuremathδ\ensuremath≤1.361.2585\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1.361.2585\ensuremath≤\ensuremathδ\ensuremath≤1.36 and becomes continuous for 1.37\ensuremath≤\ensuremathδ\ensuremath≤1.91.37\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1.91.37\ensuremath≤\ensuremathδ\ensuremath≤1.9. This result suggests the existence of a tricritical point close to \ensuremathδ=1.37\ensuremath{\delta}=1.37\ensuremathδ=1.37. When it is appropriate, the complete set of critical exponents is obtained, and in all the studied cases they depend on \ensuremathδ\ensuremath{\delta}\ensuremathδ but do not belong to the Ising universality class. Furthermore, short-time dynamic studies reveal that at the point where the mentioned lines meet the h1\ensuremath−h2{h}_{1}\ensuremath{-}{h}_{2}h1​\ensuremath−h2​ line, i.e., at \ensuremathδ=1.2585\ensuremath{\delta}=1.2585\ensuremathδ=1.2585, the critical phase corresponding to the h1\ensuremath−TL{h}_{1}\ensuremath{-}\mathrm{TL}h1​\ensuremath−TL transition coexists with the h2{h}_{2}h2​ phase.
The ferromagnetic Ising model with antiferromagnetic dipole interactions is investigated by means of Monte Carlo simulations, focusing on the characterization of the phase transitions between the tetragonal liquid and stripe of width h phases. The dynamic evolution of the physical observables is analyzed within the short-time regime for 0.5≤δ≤1.3, where δ is the ratio between the short-range exchange and the long-range dipole interaction constants. The obtained results for the interval 0.5≤δ≤1.2 indicate that the phase transition line between the h=1 stripe and tetragonal liquid phases is continuous. This finding contributes to clarifying the controversy about the order of this transition. This controversy arises from the difficulties introduced in the simulations due to the presence of long-range dipole interactions, such as an important increase in the simulation times that limits the system size used, strong finite size effects, as well as to the existence of multiple metastable states at low temperatures. The study of the short-time dynamics of the model allows us to avoid these hindrances. Moreover, due to the fact that the finite-size effects do not significantly affect the power-law behavior exhibited in the observables within the short-time regime, the results could be attributed to those corresponding to the thermodynamic limit. As a consequence of this, a careful characterization of the critical behavior for the whole transition line is performed by giving the complete set of critical exponents.
We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction δ where the phase characterized by striped configurations of width h = 1 is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between δ = 0.85 and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents ν that are clearly consistent with a single second-order phase transition line, thus excluding such a tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.
We studied a layered metamagnetic Ising model with competing ferromagnetic and antiferromagnetic interactions on a square lattice. The model is formed of ferromagnetic chains coupled by an antiferromagnetic interaction. Using Monte Carlo simulations we have determined the phase diagram of the model, which exhibits a tricritical point. By exploring the short-time scaling dynamics, we have found the dynamic and static critical exponents along the continuous transition line between the antiferromagnetic and paramagnetic phases.
Due to intrinsic frustrations of interaction, the 2d Ising model with competing ferromagnetic short-range nearest-neighbour and antiferromagnetic long-range dipole interactions possesses a rich phase diagram. The order of the phase transition from the striped h = 1 phase to the tetragonal phase that is observed in this model has been a subject of recent controversy. We address this question by using the partition function density analysis in the complex temperature plane. Our results support the second-order phase transition scenario. To measure the strength of the phase transition, we calculate the values of specific heat critical exponent a. Along with the space dimension, it appears to depend on the ratio of strengths of the short-range and long-range interactions.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0{J}_{1}<0J1​<0 (nearest neighbor, ferromagnetic) and J2>0{J}_{2}>0J2​>0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4{Z}_{4}Z4​ symmetry and is stabilized at low temperatures for g=J2/∣J1∣>1/2g={J}_{2}/|{J}_{1}|>1/2g=J2​/∣J1​∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1{J}_{1}J1​-J2{J}_{2}J2​ model has a line of very weak first-order phase transitions in the whole region 1/2<g<g∗1/2<g<{g}^{*}1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from g=g∗g={g}^{*}g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point g∗{g}^{*}g∗ in the J1{J}_{1}J1​-J2{J}_{2}J2​ model. The continuous transitions near g∗{g}^{*}g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of g>1/2g>1/2g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
The Ising fluid we have investigated is a system of hard spheres which carry an Ising spin. A short range ferromagnetic potential acts between particles. A renormalization group study of the model, based on the hierarchical reference theory of fluids, gives quantitative predictions on the phase diagram and the phase boundaries. The model can show two distinct phase transitions: a magnetic order-disorder transition and a liquid-vapor phase separation which meet at a tricritical point. The critical properties of the magnetic transition have been studied by Monte Carlo simulations at two densities in systems with up to 4000 particles. The locations of the critical points are in good agreement with the theoretical estimates. However, a finite-size scaling analysis shows effective critical exponents which deviate from the expected Ising values. These discrepancies resemble the analogous result of a previous numerical study of Heisenberg fluids. Here they are interpreted as due to crossover phenomena, partly related to Fisher exponent renormalization induced by the presence of noncritical density fluctuations.