Using the Monte Carlo Simulation (MCS) method combined with the Metropolis algorithm, we explore the magnetic properties of a two-dimensional (2D) spin system with spins 1, 2 and 3. The adapted simulation to the Ising model aims to study the impact of the correction term a of the interaction with second neighbors and the crystal field D on the critical temperature [Formula: see text] of the 2D anisotropic square lattice for different integer values of spins. The obtained results demonstrate that the critical temperature [Formula: see text] is significantly affected by the competing interactions between the anisotropy parameter, second-neighbor interaction and the crystal field. The considered model exhibits a critical phase transition regardless of the spin S value where the critical temperature increases with the value of S. Note that this phase transition occurs under the condition [Formula: see text]. Moreover, the obtained results show that the complex interactions give rise to a variety of phase diagrams, thereby revealing the richness of possible magnetic behaviors in the studied system.
Transport properties of the classical antiferromagnetic XXZ model on the square lattice have been theoretically investigated, putting emphasis on how the occurrence of a phase transition is reflected in spin and thermal transports. As is well known, the anisotropy of the exchange interaction $\mathrm{\ensuremath{\Delta}}\ensuremath{\equiv}{J}_{z}/{J}_{x}$ plays a role to control the universality class of the transition of the model, i.e., either a second-order transition at
TN into a magnetically ordered state or the Kosterlitz-Thouless (KT) transition at
TKT, which respectively occur for the Ising-type ($\mathrm{\ensuremath{\Delta}}>1$) and XY-type ($\mathrm{\ensuremath{\Delta}}<1$) anisotropies, while for the isotropic Heisenberg case of $\mathrm{\ensuremath{\Delta}}=1$, a phase transition does not occur at any finite temperature. It is found by means of the hybrid Monte Carlo and spin-dynamics simulations that the spin current probes the difference in the ordering properties, while the thermal current does not. For the XY-type anisotropy, the longitudinal spin-current conductivity ${\ensuremath{\sigma}}_{xx}^{s}$ ($={\ensuremath{\sigma}}_{yy}^{s}$) exhibits a divergence at
TKT of the exponential form ${\ensuremath{\sigma}}_{xx}^{s}\ensuremath{\propto}exp\left[B/\sqrt{T/{T}_{\text{KT}}\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}\right]$ with
B=O(1), while for the Ising-type anisotropy, the temperature dependence of ${\ensuremath{\sigma}}_{xx}^{s}$ is almost monotonic without showing a clear anomaly at
TN and such a monotonic behavior is also the case in the Heisenberg-type spin system. The significant enhancement of ${\ensuremath{\sigma}}_{xx}^{s}$ at
TKT is found to be due to the exponential rapid growth of the spin-current relaxation time toward
TKT, which can be understood as a manifestation of the topological nature of a vortex whose lifetime is expected to get longer toward
TKT. Possible experimental platforms for the spin-transport phenomena associated with the KT topological transition are discussed.
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 (nearest neighbor, ferromagnetic) and
J2>0 (second neighbor, antiferromagnetic). The striped phase breaks a
Z4 symmetry and is stabilized at low temperatures for
g=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-
J2 model has a line of very weak first-order phase transitions in the whole region
1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from
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∗ in the
J1-
J2 model. The continuous transitions near
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>1/2. 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.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
Monte Carlo simulations are used to study the location and nature of phase boundaries for Ising square lattices with antiferromagnetic coupling
JNN between nearest neighbors and additional interactions
JNNN between next-nearest neighbors and
J3NN between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R=
JNNN/JNN and R'=
J3NN/JNN. In addition to the c(2\ifmmode\times\else\texttimes\fi{}2) and (2\ifmmode\times\else\texttimes\fi{}1) phases, which also occur for R'=0, we find new (4\ifmmode\times\else\texttimes\fi{}4) and (4\ifmmode\times\else\texttimes\fi{}2) ordered states, for R'\ensuremath{\ne}0, which are separated from the disordered state by lines of first-order transitions. The nonuniversal critical behavior of the (2\ifmmode\times\else\texttimes\fi{}1) phase is studied using the block-distribution method and finite-size scaling. The possible existence of incommensurate phases is also explored.
Phase transitions in magnetic systems are studied using a Monte Carlo computer simulation method. The fundamentals of the technique are described and the strengths and limitations of this approach are discussed. Comparisons are made between the Monte Carlo data and exact results for the Ising square lattice. The properties of several different types of anisotropic systems are determined with particular care taken near the critical temperature Tc Competing interactions between neighboring sites has the effect of shifting Tc and may, in some cases, even change the nature of the ordered magnetic state. Critical field boundaries are traced out for several antiferromagnets and we show how competing interactions may give rise to tricritical points. We also present data on systems with impurities and examine the effects of free surfaces on the phase transition. Lastly we show how the transformation from the 2-dim Ising model to its lattice-gas model equivalent allows us to study adsorbed gas monolayers.