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Register a new userThe critical Binder cumulant in a two--dimensional anisotropic Ising model with competing interaction
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The Binder cumulant at the phase transition of Ising models on square lattices with ferromagnetic couplings between nearest neighbors and with competing antiferromagnetic couplings between next--nearest neighbors, along only one diagonal, is determined using Monte Carlo techniques. In the phase diagram a disorder line occurs separating regions with monotonically decaying and with oscillatory spin--spin correlations. Findings on the variation of the critical cumulant with the ratio of the two interaction strengths are compared to related recent results based on renormalization group calculations.
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We investigate the dependence of the critical Binder cumulant of the magnetization and the largest Fortuin-Kasteleyn cluster on the boundary conditions and aspect ratio of the underlying square Ising lattices. By means of the Swendsen-Wang algorithm, we generate numerical data for large system sizes and we perform a detailed finite-size scaling analysis for several values of the aspect ratio $r$, for both periodic and free boundary conditions. We estimate the universal probability density functions of the largest Fortuin-Kasteleyn cluster and we compare it to those of the magnetization at criticality. It is shown that these probability density functions follow similar scaling laws, and it is found that the values of the critical Binder cumulant of the largest Fortuin-Kasteleyn cluster are upper bounds to the values of the respective order-parameters cumulant, with a splitting behavior for large values of the aspect ratio. We also investigate the dependence of the amplitudes of the magnetization and the largest Fortuin-Kasteleyn cluster on the aspect ratio and boundary conditions. We find that the associated exponents, describing the aspect ratio dependencies, are different for the magnetization and the largest Fortuin-Kasteleyn cluster, but in each case are independent of boundary conditions.
We study the $pm J$ three-dimensional Ising model with a spatially uniaxially anisotropic bond randomness on the simple cubic lattice. The $pm J$ random exchange is applied in the $xy$ planes, whereas in the z direction only a ferromagnetic exchange is used. After sketching the phase diagram and comparing it with the corresponding isotropic case, the system is studied, at the ferromagnetic-paramagnetic transition line, using parallel tempering and a convenient concentration of antiferromagnetic bonds ($p_z=0 ; p_{xy}=0.176$). The numerical data point out clearly to a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3d random Ising model. The smooth finite-size behavior of the effective exponents describing the peaks of the logarithmic derivatives of the order parameter provides an accurate estimate of the critical exponent $1/ u=1.463(3)$ and a collapse analysis of magnetization data gives an estimate $beta/ u=0.516(7)$. These results, are in agreement with previous studies and in particular with those of the isotropic $pm J$ three-dimensional Ising at the ferromagnetic-paramagnetic transition line, indicating the irrelevance of the introduced anisotropy.
The Ising one-dimensional (1D) chain with spin $S=1/2$ and magnetoelastic interactions is studied with the lattice contribution included in the form of elastic interaction and thermal vibrations simultaneously taken into account. The magnetic energy term and the elastic (static) energy term based on the Morse potential are calculated exactly. The vibrational energy is calculated in the Debye approximation, in which the anharmonicity is introduced by the Gr{u}neisen parameter. The total Gibbs potential, including both the magnetic field, as well as the external force term, is constructed and from its minimum the equation of state is derived. From the Gibbs energy all the thermodynamic properties are calculated in a self-consistent manner. The comprehensive numerical calculations are performed in a full temperature range, i.e., from zero temperature up to the vicinity of melting. In particular, a role of magneto-elastic coupling is emphasized and examined. The numerical results are illustrated in figures and discussed.
A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of anisotropic two-layer Ising ferromagnet, on strips of r wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest neighbor couplings Kx, Ky (where Kx and Ky are the nearest neighbor interactions within each layer in the x and y directions, respectively) and with inter-layer coupling Kz, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted on a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained, are in good agreement with the accurate values reported by others.
We consider quantum Heisenberg ferro- and antiferromagnets on the square lattice with exchange anisotropy of easy-plane or easy-axis type. The thermodynamics and the critical behaviour of the models are studied by the pure-quantum self-consistent harmonic approximation, in order to evaluate the spin and anisotropy dependence of the critical temperatures. Results for thermodynamic quantities are reported and comparison with experimental and numerical simulation data is made. The obtained results allow us to draw a general picture of the subject and, in particular, to estimate the value of the critical temperature for any model belonging to the considered class.
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