Dynamical phase transition in the Ising model on a Barabási–Albert network under the influence of periodic magnetic field is studied using Monte-Carlo simulations. For a wide range of the system sizes N and the field frequencies, approximate phase borders between dynamically ordered and disordered phases are obtained on a plane h (field amplitude) versus T/Tc (temperature normalized to the static critical temperature without external field, Tc∝ ln N). On these borders, second- or first-order transitions occur, for parameter ranges separated by a tricritical point. For all frequencies of the magnetic field, position of the tricritical point is shifted toward higher values of T/Tc and lower values of h with increasing system size, i.e. the range of critical parameters corresponding to the first-order transition is broadened.
The form of the field distribution, in a random-field Ising model, is known to play an important role in determining the order of the phase transition and the existence of a tricritical point. We have studied a random-field Ising model with a trimodal distribution (sum of three \ensuremath{\delta} functions), which simulates a Gaussian distribution for a certain range of its parameters, within the mean-field renormalization-group scheme. We have considered clusters of one, two, and four spins, and the recursion relations for the determination of the fixed points are numerically analyzed. Using this analysis we have constructed the phase diagram of the system in the space of coupling (K), field (h), and zero-field probability (p). It is shown that the line of tricritical points ends for values of p near (2/3, very different from that obtained in the usual mean-field approximation. Mean-field renormalization-group (p\ensuremath{\simeq}(2/3) results are found to be consistent with the equivalence of the trimodal and Gaussian (absence of tricritical points) distributions until the fourth-order moments for p=(2/3.
Phase diagrams and magnetization curves of a diluted spin-1 transverse Ising model in a random field on honeycomb, square, and simple-cubic lattices, respectively, are investigated by the use of an effective-field theory with correlations. The tricritical point is found in the system, in contrast to the corresponding spin-1/2 Ising counterpart. The behavior of the tricritical point is also examined as a function of applied transverse field. The possible reentrant phenomena displayed by the system due to the competition effects that occur for appropriate ranges of the random and transverse fields are investigated.
The first phase diagram explicitly exhibiting intersecting lines of tricritical points is presented. Their point of intersection is a critical point of order 4. The system is a simple assembly of ferromagnetic Ising planes coupled with an arbitrary interplanar interaction, with Hamiltonian $\mathcal{H}=\ensuremath{-}J\left(\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}+\mathcal{R}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}\right)\ensuremath{-}\ensuremath{\mu}H\ensuremath{\Sigma}\stackrel{}{i}{s}_{i}\ensuremath{-}\ensuremath{\mu}{H}^{\ensuremath{'}}{(\ensuremath{-}1)}^{\ensuremath{\eta}}\ensuremath{\Sigma}\stackrel{}{i}{s}_{i},$ where
J>0, ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, and $\ensuremath{\mu}$ is the magnetic moment per spin. The first sum is over nearestneighbor (nn) spins in an $x\ensuremath{-}y$ plane, the second sum over nn spins coupled along the
z direction,
H is the magnitude of a uniform magnetic field, and ${H}^{\ensuremath{'}}$ is a staggered magnetic field which acts oppositely on adjacent planes of constant
z ($\ensuremath{\eta}=0$ on even planes, +1 on odd planes). The Hamiltonian is invariant under $\mathcal{R}\ensuremath{\rightarrow}\ensuremath{-}\mathcal{R}$, $H\ensuremath{\rightarrow}{H}^{\ensuremath{'}}$, and ${s}_{i}\ensuremath{\rightarrow}{(\ensuremath{-}1)}^{\ensuremath{\eta}}{s}_{i}$, so that the Gibbs potential is also invariant, $G(T, H, {H}^{\ensuremath{'}}, \mathcal{R})=G(T, {H}^{\ensuremath{'}}, H, \ensuremath{-}\mathcal{R})$. Using this symmetry, we make a scaling hypothesis about the special point $\ensuremath{\tau}[\ensuremath{\equiv}T\ensuremath{-}{T}_{c} (R=0)]=H={H}^{\ensuremath{'}}=R=0$, namely, that the Gibbs potential obeys the functional equation $G({\ensuremath{\lambda}}^{{a}_{\ensuremath{\tau}}}\ensuremath{\tau}, {\ensuremath{\lambda}}^{\mathrm{aH}}H, {\ensuremath{\lambda}}^{a{H}^{\ensuremath{'}}}{H}^{\ensuremath{'}}, {\ensuremath{\lambda}}^{a\mathcal{R}}\mathcal{R})=\ensuremath{\lambda}G(\ensuremath{\tau}, H, {H}^{\ensuremath{'}}, \mathcal{R})$; the four scaling powers are found to be ${a}_{\ensuremath{\tau}}=\frac{1}{2}$,
aH=aH=1615,
aR=87.
Thirteen terms are presented of the low-temperature series for the free energy of the three-state Potts model in arbitrary external fields. Extrapolation of the series for specific heat, order parameter, and susceptibilities indicates that the transition in zero field is continuous (in contrast to the Landau prediction) with exponents alpha = alpha '=0.05+or-0.10, beta =0.10+or-0.01, gamma ' approximately=1.5+or-0.2 and gamma ' perpendicular to =1.1+or-0.1. These conclusions suggest the presence of an 'anomalous' tricritical point in the model; the associated phase diagram is discussed and contrasted with phenomenological predictions.
The microscopic approach to calculating the free energy of a three-dimensional Ising-like system in a homogeneous external field is developed in the higher non-Gaussian approximation (the
ρ6 model) at temperatures above the critical value of
Tc (
Tc is the phase-transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short- and long-wave spin-density oscillation modes taking into account both the temperature and field fluctuations of the order parameter. Our analytical calculations do not involve power series in the scaling variable and are valid in the whole field-temperature plane near the critical point including the region in the vicinity of the limiting field
h~c, which divides external fields into the weak and strong ones (i.e., the crossover region). In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity, and power series are not efficient. The obtained expression for the free energy contains the leading terms and terms determining the temperature and field confluent corrections.