Fourth-order Critical Points Research Articles

Critical properties of the spin-1 Heisenberg model with random magnetic field and Dzyaloshinskii–Moriya interaction

The spin-1 Heisenberg model with Dzyaloshinskii–Moriya interaction and in presence of a random magnetic field is studied by using the framework of the two-spin cluster approximation. The formalism is applied to the simple cubic lattice (q=6). At finite temperature, the phase diagrams of the system reveal the presence of tricritical points, implying the presence of second and first-order phase transitions. In addition, the occurrence of fourth-order critical points is also investigated.

Ensemble inequivalence in the Blume-Emery-Griffiths model near a fourth-order critical point.

The canonical phase diagram of the Blume-Emery-Griffiths model with infinite-range interactions is known to exhibit a fourth-order critical point at some negative value of the biquadratic interaction K<0. Here we study the microcanonical phase diagram of this model for K<0, extending previous studies which were restricted to positive K. A fourth-order critical point is found to exist at coupling parameters which are different from those of the canonical ensemble. The microcanonical phase diagram of the model close to the fourth-order critical point is studied in detail revealing some distinct features from the canonical counterpart.

Multicriticality in the Blume–Capel model under a continuous-field probability distribution

The multicritical behavior of the Blume–Capel model with infinite-range interactions is investigated by introducing quenched disorder in the crystal field Δi, which is represented by a superposition of two Gaussian distributions with the same width σ, centered at Δi = Δ and Δi = 0, with probabilities p and (1 − p), respectively. A rich variety of phase diagrams is presented, and their distinct topologies are shown for different values of σ and p. The tricritical behavior is analyzed through the existence of fourth-order critical points, as well as how the complexity of the phase diagrams is reduced by the strength of the disorder.

Ground-state phase diagrams of Ising metamagnet in a mixed longitudinal and transverse magnetic field

The ground-state magnetic properties of a two-sublattice Ising metamagnet in a mixed longitudinal and transverse magnetic field are studied within the effective-field theory. A parameter j 2= J 2/ J 1 is introduced, which reflects the strength ratio of spin coupling between adjacent planes and in each plane. In addition to the second-order transition lines, the first-order transition lines are also presented, since the ground-state energy can be calculated numerically. The ground-state phase diagrams in h x – h z are presented. The results show that when j 2<0 the phase transition of the system is always first-order for h x <2.751, and when −1000⩽ j 2<0 it is always second-order for h x >4.36. For the given h x (0< h x <14.71), the longitudinal critical magnetic field increases as j 2 decreases. The reentrant phenomenon occurs in the range of j 2<−11.89, h x >14.71. There is no fourth-order critical point in the phase diagrams given by using EFT as found by using mean field theory (MFT).

An effective-field study of Ising metamagnet in an external field

The phase diagrams and magnetization curves of a two-sublattice Ising metamagnet at finite temperature with longitudinal crystal field H are investigated by the use of an effective-field theory (EFT) with correlations. In addition to the second-order transition lines, the first-order transition lines are also presented, since a method to calculate the Gibbs free energy numerically at finite temperature within EFT is found in this work. The results show that there is no fourth-order critical point or reentrant phenomenon in the phase diagrams given by using EFT as found by using mean-field theory (MFT).

Multicritical behavior of the Ising metamagnet in both external longitudinal and transverse fields

A two-sublattice Ising metamagnet in both external longitudinal and transverse fields is studied within the mean-field approach based on Bogoliubov's inequality for the Gibbs free energy. At finite temperatures, by changing values of the parameters of the model many different types of phase diagrams in the longitudinal field-temperature plane and in the transverse field-temperature plane are determined. The results show that the tricritical point can occur and decomposes into a critical end point and a double critical end point in a certain small region of the external longitudinal and transverse fields. The temperature of the tricritical point monotonically increases with decreasing the transverse magnetic field $\ensuremath{\Omega}$ and increasing the longitudinal magnetic field $h$. A line of fourth-order critical points is also determined.

Tricritical points in the Sherrington-Kirkpatrick model in the presence of discrete random fields

The infinite-range-interaction Ising spin glass is considered in the presence of an external random magnetic field following a trimodal (three-peak) distribution. The model is studied through the replica method and phase diagrams are obtained within the replica-symmetry approximation. It is shown that the border of the ferromagnetic phase may present first-order phase transitions, as well as tricritical points at finite temperatures. Analogous to what happens for the Ising ferromagnet under a trimodal random field, it is verified that the first-order phase transitions are directly related to the dilution in the fields (represented by $p_{0}$). The ferromagnetic boundary at zero temperature also exhibits an interesting behavior: for $0<p_{0}<p_{0}^{*} \approx 0.30856$, a single tricritical point occurs, whereas if $p_{0}>p_{0}^{*}$ the critical frontier is completely continuous; however, for $p_{0}=p_{0}^{*}$, a fourth-order critical point appears. The stability analysis of the replica-symmetric solution is performed and the regions of validity of such a solution are identified; in particular, the Almeida-Thouless line in the plane field versus temperature is shown to depend on the weight $p_{0}$.

Multicritical behavior in the antiferromagnetic Blume–Capel model

The existence of the fourth-order critical point has been found in the spin-1 Ising antiferromagnet with a single-ion potential and an external magnetic field involved by using the mean-field approximation calculation. The Monte Carlo simulation gives more accurate results and shows that no such phase transition occurs in the square lattice system. To show definitely that the fourth-order phase transition appears in three-dimensional systems, the simple cubic system is investigated by using Monte Carlo simulations. It was found that the decomposition of the tricritical point into a critical end point and a double critical end point appears in the phase diagram within a small range of the single-ion potential parameter. The location of the fourth-order critical point can also be estimated.

A new multicritical point in anisotropic magnets: II. Ferromagnetic in a random skew field

The phase diagram of an anisotropic ferromagnetic in a skew random magnetic field is studied as a function of the temperature, anisotropy and two random field components. At fixed high anisotropy, there exists a single ordered phase, bounded by surfaces of second- and first-order transitions separated by a line of tricritical points. As the anisotropy is decreased, the low-temperature tricritical points split into pairs of critical points (inside the ordered phase) and critical end points. The lines of critical points, critical end points and tricritical points meet at a critical point of fourth order. As the anisotropy decreases, this fourth-order critical point moves to lower transverse fields and higher temperatures until it reaches the temperature-longitudinal-field plane exactly at the new multicritical point found in the first paper in this series. For lower anisotropies there is no longer any first-order transition between the ordered and disordered phases. Instead, there exists a spin-flop 'shelf', inside the ordered phase, ending at a bicritical point. In the general four-parameter (temperature, anisotropy, longitudinal random field, transverse random field) space, the new multicritical point occurs where the lines of bicritical points and of fourth-order critical points meet. All of these results have been calculated explicitly using mean-field theory. Due to the random fields, the mean-field predictions for the fourth-order critical points break at 42/3 dimensions, and are replaced by renormalisation-group estimates. The results may be observed experimentally in random exchange antiferromagnets with a uniform magnetic field.

Crossover from First-Order to Continuous Phase Transition Induced by Symmetry-Breaking Fields

The crossover from first order to continuous transition induced by a symmetry-breaking field $g$ in an $n=2$ cubic model, for which a stable fixed point is not accessible, is studied. It is shown that unlike previously studied cases, the ($g$,$T$) phase diagram is rather complicated, exhibiting critical end points, tricritical, and fourth-order critical points. It is suggested that this phase diagram be studied experimentally in ferroelectric ${\mathrm{Tb}}_{2}$(${\mathrm{Mo}{\mathrm{O}}_{4})}_{3}$ by applying uniaxial and shear stresses.

Multicritical behavior and global phase diagrams of tetragonalXY-like antiferromagnets

The global $T\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$phase diagram of a model Hamiltonian associated with $\mathrm{XY}$-like antiferromagnetic transitions in tetragonal crystals is studied. It is found that for sufficiently large fourth-order anisotropy ($\frac{K}{J\ensuremath{\gtrsim}0.0814}$), the model exhibits tricritical points at a finite nonzero magnetic field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$, in addition to the tetracritical point ($T={T}_{N}, \stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}=0$) predicted previously on the basis of a Landau-Ginzburg-Wilson analysis. The wing critical lines associated with one of the tricritical points are physically accessible in the $T\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ space. In a closely related model appropriate for uniaxial antiferromagnets, we find tricritical and fourth-order critical points at low temperatures.

Global phase diagram for a three-component model

The global phase diagram in a five-dimensional parameter space is described for a model which can be thought of as the regular-solution model of a ternary mixture or the mean-field approximation to a spin-1 Ising ferromagnet with a general nearest-neighbor interaction (the Blume-Capel model). The model possesses three fourth-order critical points (known from previous work) which are connected to a total of nine lines of tricritical points. Four manifolds of four-phase coexistence occur along with three manifolds of double critical points and six manifolds of critical double-end points. The locations of all significant features of the phase diagram are described qualitatively, and quantitative results are provided for some of the manifolds of lower dimension. Computational procedures are described which permit a detailed exploration of any portion of the phase diagram which may be of interest.

Multicritical behavior in an ising antiferromagnet with zero field splitting

We show that a fourth order critical point along with other critical points of lower order prevails in a simple Ising antiferromagnet with zero field splitting. The observation of the fourth criticality is also discussed.

Spin-32Ising model for tricritical points in ternary fluid mixtures

We present a lattice-gas model of ternary fluid mixtures. Within the mean-field approximation, we study a nonsymmetric tricritical point in this model. We compare our results to the experimental observations on the system ethanol-water-carbon-dioxide. In the course of our work, we have studied a Landau theory describing the neighborhood of a fourth-order critical point. Also, we have noted that mean-field theory indicates the existence of a fourth-order critical point in the spin-1 model of Blume, Emery, and Griffiths, corresponding to $K&lt;0$, $J+K&gt;0$.