Double Critical Point Research Articles

Coexistence of topological semimetal states in holography

We introduce a holographic model that exhibits a coexistence state of the Weyl semimetal and the topological nodal line state, providing us with a valuable tool to investigate the system’s behavior in the strong coupling regime. Nine types of bulk solutions exhibiting different IR behaviors have been identified, corresponding to nine different types of boundary states. These nine states include four distinct phases, namely the Weyl-nodal phase, the gap-nodal phase, the Weyl gap phase and the gap-gap phase, four phase boundaries, which are the Weyl-Dirac phase, the gap-Dirac phase, the Dirac-gap phase and the Dirac-nodal phase, and finally a double critical point. A phase diagram is plotted that exhibits qualitative similarity to the one obtained in the weak coupling limit. The anomalous Hall conductivity, which serves as an order parameter, and the free energy are calculated, with the latter showing the continuity of the topological phase transitions within the system. Our study highlights the similarities and differences in such a topological system between the weak and strong coupling regimes, paving the way for further experimental observations.

Light scattering investigations in the system methyl ethyl ketone + water + secondary butyl alcohol with traces of potassium thiocyanate

The effect of trace amounts of potassium thiocyanate (KSCN) at weight fractions XKSCN = 0.0300, 0.0319 and 0.0320 – the weight fractions at which no particular structural peculiarities exist in the phase diagram of MEK + W + sBA + KSCN; presence of critical endpoints (CEPs) are detected and coexistence domes with minimal separation between the vertices containing double critical points (DCPs) are formed in the phase diagram, respectively [Naseer A. P. et. al., J. mol. Liq. 296 (2019)] – on the critical behaviour of the system methyl ethyl ketone (MEK) + water(W) + secondary butyl alcohol (sBA) are investigated through thelight scattering technique. For MEK weight fractions (x) in the range of 0.1000 ≤ x ≤ 0.1800 at constant XKSCN = 0.0300 near Ising-type behavior is observed for experimental paths approaching both TL from below and TU from the above, while in the range of 0.2000 ≤ x ≤ 0.2100 the samples show a clear mean-field type of behavior both at the TL as well as TU. Investigations near the DCPs corresponding to the minimum approach of the two coexistence domes (i.e., at x = 0.1419 and 0.1608 at XKSCN = 0.0320) do not result in the well-established, doubling of critical exponents γ and ν due to inherent chemical complexities in the system MEK + W + sBA + KSCN. FTIR and LC-HRMS studies establish that the formation of the isothiocyanate-isomer of α-thiocyanated methyl-ethyl-ketone resulting from prolonged exposure to elevated temperatures during the course of the light-scattering experiments, is responsible for the results.

Dynamic Magnetic Features of a Mixed‐Spin‐2 and Spin‐5/2 Ising Ferrimagnetic System under a Time‐Dependent Oscillating Magnetic Field: Path Probability Method Approach

A study, within the path probability method, of the dynamic magnetic features of the mixed spin (2, 5/2) Ising ferrimagnetic system on a hexagonal lattice under the presence of a sinusoidal external magnetic field is presented. The Hamiltonian of the system contains intersublattice, intrasublattice, and crystal‐field interactions. The intersublattice interaction is taken as antiferromagnetic. The time variations of average dynamic magnetizations and the thermal behavior of the dynamic magnetizations are investigated to obtain the phases in the system and to characterize the nature of the dynamic phase transitions and find the dynamic phase transition temperature points. The dynamic phase diagrams in six different phase planes are presented, in which they contain the paramagnetic (p) phase, three different ferrimagnetic (i 1, i 2, i 3) fundamentals, and the p + i 1, i 1 + i 2, and i 2 + i 3 mixed phases as well as the dynamic tricritical point and the dynamic double critical end point, depending on the interaction parameters. The dynamic compensation behavior is also examined and only N‐type behaviors are observed. Some of the obtained dynamic phase diagrams are in qualitatively good agreement within some works on the dynamics of mixed‐spin Ising system.

Nonequlibrium magnetic features in a mixed spin (2, 5/2) Ising system driven by the external oscillating magnetic field by path probability method

Utilizing the path probability method, we investigated nonequilibrium magnetic features in a mixed spin (2, 5/2) Ising model Hamiltonian composed of bilinear and crystal-field interactions in the presence of an external oscillating magnetic field. We numerically solved the time dependence of average magnetizations to find the phases in the system. We examined the dynamic magnetizations to obtain dynamic phase transition (DPT) temperatures, the nature of the DPTs, and phases in the system. The dynamic phase diagrams (DPDs) were constructed in reduced temperature and the amplitude of oscillating magnetic field plane for various interaction parameters. We observed that the system gives very rich and interesting topological behaviors of DPDs, such as up to two dynamic tricritical points, six critical end points, a double critical end points, a zero-temperature critical point, one inverse critical end point and a quadruple point depending on interaction parameters. The system also exhibits paramagnetic, six distinct ferrimagnetic and three different nonmagnetic phases as well as up to eleven different mixed or hybrid phases. In addition, the system exhibits the reentrant behavior.

Dynamic multicritical phase diagrams of the mixed spin (2, 5/2) Blume-Emery-Griffiths model with repulsive biquadratic coupling

We investigated the dynamic phase transitions (DPTs) in the mixed spin (2, 5/2) Blume-Emery Griffiths model with repulsive biquadratic interaction in the presence of a time-varying magnetic field. We used the path probability method to obtain the set of the dynamic equations. We numerically solved these dynamic equations to characterize the nature of first- and second-order phase transitions and to find the DPT temperatures as well as obtain the phases in the system. We constructed the dynamic phase diagrams (DPDs) in reduced temperature and amplitude of oscillating magnetic field plane. We observed that the DPDs display richer, complex and more topological various type of phase diagrams. In particular, DPDs exhibit the disordered phase, antiquadrupolar or staggered phase, six different ferrimagnetic phases, three different nonmagnetic phases, and numerous mixed phases. DPDs also display two dynamic tricritical points for only smaller values of crystal-field interactions, multiple critical end and double critical end points, one zero-temperature critical point, one inverse critical end point, and a quadruple point depending on interaction parameters. The system always shows the reentrant behaviors for the higher values of magnetic field amplitude, but it does not exhibit the dynamic tricritical behavior for higher values of crystal-field parameter.

Random crystal field effects on antiferromagnetic spin-1 Blume–Capel model

The outcome of the random crystal field effects on the antiferromagnetic spin-1 Blume–Capel model and external magnetic field are examined on the Bethe Lattice in terms of exact recursion relations. It is assumed that the crystal field is either turned on or off randomly with probability [Formula: see text] and [Formula: see text], respectively. The phase diagrams are constructed from the thermal analysis of the order parameters with the coordination number [Formula: see text] which corresponds to honeycomb lattice. It is explored that the system goes both second- and first-order phase transitions, along with the reentrant behavior and a few critical points. The reentrant behavior is stronger for lower values of [Formula: see text] and disappears as [Formula: see text] gets closer to 1.0. The first-order lines are observed to be either linked to the tricritical points or decomposed. The critical end points and double critical points are also observed.

Thermodynamic properties of the spin S=3/2 quantum ferromagnetic Blume-Capel model in a transverse crystal field.

The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.

Dynamic phase of transitions of the mixed spin (1/2, 3/2) Ising model in the presence of a time-varying magnetic field by using the path probability method

Dynamic phase transitions (DPTs) of the mixed spin (1/2, 3/2) Ising ferrimagnetic model on the two interpenetrating square lattice with the bilinear and crystal-field interactions under a time-varying (sinusoidal) magnetic field were examined by using the path probability method (PPM). We studied stationary solutions of dynamic average magnetizations and obtained phases and then we investigated thermal behaviors of dynamic magnetizations to designate the nature of the DPTs and find the DPT temperatures. We also constructed the dynamic phase diagrams (DPDs) in two different planes. DPDs exhibit the paramagnetic (p), ferrimagnetic (i), and mixed phases (i+p), and one or two dynamic tricritical points and dynamic double critical end points. The reentrant behavior is also seen. We observed that the PPM is a more appropriate dynamic method to study the dynamic features of ferrimagnetism.

Antiferromagnetic Spin-3/2 Ising Model Under the Influence of Random Crystal Field

The critical behaviors of the antiferromagnetic spin-3/2 Ising model are studied by using the exact recursion relations (ERR) in the presence of a random crystal field on the Bethe lattice (BL). The sublattice order parameters, magnetizations, and quadrupolar moments are obtained in terms of ERRs under the effect of random crystal field which was either turned on or off with probability p and 1 − p, respectively, in a bimodal form. The nature of phase transitions, thus, the phase diagrams of the model are calculated on the (H, T) planes for the randomly changing crystal field values with probability p for the coordination numbers q = 3 and 4. It is found that the model exhibits both second- and first-order phase transitions in addition to some critical points, i.e., tricritical point (TCP), critical end point, double critical point, and isolated end point. We have also observed the reentrant behavior for some values of our system parameters.

Effect of potassium thiocyanate on the phase diagram of Methyl Ethyl Ketone + secondary Butyl Alcohol + Water: Search for the Quadruple Critical Point

The effect of addition of small amounts of the ionic impurity Potassium Thiocyanate (KSCN) on the tunnel-like region in the phase diagram of Methyl Ethyl Ketone (MEK) + secondary Butyl Alcohol (sBA) + Water (W) is investigated in detail. The KSCN weight fraction range, 0 ≤ XKSCN ≤ 0.0318, results in a systematic decrease in the diameter of the critical-tunnel from ΔT = 71.416 K at XKSCN = 0, to ΔT = 5.275 K at XKSCN = 0.0315. For 0.03190 ≤ XKSCN ≤ 0.03195 the line of critical points end abruptly at specific values of water weight fractions (XW) resulting in the formation of several critical end points. For XKSCN ≥ 0.0320 the critical-tunnel splits into several pairs of parabolic critical lines, each carrying double critical points (DCPs) at their vertices, with one-phase region in-between. The distance between the DCPs corresponding to the low- and high-MEK regions can be made to approach or recede from each other depending on the KSCN weight fraction. The QCP in this system is pushed into a hypothetical region due to curvature effects of the coexistence surface in MEK + W + sBA + KSCN.

Quantum interference independence of the heteroatom position

We introduce a novel calculation to the transmission coefficient of molecules with single impurity. The study employs tight binding Green’s function method and new derived analytical formula to investigate chromene-like (even) and doped azulene-like (odd) model systems. Under certain circumstances, these specific even and odd models are respectively shown single and double critical points, at which the transmission becomes independent of the heteroatom position. Moreover, a counterintuitive correlation between the transmission and the coupling strength has analytically and numerically been approved: the transmission decreases with strengthening lead-nodes couplings.

Solubility and Critical Phenomena in Ternary Aqueous Solutions Containing Type-2 Salt and Alkali

Supercritical phase equilibria in the ternary system K2SO4–KOH–H2O at 420–500°C and up to 130 MPa pressure with binary boundary subsystems of different types are studied. The binary subsystem of type 1 features no critical phenomena in saturated (l = g) aqueous solution and no phase separation (l1–l2) (KOH–H2O); the binary subsystem of type 2 is characterized by immiscibility of the liquid phase and has two critical end-points \(p(g = l-_{S_{K_{2}SO_{4}}})\) and \(Q(l_{1} = l_{2}-_{S_{K_{2}SO_{4}}})\) in saturated aqueous solution (K2SO4–H2O). The ternary system has two three-phase equilibria (g–l–s) and (l1–l2–s), separated by a two-phase supercritical fluid region \((fl-_{S_{K_{2}SO_{4}}})\), and two types of monovariant critical curves \((g=l-_{S_{K_{2}SO_{4}}})\) and \((l_{1}=l_{2}-_{S_{K_{2}SO_{4}}})\). The three-phase regions approach each other upon temperature increase up to the point where the two-phase supercritical equilibrium disappears, and the two mentioned monovariant critical curves are joined into a double homogeneous critical point \((g=l-_{S_{K_{2}SO_{4}}} \leftrightarrow l_{1} = l_{2}-_{S_{K_{2}SO_{4}}})\) at maximum temperature ~445°C and 51–52 MPa.

Stepwise positional-orientational order and the multicritical-multistructural global phase diagram of the s=3/2 Ising model from renormalization-group theory.

The spin-3/2 Ising model, with nearest-neighbor interactions only, is the prototypical system with two different ordering species, with concentrations regulated by a chemical potential. Its global phase diagram, obtained in d=3 by renormalization-group theory in the Migdal-Kadanoff approximation or equivalently as an exact solution of a d=3 hierarchical lattice, with flows subtended by 40 different fixed points, presents a very rich structure containing eight different ordered and disordered phases, with more than 14 different types of phase diagrams in temperature and chemical potential. It exhibits phases with orientational and/or positional order. It also exhibits quintuple phase transition reentrances. Universality of critical exponents is conserved across different renormalization-group flow basins via redundant fixed points. One of the phase diagrams contains a plastic crystal sequence, with positional and orientational ordering encountered consecutively as temperature is lowered. The global phase diagram also contains double critical points, first-order and critical lines between two ordered phases, critical end points, usual and unusual (inverted) bicritical points, tricritical points, multiple tetracritical points, and zero-temperature criticality and bicriticality. The four-state Potts permutation-symmetric subspace is contained in this model.

Dynamic Magnetic Properties of the Spin-3/2 Ising Model on a Cylindrical Nanowire in an Oscillating Magnetic Field

Dynamic magnetic properties of the spin-3/2 model on a cylindrical nanowire in an oscillating magnetic field are investigated, within the mean-field theory, by utilizing Glauber-type stochastic dynamics. The thermal variations of the dynamic core, shell, and total magnetizations are examined and then dynamic phase diagrams are calculated in four different planes which exhibit several ordered and mixed phases, as well as dynamic tricritical and compensation behaviors. The double and zero-temperature critical points are also observed. Results are compared with the published works and find a quantitatively good agreement.

Bifurcation analysis in a predator–prey model for the effect of delay in prey

In this paper, we study dynamics in a predator–prey model with delay, in which predator can be infected, with particular attention focused on nonresonant double Hopf bifurcation. By using center manifold reduction methods, we obtain the equivalent normal forms near a double Hopf critical point in this system. Moreover, bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results.

Investigation of critical lines and global phase behavior of unequal size of molecules in binary gas–liquid mixtures in the combined pressure–temperature–concentration planes around the van Laar point

Abstract We investigate critical curves and global phase behavior of unequal size of molecules in binary gas–liquid mixtures at the van Laar point and its vicinity. The van Laar point is only point at which the mathematical double point curve is stable, and also the intersection of the tricritical point and the double critical end point. The critical line structure is displayed for various combinations of the chain length and system parameters in the reduced pressure ( P ∗ ) temperature ( T ∗ ) plane, as is usually done with experimental results and temperature–concentration ( T , x ) plane. The P ∗ , T ∗ diagrams are discussed in accordance with the Scott and van Konynenburg binary phase diagram classification. We found that our P ∗ , T ∗ plots correspond to the type II, type III, type IV phase diagram behaviors and they are in good agreement with the theoretical and experimental studies. It is also found that the critical lines and phase behavior are extremely sensitive to small modifications in the system parameters.

The dynamic critical properties of the spin-2 Ising model on a bilayer square lattice

The spin-2 Ising model is investigated for the ferromagnetic/ferromagnetic (FM/FM), antiferromagnetic/ferromagnetic (AFM/FM) and antiferromagnetic/antiferromagnetic (AFM/AFM) interactions on the two-layer square lattice by using the Glauber-type stochastic dynamics. The system is in contact with a heat bath at temperature T, and the exchange of energy with the heat bath occurs via one-spin flip. By employing the Master equation and Glauber transition rates, the dynamic equations of the system are obtained. These equations are solved by using the numerical methods. First, we investigate the average order parameters as a function of the time to find the phases in the system. Then, the temperature-dependence of the dynamic order parameters is examined to obtain the dynamic phase transition temperatures. The dynamic phase diagrams are presented on the different planes. According to the values of the system parameters, a variety of dynamic critical points such as tricritical point, triple point, quadruple point, critical end point, double critical end point, zero-temperature critical point, multicritical point and tetracritical point are obtained. The reentrant behavior is seen in the system for the AFM/AFM interaction. Finally, we also investigate the influence of the oscillating field frequency on the dynamic phase diagrams in detail.

Double Hopf bifurcation in a four-neuron delayed system with inertial terms

In this paper, a four-neuron delayed system with inertial terms is considered. By studying the distribution of the eigenvalues of the associated characteristic equation, we derive the critical values where double Hopf bifurcation occurs. Then by employing the perturbation-incremental scheme for the system, bifurcation diagrams are obtained. Furthermore, we carry out bifurcation analysis showing that there exist a stable fixed point, two stable periodic solutions, co-existence of a pair of stable periodic solutions and quasi-periodic motion in the neighborhood of the double Hopf critical point. We also find some interesting phenomena that the dynamical period switching occurs in some delayed regions. Finally, some numerical simulations are performed to support the theoretical analysis.

Dynamic phase diagrams of a ferrimagnetic mixed spin (1/2, 1) Ising system within the path probability method

In this study we used the path probability method (PPM) to calculate the dynamic phase diagrams of a ferrimagnetic mixed spin-(1/2, 1) Ising system under an oscillating magnetic field. One of the main advantages of the PPM over the mean-field approximation and the effective-field theory based on Glauber-type stochastic dynamics is that it contains two rate constants which are very important for studying dynamic behaviors. We present the dynamic phase diagrams in the reduced magnetic field amplitude and reduced temperature plane and the twelve main different topological types of the phase diagrams are obtained. The phase diagrams contain paramagnetic (p), ferrimagnetic (i) and i+p mixed phases. They also exhibit a dynamic tricritical and reentrant behavior as well as the dynamic double critical end point (B), critical end point (E), quadruple point (QP) and triple point (TP). The dynamic phase diagrams are compared and discussed with the phase diagrams obtained in previous works within the mean-field approximation and the effective-field theory based on Glauber-type stochastic dynamics.

Magnetic properties of mixed spin (1, 3/2) Ising nanoparticles with core–shell structure

The magnetic properties of mixed spin-1 and spin-3/2 Ising nanoparticles with core/shell structure are studied by using the effective-field theory with correlations. We investigate the thermal variations of the core, shell and total magnetizations and the Q-, R-, P-, S-, N- and L-types of compensation behavior in Néel classification nomenclature exists in the system. The effects of the crystal-field, core and shell interactions and interface coupling, on the phase diagrams are investigated in detail and the obtained phase diagrams are presented in three different planes. The system exhibits both second- and first-order phase transitions besides tricritical point, double critical end point, triple point and critical end point depending on the appropriate values of the interaction parameters. The system strongly affected by the surface situations and some characteristic phenomena are found depending on the ratio of the physical parameters in the surface shell and the core.