Tetracritical Points Research Articles

Spontaneous breaking of finite group symmetries at all temperatures

We study conformal field theories with finite group symmetries with spontaneous symmetry breaking (SSB) phases which persist at all temperatures. We work with two \lambda \phi^4λϕ4 theories coupled through their mass terms. The two \lambda \phi^4λϕ4 theories are chosen to preserve either the cubic symmetry group, or the tetrahedral symmetry group. A one-loop calculation in the 4-\epsilon4−ϵ expansion shows that there exist infinitely many fixed points that can host an all temperature SSB phase. An analysis of the renormalization group (RG) stability matrix of these fixed points, reveals that their spectrum contains at least three relevant operators. In other words, these fixed points are tetracritical points.

Formula omitted] Blume–Capel Model with external magnetic field in the cluster variation method

The thermal variations of the order-parameters are investigated to study the phase diagrams of the ±J Blume–Capel (BC) model in the lowest approximation of the cluster variation method (LACVM). The bilinear exchange interaction parameter J is either turned on ferromagnetically with probability p or antiferromagnetically with 1−p on the sublattices to simulate the ±J BC model. The inclusion of external magnetic field reveals many extra features which is absent otherwise. The model presents up to two first-order and three second-order phase transitions. In addition to the ferromagnetic (FM), antiferromagnetic (AFM), ferrimagnetic (FI) and paramagnetic (PM) phases, chaotic regions are also observed which maybe signaling the existence of spin glass phase. The model displays tetracritical points in addition to the double reentrant behavior.

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.

Phase transitions of heliconical smectic-C and heliconical nematic phases in banana-shaped liquid crystals.

A mean-field theory is introduced to describe heliconical nematic (N_{TB}), heliconical smectic-C (S_{m}C_{TB}), and biaxial heliconical smectic-C (S_{m}C_{TB,b}) phases with mirror symmetry breaking. We extend our previous theories of the N_{TB} phase to the heliconical smectic-C phases, by taking into account one-dimensional spatial ordering of smectic layers. The calculated phase diagrams on the temperature-alkyl chain length plane show a rich variety of phase transitions: first- and second-order N_{TB}-S_{m}C_{TB} transitions, etc., including tricritical, tetracritical, and multicritical points. Our theory is qualitatively consistent with an experimental phase diagram.

Exact solutions and topological phase diagram in interacting dimerized Kitaev topological superconductors

It was recently shown that an interacting Kitaev topological superconductor model is exactly solvable based on two-step Jordan-Wigner transformations together with one spin rotation. We generalize this model by including the dimerization, which is shown also to be exactly solvable. We analytically determine the topological phase diagram containing seven distinct phases. It is argued that the system is topological when a fermionic many-body Majorana zero-energy edge state emerges. It is intriguing that there are two tetra-critical points, at each of which four distinct phases touch.

Classical Monte Carlo Study for Antiferro Quadrupole Orders in a Diamond Lattice

We investigate antiferro quadrupole orders in a diamond lattice under magnetic fields by Monte Carlo simulations for two types of classical effective models. One is an XY model with Z_3 anisotropy, and the other is a two-component phi^4 model with a third-order anisotropy. We confirm that the universality class of the zero-field transition is that for the three-dimensional XY model. Magnetic field corresponds to a Z_3 field in the effective model, and under this field, we find that collinear and canted antiferro-quadrupole orders compete. Each phase is characterized by symmetry breaking in the sector of (sublattice Z_2)x(reflection Z_2 for the order parameter). When Z_3 anisotropy and magnetic field vary, it turns out that this system is a good playground for various multicritical points; bicritical and tetracritical points emerge in a finite field. Another important finding is about the scaling of parasitic ferro quadrupole order at the zero-field critical point. This is the secondary order parameter induced by the primary antiferro order, and its critical exponent beta'=0.815 clearly differs from the expected value that is twice the value for the primary order parameter. The corresponding correlation length exponent is also different, nu'=0.597(12). We also discuss relation of the present effective quadrupole models with the 3-state Potts model as well as implication to undertanding of orbital orders in Pr-based 1-2-20 compounds.

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.

Magnetic properties of a spin-1 triangular Ising system

We studied some magnetic behaviors of the Blume-Capel (BC) model in a site diluted triangular lattice by means of the effective-field theory (EFT) with correlations. The effects of the exchange interaction (J), crystal field (D), concentration (p) and temperature (T) on the magnetic properties of the spin-1 BC model in a triangular lattice, such as magnetization, susceptibility, phase diagram and hysteresis behaviors, are investigated in detail. The phase diagrams of the system are presented in two different planes. The tricritical point as well as the tetracritical and critical end special points are found to depend on the physical parameters of the system. Moreover, when the hysteresis behaviors of the system are examined, single and double hysteresis loops are observed for various values of the physical parameters. We show that the hysteresis loops have different coercive field points in which the susceptibility makes a peak at these points.

Molecular field theory for biaxial smectic A liquid crystals

Thermotropic biaxial nematic phases seem to be rare, but biaxial smectic A phases less so. Here we use molecular field theory to study a simple two-parameter model, with one parameter promoting a biaxial phase and the second promoting smecticity. The theory combines the biaxial Maier-Saupe and McMillan models. We use alternatively the Sonnet-Virga-Durand (SVD) and geometric mean approximations (GMA) to characterize molecular biaxiality by a single parameter. For non-zero smecticity and biaxiality, the model always predicts a ground state biaxial smectic A phase. For a low degree of smectic order, the phase diagram is very rich, predicting uniaxial and biaxial nematic and smectic phases, with the addition of a variety of tricritical and tetracritical points. For higher degrees of smecticity, the region of stability of the biaxial nematic phase is restricted and eventually disappears, yielding to the biaxial smectic phase. Phase diagrams from the two alternative approximations for molecular biaxiality are similar, except inasmuch that SVD allows for a first-order isotropic-biaxial nematic transition, whereas GMA predicts a Landau point separating isotropic and biaxial nematic phases. We speculate that the rarity of thermotropic biaxial nematic phases is partly a consequence of the presence of stabler analogous smectic phases.

Quantum spin-1 anisotropic ferromagnetic Heisenberg model in a crystal field: A variational approach

A variational approach based on Bogoliubov inequality for the free energy is employed in order to treat the quantum spin-1 anisotropic ferromagnetic Heisenberg model in the presence of a crystal field. Within the Bogoliubov scheme an improved pair approximation has been used. The temperature-dependent thermodynamic functions have been obtained and provide much better results than the previous simple mean-field scheme. In one dimension, which is still nonintegrable for quantum spin-1, we get the exact results in the classical limit, or near-exact results in the quantum case, for the free energy, magnetization, and quadrupole moment, as well for the transition temperature. In two and three dimensions the corresponding global phase diagrams have been obtained as a function of the parameters of the Hamiltonian. First-order transition lines, second-order transition lines, tricritical and tetracritical points, and critical endpoints have been located through the analysis of the minimum of the Helmholtz free energy and a Landau-like expansion in the approximated free energy. Only first-order quantum transitions have been found at zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capel, and Ising models, have been analyzed and compared to previous results obtained from other analytical approaches as well as from Monte Carlo simulations.

Field theory of bicritical and tetracritical points. IV. Critical dynamics including reversible terms

This article concludes a series of papers [Folk, Holovatch, and Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)] where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here we give the complete two-loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described, in distances from the critical point accessible to experiments, by the critical behavior qualitatively found in one-loop order. Although one may conclude from the effective dynamic exponents (taking almost their one-loop values) that weak scaling for the order parameter components is valid, the flow of the time-scale ratios is quite different, and they do not reach their asymptotic values.

Polydispersity Stabilizes Biaxial Nematic Liquid Crystals

Inspired by the observations of a remarkably stable biaxial nematic phase [vandenPol etal., Phys. Rev. Lett. 103, 258301 (2009)], we investigate the effect of size polydispersity on the phase behavior of a suspension of boardlike particles. By means of Onsager theory within the restricted orientation (Zwanzig) model we show that polydispersity induces a novel topology in the phase diagram, with two Landau tetracritical points in between which oblate uniaxial nematic order is favored over the expected prolate order. Additionally, this phenomenon causes the opening of a huge stable biaxiality regime in between uniaxial nematic and smectic states.

Charge-density wave and superconductor competition in stripe phases of high-temperature superconductors

We discuss the problem of competition between a superconducting (SC) ordered state with a charge density wave (CDW) state in stripe phases of high $T_c$ superconductors. We consider an effective model for each stripe motivated by studies of spin-gapped electronic ladder systems. We analyze the problem of dimensional crossover arising from inter-stripe SC and CDW couplings using non-Abelian bosonization and renormalization group (RG) arguments to derive an effective $O(4)$-symmetric nonlinear $\sigma$-model in $D=2+1$ for the case of when both inter-stripe couplings are of equal magnitude as well as equally RG relevant. By studying the effects of various symmetry lowering perturbations, we determine the structure of the phase diagram and show that, in general, it has a broad regime in which both orders coexist. The quantum and thermal critical behavior is discussed in detail, and the phase coexistence region is found to end at associated $T=0$ as well as $T>0$ tetracritical points. The possible role of hedgehog topological excitations of the theory is considered and argued to be RG irrelevant at the spatially anisotropic higher dimensional low-energy fixed point theory. Our results are also relevant to the case of competing N\'eel and valence bond solid (VBS) orders in quantum magnets on 2D isotropic square as well as rectangular lattices interacting via nearest neighbor Heisenberg exchange interactions.

Publisher's Note: Field theory of bicritical and tetracritical points. III. Relaxational dynamics including conservation of magnetization (model C) [Phys. Rev. E79, 031109 (2009)

Publisher's Note: Field theory of bicritical and tetracritical points. III. Relaxational dynamics including conservation of magnetization (model C) [Phys. Rev. E<b>79</b>, 031109 (2009)

Field theory of bicritical and tetracritical points. III. Relaxational dynamics including conservation of magnetization (model C)

We calculate the relaxational dynamical critical behavior of systems of O(n_{ parallel}) plus sign in circleO(n_{ perpendicular}) symmetry including conservation of magnetization by renormalization group theory within the minimal subtraction scheme in two-loop order. Within the stability region of the Heisenberg fixed point and the biconical fixed point, strong dynamical scaling holds, with the asymptotic dynamical critical exponent z=2varphinu-1 , where varphi is the crossover exponent and nu the exponent of the correlation length. The critical dynamics at n_{ parallel}=1 and n_{ perpendicular}=2 is governed by a small dynamical transient exponent leading to nonuniversal nonasymptotic dynamical behavior. This may be seen, e.g., in the temperature dependence of the magnetic transport coefficients.

Field theory of bicritical and tetracritical points. I. Statics

We calculate the static critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by the renormalization group method within the minimal subtraction scheme in two-loop order. Summation methods lead to fixed points describing multicritical behavior. Their stability border lines in the space of the order parameter components n_|| and n_perpendicular and spatial dimension d are calculated. The essential features obtained already in two-loop order for the interesting case of an antiferromagnet in a magnetic field ( n_|| =1, n_perpendicular =2 ) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points, leading to very small transient exponents. We are also able to calculate the flow of static couplings, which allows us to consider the attraction region. Depending on the nonuniversal background parameters, the existence of different multicritical behavior (bicritical or tetracritical) is possible, including a triple point.

Field theory of bicritical and tetracritical points. II. Relaxational dynamics

We calculate the relaxational dynamical critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by renormalization group method within the minimal subtraction scheme in two-loop order. The three different bicritical static universality classes previously found for such systems correspond to three different dynamical universality classes within the static borderlines. The Heisenberg and the biconical fixed point lead to strong dynamic scaling whereas in the region of stability of the decoupled fixed point weak dynamic scaling holds. Due to the neighborhood of the stability border between the strong and the weak scaling dynamic fixed point to the dynamical stable fixed point a very small dynamic transient exponent of omega(Beta)_(v) =0.0044 is present in the dynamics for the physically important case n_|| =1 and n_perpendicular =2 in d=3 .

Dynamic phase transition and multicritical dynamic phase diagrams of the kinetic spin-3/2 Blume–Emery–Griffiths model with repulsive biquadratic coupling under a time-dependent oscillating external field

We extend our recent paper [O. Canko, B. Deviren, M. Keskin, J. Phys.: Condens. Mater 118 (2006) 6635] to present a study, within a mean-field approach, the stationary states of the kinetic spin-3/2 Blume–Emery–Griffiths model with repulsive biquadratic interaction under the presence of a time varying (sinusoidal) magnetic field. We found that the dynamic phase diagrams of the present work exhibit more complex, richer and more topological different types of phase diagrams than our recent paper. Especially, the obtained dynamic phase diagrams show the ferrimagnetic ( i) phase in addition to the ferromagnetic ± 3 / 2 ( f 3 / 2 ), ferromagnetic ± 1 / 2 ( f 1 / 2 ), antiquadrupolar or staggered ( a) and disordered ( d) phases, and the f 3 / 2 + i , f 1 / 2 + d , i + d , f 3 / 2 + i + d , a + d and/or f 3 / 2 + i + a coexistence regions in addition to the f 3 / 2 + f 1 / 2 , f 3 / 2 + d , f 3 / 2 + a , f 1 / 2 + d and/or f 3 / 2 + a + d coexistence regions, depending on interaction parameters. Moreover, the phase diagrams exhibit dynamic zero-temperature critical, critical end, double critical end, multicritical, and/or pentacritical special points in addition to the dynamic tricritical, double critical end point, triple, quadruple and/or tetracritical special points that depending on the interaction parameters.

Effects of long-range interaction on critical and multicritical behavior of compressible disordered systems

A field-theoretic approach is applied to describe behavior of homogeneous three-dimensional systems with long-range interactions defined by two order parameters at bicritical and tetracritical points. Renormalization- group equations are analyzed in the two-loop approximation by using the Pade-Borel summation technique. The fixed points corresponding to various types of multicritical behavior are determined. It is shown that effects due to long-range interactions can be responsible for a change from bicritical to tetracritical behavior.

Effect of long-range interactions on the multicritical behavior of homogeneous systems

A field-theoretic approach is applied to describe behavior of homogeneous three-dimensional systems with long-range interactions defined by two order parameters at bicritical and tetracritical points. Renormalization-group equations are analyzed in the two-loop approximation by using the Pade-Borel summation technique. The fixed points corresponding to various types of multicritical behavior are determined. It is shown that effects due to long-range interactions can be responsible for a change from bicritical to tetracritical behavior.