Bicritical Point Research Articles

On the scaling properties of two unidirectionally coupled period-doubling systems in the presence of noise

The results of numerical experiments reveal and illustrate the scaling properties manifested under the noise action at a bicritical point situated at the chaos boundary in a system of two unidirectionally coupled subsystems featuring the period doubling.

Critical behaviour of driven bilayer systems: a field-theoretic renormalization group study

We investigate the static and dynamic critical behaviour of a uniformly driven bilayer Ising lattice gas at half filling. Depending on the strength of the interlayer coupling J, phase separation occurs across or within the two layers. The former transitions are controlled by the universality class of model A (corresponding to an Ising model with Glauber dynamics), with upper critical dimension d_c=4. The latter transitions are dominated by the universality class of the standard (single-layer) driven Ising lattice gas, with d_c=5 and a non-classical anisotropy exponent. These two distinct critical lines meet at a non-equilibrium bicritical point which also falls into the driven Ising class. At all transitions, novel couplings and dangerous irrelevant operators determine corrections to scaling.

Critical behavior of a lattice prey-predator model.

The critical properties of a simple prey-predator model are revisited. For some values of the control parameters, the model exhibits a line of directed percolationlike transitions to a single absorbing state. For other values of the control parameters one finds a second line of continuous transitions toward an infinite number of absorbing states, and the corresponding steady-state exponents are mean-field-like. The critical behavior of the special point T (bicritical point), where the two transition lines meet, belongs to a different universality class. A particular strategy for preparing the initial states used for the dynamical Monte Carlo method is devised to correctly describe the physics of the system near the second transition line. Relationships with a forest fire model with immunization are also discussed.

Bicritical and Tetracritical Phenomena and Scaling Properties of the SO(5) Theory

By Monte Carlo simulations on a 3D SO(5) rotator model it is shown that, in contrast to the epsilon expansions of the renormalization group, the bicritical point is stable to biquadratic perturbations of AF-SC (antiferromagnetism-superconductivity) repulsions, which are produced by quantum fluctuations originated from the Gutzwiller projection. Therefore, the present work completes the link from a physical projected SO(5) starting point to an asymptotic SO(5) symmetry point. The tetracritical point is stable for attractive AF-SC interactions. Critical exponents and ratios are evaluated by scaling analysis. Bicritical and tetracritical scaling functions are derived for the first time. Suggestions on experiments are given.

Electronic Isomerism: Symmetry Breaking and Electronic Phase Diagrams for Diatomic Molecules at the Large-Dimension Limit

We present symmetry-breaking and electronic-structure phase diagrams for two-center molecules with one and two electrons in the limit of a space of large dimensions. For one electron, the phase diagram in the internuclear distance-nuclear charge (R-Z) plane has two different stable phases. One corresponds to the electron equidistant from the two nuclei; the other where the electron is localized on one of the nuclei. The phase diagram for two electrons with two equally charged centers shows three different stable phases corresponding to different electronic-structure configurations. This phase diagram is characterized by a bicritical point. When the charges are unequal, the phase diagram shows only two stable phases, covalent and ionic. This phase diagram is characterized by a tricritical point, where the first-order transition line meets with the second-order transition line. The role of the inter-electron Coulombic repulsion in giving rise to different electronic structures and the distinction between a continuous deformation of one structure into another versus a discontinuous, so-called first-order, transition, where two isomers can coexist, are emphasized. The connection to the spectroscopic notion of intersecting potential energy curves is discussed.

Universal Bicritical Behavior of Period Doublings in Unidirectionally Coupled Oscillators

We study the bicritical behavior of period doublings in unidirectionally coupled oscillators to confirm the universality of the bicriticality in an abstract system of two unidirectionally coupled one-dimensional (1D) maps. A transition to hyperchaos occurs (i.e., a hyperchaotic attractor with two positive Lyapunov exponents appears) when crossing a bicritical point where two Feigenbaum critical lines of a period-doubling transition to chaos in the two subsystems meet. Using both a “residue-matching” renormalization group method and a direct numerical method, we make an analysis of the scaling behavior near the bicritical point. It is thus found that the second response subsystem exhibits a new type of non-Feigenbaum critical behavior, while the first drive subsystem is in the usual Feigenbaum critical state. Note that the bicritical scaling behavior is the same as that in the unidirectionally coupled 1D maps. We thus suppose that bicriticality may be observed generally in real systems, consisting of period-doubling subsystems with a unidirectional coupling.

Bicritical scaling behavior in unidirectionally coupled oscillators.

We study the scaling behavior of period doublings in a system of two unidirectionally coupled parametrically forced pendulums near a bicritical point where two critical lines of period-doubling transition to chaos in both subsystems meet. When crossing a bicritical point, a hyperchaotic attractor with two positive Lyapunov exponents appears, i.e., a transition to hyperchaos occurs. Varying the control parameters of the two subsystems, the unidirectionally coupled parametrically forced pendulums exhibit multiple period-doubling transitions to hyperchaos. For each transition to hyperchaos, using both a "residue-matching" renormalization group method and a direct numerical method, we make an analysis of the bicritical scaling behavior. It is thus found that the second response subsystem exhibits a new type of non-Feigenbaum scaling behavior, while the first drive subsystem is in the usual Feigenbaum critical state. The universality of the bicriticality is also examined for several different types of unidirectional couplings.

Renormalization-group approach to surface critical behavior in the semi-infinite spin-32Blume-Emery-Griffiths model

Using a renormalization-group transformation in position space based on the Migdal-Kadanoff recursion relations, we investigate the three-dimensional semi-infinite cubic spin-$\frac{3}{2}$ Blume-Emery-Griffiths model with single-site uniaxial anisotropy and nearest-neighbor pair interactions, both bilinear and biquadratic. The fixed-point topologies and corresponding phase diagrams predicted for this model are presented. Attention is focused on the renormalization-group description of the surface critical behaviors, in comparison with the mean-field approximation results. Three main classes of system phase diagrams are determined, featuring a rich variety of phase transitions associated with the surface with type being according to the values of bulk and surface interaction parameters. Surface critical, tricritical, bicritical, and multicritical points are also found.

How to experimentally measure the number 5 of the SO(5) theory?

According to Wilson's theory of critical phenomena, critical exponents are universal functions of d, the dimension of space, and n, the dimension of the symmetry group. SO(5) theory of antiferromagnetism and superconductivity predicts a bicritical point where T N and T c intersect. By measuring critical exponents close to the bicritical point, and knowing that d = 3, one can experimentally measure the number 5 of the SO(5) theory.

Phase diagram of the quantum SO(5) symmetry model for high-T C superconductivity

We have performed quantitative analysis of the SO(5) quantum model describing low energy degrees of freedom in the high-T C superconductors. Employing spherical approximation, we have calculated temperature vs. chemical-potential phase diagram of the system. On the simple cubic lattice it exhibits in general case two second-order quantum phase transition lines (corresponding to antiferromagnetic (AF) and superconducting (SC) phases with an intermediate quantum disordered phase). Depending on system parameters AF and SC states may meet in the zero-temperature quantum critical point or in the bicritical point with first-order phase transition separating AF and SC states. The nature of the system ground state was also investigated.

Subcritical bifurcations and nonlinear balloons in faraday waves

Bicritical points at wave numbers k(b) larger than the critical wave numbers k(c) are found in parametric surface waves (Faraday waves) using both numerical simulations and nonlinear analysis. Because k(b)-k(c) is small, it is argued that subcritical bifurcations at k>k(b) can be easily observed in experiments. In the second part we present a generic argument predicting the existence of nonlinear states resembling a balloon outside the instability region. The prediction is confirmed in simulations and it is argued to apply to other systems with similar instability curves.

Multicritical Phenomena of Superconductivity and Antiferromagnetism in Organic Conductorκ-(BEDT-TTF)2X

We study theoretically the multicritical phenomena of the superconductivity (SC) and antiferromagnetism (AF) in organic conductors $\kappa$-BEDT salts. The phase diagram and the experimental data on the NMR relaxation rate $1/T_1$ is analysed in terms of the renormalization group method. The bicritical phenomenon observed experimentally indicates the rotational symmetry, i.e., SO(5) symmetry, of the SC and the AF. The critical exponent $x$ for the divergence of $1/T_1$ is well explained by $ x = \nu (z - 1 - \eta)$ with the dynamical exponent $z = 3/2$ for the AF region while $z = \phi/\nu \sim 1.84 $ at the bicritical point. These results strongly suggest that the origin of the SC is in common with that of the AF and that its symmetry is d-wave.

Spin-gap phenomenon by an SO(5) model

Thermodynamic properties of the SO(5) theory for the high- T c superconductivity (SC) are explored by means of Monte Carlo simulations on a classical model hamiltonian. Temperature versus doping rate phase diagram with a bicritical point is obtained. Enhancement of antiferromagnetism (AF) correlations is observed above the SC critical temperature. Its relation with the spin-gap phenomenon is addressed.

Monte Carlo investigation of the tricritical point stability in a three-dimensional Ising metamagnet

We use Monte Carlo simulations to study multicritical properties of an Ising metamagnet in an external field. According to the mean field theory predictions, a three-dimensional layered metamagnet is expected to display a tricritical point decomposition to a critical and a bicritical end point, when a ratio between intralayer ferromagnetic and interlayer antiferromagnetic couplings becomes sufficiently small. Our simulations show no evidence of such a decomposition and produce a tricritical behavior even for a coupling ratio as small as $R=0.01.$

Biaxial nematic liquid crystal phases of a soft ellipsoid-string fluid

We have calculated the phase diagram of a fluid, the molecules of which are composed of a string of partially overlapping oblate soft ellipsoid interaction sites, by molecular dynamics simulation. The molecules are elongated in one direction and flattened in the perpendicular direction so that they become biaxial. We find that when the fluid consisting of 8-site or 9-site molecules is compressed, a discotic uniaxial nematic phase is formed. Further compression causes a transition to a biaxial phase. The same phase behaviour is found by compressing a fluid composed of molecules with 10 or 11 sites, but in this case the uniaxial phase is calamitic. The 9-site and 10-site molecules consequently bracket the Landau bicritical point where there is a continuous transition between the isotropic phase and the biaxial nematic phase. This behaviour is consistent with the Landau–de Gennes theory and mean field theories. However, the molecular dimensions for which the Landau point appears differ from the theoretical predictions of the mean field theories.

Bicritical behavior of period doublings in unidirectionally coupled maps.

We study the scaling behavior of period doublings in two unidirectionally coupled one-dimensional maps near a bicritical point where two critical lines of period-doubling transition to chaos in both subsystems meet. Note that the bicritical point corresponds to a border of chaos in both subsystems. For this bicritical case, the second response subsystem exhibits a type of non-Feigenbaum critical behavior, while the first drive subsystem is in the Feigenbaum critical state. Using two different methods, we make the renormalization-group analysis of the bicritical behavior and find the corresponding fixed point of the renormalization transformation with two relevant eigenvalues. The scaling factors obtained by the renormalization-group analysis agree well with those obtained by a direct numerical method.

Landau theory of bicriticality in a random quantum rotor system

We consider here a generalization of the random quantum rotor model in which each rotor is characterized by an M-component vector spin. We focus entirely on the case not considered previously, namely when the distribution of exchange interactions has non-zero mean. Inclusion of non-zero mean permits ferromagnetic and superconducting phases for M=1 and M=2, respectively. We find that quite generally, the Landau theory for this system can be recast as a zero-mean problem in the presence of a magnetic field. Naturally then, we find that a Gabay-Toulouse line exists for $M>1$ when the distribution of exchange interactions has non-zero mean. The solution to the saddle point equations is presented in the vicinity of the bi-critical point characterized by the intersection of the ferromagnetic (M=1) or superconducting (M=2) phase with the paramagnetic and spin glass phases. All transitions are observed to be second order. At zero temperature, we find that the ferromagnetic order parameter is non-analytic in the parameter that controls the paramagnet/ferromagnet transition in the absence of disorder. Also for M=1, we find that replica symmetry breaking is present but vanishes at low temperatures. In addition, at finite temperature, we find that the qualitative features of the phase diagram, for M=1, are {\it identical} to what is observed experimentally in the random magnetic alloy $LiHo_xY_{1-x}F_4$.

Phase Diagram of a Superconducting and Antiferromagnetic System with SO(5) Symmetry

Temperature vs. chemical-potential phase diagrams of an SO(5) model for high-(T_c) cuprates are calculated by Monte Carlo simulation. There is a bicritical point where the second-order antiferromagnetism (AF) and superconductivity transition lines merge tangentially into a first-order line, and the SO(5) symmetry is achieved. In an external magnetic field, the AF ordering is first order in the region where the first-order melting line of flux lattice joins in. There is a tricritical point on the AF transition line from which the AF ordering becomes second order.

Spin Correlations in an Isotropic Spin-5/2Two-Dimensional Antiferromagnet

We report a neutron scattering study of the spin correlations for the spin- $5/2$ two-dimensional antiferromagnet $\mathrm{Rb}{}_{2}\mathrm{MnF}{}_{4}$ in an external magnetic field. Choosing fields near the system's bicritical point, we tune the effective anisotropy in the spin interaction to zero, constructing an ideal $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5/2$ Heisenberg system. The correlation length and structure factor amplitude are closely described by the semiclassical theory of Cuccoli et al. over a broad temperature range, but show no indication of approaching the low-temperature renormalized classical regime of the quantum nonlinear sigma model.

PHASE DIAGRAMS OF SYSTEMS WITH TWO COUPLED ORDER PARAMETERS

Within the framework of an exactly solvable model, which takes into account only the interaction of fluctuating modes with equal and opposite momenta, we study phase diagrams in systems with coupled scalar order parameters. We show that, in agreement with renormalization group theory, the fluctuation interaction can split the continuous order–disorder transition locus into two phase transitions of first order, replacing a bicritical point by a triple point and two tricritical points. The effect disappears when the fluctuation interaction is reduced.