Bicritical Point Research Articles

Spherical model for a quantum spin glass.

Zero temperature phase transitions in the quantum rotor model of disordered, frustrated, and unfrustrated $n$-component short-range-coupled spin systems are investigated in the spherical limit, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We find a transition to a spin glass ordered phase in three dimensions but not in two. We map out the phase diagram as a function of frustration, finding that the ferromagnetic and spin glass critical lines meet at a bicritical point which also terminates a first order line separating the spin glass and ferromagnetic ordered phases. Finite size scaling of numerically obtained exact solutions on finite lattices is used to estimate critical exponents.

Mode interactions in reaction-driven convection in a porous medium

Convective-mode interaction occurring in reaction-driven convection in a porous medium is analyzed using local bifurcation theory. The system considered is a porous medium in the shape of a rectangular box with nonflux boundary conditions at the side walls and the bottom. From linear stability analysis it is found that vertical mode 1 (with one roll in the vertical direction) was always the first to destabilize. Hence, mode interactions occurring among various horizontal modes and the first vertical mode are considered. Center manifold theory is used to derive formulae for evaluating the coefficients of the amplitude equations of the normal form for a double zero singularity, directly from the original system of PDEs. Three different kinds of bicritical points are considered. For the case of interaction between mode 1 and mode 2 solutions, pure mode 1 solution does not exist and the presence of a secondary Hopf bifurcation and hence, oscillatory convection is predicted. From the simple zero and limit point interaction, it is shown that when the convective solution branches from the middle (unstable) conduction branch, a secondary Hopf bifurcation point exists on the convective branch leading to stable oscillatory convective solutions. Similar analysis done for mode 2 and mode 3 interaction shows that three types of solutions, pure mode 2, pure mode 3 along with a mixed mode 2 3 solution exist. These results indicate that mode interactions occurring in reaction-driven convection are profoundly different from those in Lapwood or Benard convection. A simple lower-dimensional, discretized version of the original model was developed and shown to have the same local behavior as the infinite-dimensional model. Finally, global bifurcation diagrams were calculated for the full system of equations using the orthogonal collocation method. It was found that even though the collocation method breaks the symmetry present in the problem, spurious solutions can be avoided by taking enough terms in the direction where symmetry is present.

Pressure Induced Twist Grain Boundary Phase

Abstract We report high pressure studies on a binary system of 4-(2′-methyl butyl)phenyl 4′-n-octyl biphenyl-4-carboxylate (CE8 from BDH) and 4-n- dodecyloxy biphenyl-4′-(2′-methyl butyl)benzoate (C12). For the concentration range 0.32<X<0.62, where X is the weight fraction of C12, the system shows the following sequence of transitions at room pressure: Isotropic- Cholesteric (Ch)-Twist Grain Boundary (TGB)-Smectic A (A)Smectic C− (C*). For X=0.25, the Ch phase transforms directly to the A phase at 1 bar, but there is an induced TGB phase between the Ch and A phases at elevated pressures. Analysis of the topology of the pressure-temperature diagram in the vicinity of the Ch-TGB-A meeting point indicates that is a critical end point rather than a bicritical point as expected from the mean field theory. At X=0.64 there is a direct TGB-C* transition at 1 bar, but there appears a pressure induced A phase between the TGB and C* phases. The topology of the phase diagram suggests that the TGB-A-C* meeting point ...

Bicritical point and crossover in a two-temperature, diffusive kinetic Ising model.

The phase diagram of a two-temperature kinetic Ising model which evolves by Kawasaki dynamics is studied using Monte Carlo simulations in dimension $d=2$ and solving a mean-spherical approximation in general $d$. We show that the equal-temperature (equilibrium) Ising critical point is a bicritical point where two nonequilibrium critical lines meet a first-order line separating two distinct ordered phases. The shape of the nonequilibrium critical lines is described by a crossover exponent, $\ensuremath{\phi}$, which we find to be equal to the susceptibility exponent, $\ensuremath{\gamma}$, of the Ising model.

Singularities near critical and bicritical end points: applications to an isomorphous transition

We study a system that has an isomorphous transition using an eighth-order Landau-type free energy function. The phase diagram of such system has a critical line that meets a first-order phase boundary at a critical end point. This phase boundary is also limited by two other critical lines that meet at a bicritical end point. We showed that this phase boundary close to the end point exhibits a nonanalytic behavior associated with the singularities of the critical line, confirming similar result we previously obtained using scaling arguments. Related universal amplitude ratios are then computed. Next, the critical lines near the bicritical end point are also analyzed and checked for singularities.

Magnetic neutron-scattering studies of RbMnBr3.

${\mathrm{RbMnBr}}_{3}$ is an XY-like antiferromagnet on a distorted stacked triangular lattice. At low temperatures and fields we show by neutron scattering that there is an incommensurate ordered structure with magnetic Bragg peaks at (h/8\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta},h/8\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta},l) where h and l are odd integers and \ensuremath{\delta}=0.0183\ifmmode\pm\else\textpm\fi{}0.0004. At low temperatures, a field of 3 T applied along the a axis produces a first-order phase transition to a commensurate structure corresponding to \ensuremath{\delta}=0, so that the lattice parameter of the magnetic cell is eight times that of the nuclear cell. In addition to these two ordered structures, there is another incommensurate ordered phase found at high temperatures. The magnetic phase diagram shows a tetracritical point at zero field and T=8.5\ifmmode\pm\else\textpm\fi{}0.1 K and a bicritical point at H=2.55\ifmmode\pm\else\textpm\fi{}0.05 T and T=7.8\ifmmode\pm\else\textpm\fi{}0.1 K. Inelastic neutron scattering was used to measure spin waves along the c axis and in the easy plane. The dispersion relations confirm that ${\mathrm{RbMnBr}}_{3}$ is quasi-one-dimensional, with much stronger exchange along c than in the ab plane.

Histogram Monte Carlo study of multicritical behavior in the hexagonal easy-axis Heisenberg antiferromagnet.

The results of a detailed histogram Monte-Carlo study of critical-fluctuation effects on the magnetic-field temperature phase diagram associated with the hexagonal Heisenberg antiferromagnet with weak axial anisotropy are reported. The multiphase point where three lines of continuous transitions merge at the spin-flop boundary exhibits a structure consistent with scaling theory but without the usual umbilicus as found in the case of a bicritical point.

Time-dependent patterns in counterrotating eccentric cylinders.

Nonlinear mode interactions and pattern formation occurring along the primary stability boundary for counterrotating eccentric cylinders have been studied experimentally. Counterrotating concentric cylinders produce either Taylor vortex flow or spiral vortices with various azimuthal wave numbers. Eccentric cylinders break the rotational symmetry of the base flow, introducing a delay of the onset of the primary instability and changes in the locations of the bicritical points separating patterns with differing azimuthal wave number. A theoretical model for this system is in qualitative agreement with the observations.

CRITICAL BEHAVIOR OF TARGET SPACE MEAN FIELD THEORY

Recently, the free energy of the target space mean field (TSMF) matrix model has been calculated in the low temperature phase, order-by-order in a low temperature expansion. The TSMF model is a matrix model whose discrete target space has an infinite coordination number, and whose free energy assumes a universal form, corresponding to baby universes joined into a tree. Here the free energy is summed to all orders, and expressed through a transcendental algebraic equation, using which we analyze the critical phenomena that occur in the TSMF model. There are two critical curves, at which the matter and the geometry become critical, and for which the critical exponents are [Formula: see text] and [Formula: see text], respectively. There is a bi-critical point where the curves meet, at which [Formula: see text].

New broken-parity state and a transition to anomalous lamellae in eutectic growth.

We discover that a eutectic system exhibits pseudobicritical features corresponding to the simultaneous birth of two different broken-parity states. The first branch is the known tilted-growth mode, global parity breaking; it bifurcates from the usual lamellar symmetric state. At approximately the same criticial point, a second brach merges close to the second fold we found previously. The two states thus bifurcate from different basic states. This new branch owes its existence to the underlying degeneracy of the usual broken-parity state. It consists of a structure where one lamella assumes a right-traveling and the other a left-traveling state. As a consequence, the drift velocity (or equivalently the tilt angle) is smaller for the new branch. Close to the bicritical point, the bifurcation is described by a Landau theory with the tilt angle being the order parameter. From general considerations we can state that the new branch is locally stable in the kinetic sense, but less stable than the usual branch. This result is consistent with the conventional criterion based on comparison of undercooling, since the new branch has a higher undercooling. We propose a simple experimental protocol to have access to the new state. Finally, when the two lamellae have equivalent properties (a symmetric case) the new state is not traveling while each lamella is asymmetric with respect to its center and is a mirror image of the other lamella. This corresponds to the so-called anomalous cells observed in noneutectic systems. We develop an analytic theory, in the same spirit as the one used for the parity breaking, to account for the transition to the anomalous state. The results are in qualitative agreement with the full numerical calculation.

The bicritical phase diagram of two-dimensional antiferromagnets with and without random fields

Neutron scattering measurements have been made of the phase diagrams of the nearly two-demensional antiferromagnets Rb2MnF4 and Rb2Mn0.7Mg0.3F4 in a magnetic field applied along thec-axis. In Rb2MnF4 there is at low temperatures a spin-flop phase at fields above 5.5 T which has long range order. The observation of true long range order rather than the algebraic decay of the order characteristic of the two-dimensional XY model is presumably due to subtle anisotropy effects in the plane as well as weak three-dimensional coupling. The phase boundaries of the uniaxial and transverse phases are shown to be consistent with renormalization group predictions for two-dimensional systems. The two lines become exponentially close to each other at low temperatures. The weak three-dimensional coupling moves the bicritical point fromT=0 to a non-zero temperature. The situation is more complex in Rb2Mn0.7Mg0.3F4 because of Ising random field effects. At low fields we observe typical random field metastable behavior with a sharp metastability boundary and a gange of length scales which are time independent below that boundary. At higher fields there are substantial uniaxial fluctuations. The transverse phase boundary and the metastability line appear to intercept atT=0 showing that the random field fluctuations do have a large effect on the phase diagram. The theory of the phase diagrams has been extended to include the random field fluctuations and good agreement is obtained with the observed transverse phase boundary. Unfortunately, there is as yet no theory of the metastable uniaxial phase with which to compare our results.

Bicritical points in an exactly solvable model

Within the framework of the exactly solvable model, which takes into account the interaction of fluctuating modes with equal and opposite momenta, we show the possibility of fluctuation induced phase transitions of the first order in systems with coupled scalar order parameters.

Stability of high-speed compressible rotating Couette flow

In this study, the linear stability of high-speed, rotating Couette flow to two- and three-dimensional disturbances in finite-gap spacings, including the full effects of compressibility and viscosity, is considered. Particularly, the combined effects of Mach number, Reynolds number, radial heating, and gap spacing are investigated. For a stationary outer cylinder, the primary instability is an axisymmetric mode independent of the Mach number. Increasing Mach numbers have a destabilizing effect for wide gaps, and a stabilizing effect for narrow gaps. For a sufficiently fast, counter-rotating outer cylinder, the primary instability becomes a three-dimensional traveling wave. Compressibility has a stabilizing effect on these modes regardless of the gap width; also, heating at the outer cylinder stabilizes the flow. Bicritical points for the primary instability corresponding to the crossover of the azimuthal wave numbers are determined for cylinders counter-rotating with equal angular speed.

Instabilities in two-dimensional spatially periodic flows. Part II: Square eddy lattice

A numerical study is performed on the linear stability of square arrays of alternating vortices with emphasis on confinement effects and linear friction. The appearance of a global rotation as a result of instability in a strongly confined system consisting of four vortices is explained in terms of the first unstable mode and the calculated critical parameter is found to be in agreement with magnetohydrodynamic experiments. The existence of bicritical points in systems with more than four vortices is demonstrated. Stability of the unbounded system is treated in the framework of Floquet theory. It is found that, in the presence of linear friction, the basic flow is unstable with respect to quasiperiodic perturbations with generally incommensurable spatial frequencies. This lays the foundation for a hypothesis about transition to turbulence via spatial quasiperiodicity.

Dynamics of feigenbaum systems with unidirectional coupling near onset of chaos. The bicritical attractor

It is proposed that a bicritical point exists in the plane of the control parameters of two logistic maps with unidirectional coupling where the lines of transition to chaos in both subsystems converge. The global scaling properties (σ-functions, f(α) spectra, and generalized dimensions) of bicritical dynamics are examined. It is shown that bicriticality can also be observed in chains of more than two cells.

Faraday-optical study of the tri-to-bicritical crossover in anisotropic heisenberg antiferromagnets Fe1-xNixCl2, 0.18 ≤x≤ 0.67

Abstract The magnetic phase diagrams, H-vs-T and M-vs-T, of solid solutions Fe1-x Ni x Cl2, 0.18 ≤ x ≤ 0.67, are investigated using Faraday rotation measurements. Bicritical behavior is observed for x > 0.57, whereas tricritical behavior is encountered for x < 0.57. In the low-x regime a critical end-point of an additional low-T second-order spin flop-to-paramagnetic phase boundary seems to emerge linearly with x. It coalesces with the tricritical point at x = 0.57, forming a “new multicritical point” of higher order as predicted by Galam and Aharony. This merges into the bicritical point in the high-x regime.

Lattice instabilities and the effect of copper-oxygen-sheet distortions on superconductivity in doped La2CuO4.

Synchrotron x-ray and neutron-diffraction measurements demonstrate that ${\mathrm{Nd}}^{3+}$ substitution at the ${\mathrm{La}}^{3+}$ site in both metallic and insulating ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ induces low-temperature structural instabilities which reorient the tilting of the copper-oxygen octahedra. These structural transformations strongly affect superconductivity. In addition, there is a bicritical point involving phases of Bmab, Pccn, and P${4}_{2}$/ncm space-group symmetries in the structural phase diagram of ${\mathrm{La}}_{1.6\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Nd}}_{0.4}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ at x\ensuremath{\sim}0.15. There are also anomalies in the structural transformation temperatures and the superconducting transition temperatures when x\ensuremath{\sim}1/8, providing evidence for an electronic instability at this doping level.

Mn1-scrt Crscrt As under pressure: Competition between magnetic orderings.

Pressure phase diagrams for the metallic magnet system ${\mathrm{Mn}}_{1\mathrm{\ensuremath{-}}\mathit{s}\mathit{c}\mathit{r}\mathit{t}}$ ${\mathrm{Cr}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{t}}$ As are determined for 0.25\ensuremath{\le}scrt\ensuremath{\le}0.42, T&gt;77 K, and scrP10 kbar under fully hydrostatic conditions, and with a pressure resolution of ${10}^{\mathrm{\ensuremath{-}}2}$ kbar. Three magnetically ordered phases [ferromagnetic F, helimagnetic ${\mathit{H}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{c}}$, and helimagnetic ${\mathit{H}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{a}}$ (close to F in behavior)] exist within a certain, narrow (scrP,T,scrt) domain. The ordering temperatures of the F and ${\mathit{H}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{a}}$ phases increase with pressure at a rate of up to 11 K/kbar. All the four possible types of triple points are experimentally studied. The ${\mathit{H}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{a}}$--F--P triple point is an approximate realization of a Lifshitz point, as verified by an incomplete divergence of the susceptibility on approaching the F phase. The ${\mathit{H}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{c}}$--F--P and ${\mathit{H}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{c}}$--${\mathit{H}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{a}}$--P triple points show large umbilicuslike anomalies, qualitatively similar to those of bicritical points in the field phase diagrams of antiferromagnets.

Thermodynamic considerations and the phase diagram of superconducting UPt3.

In this paper we study the thermodynamics of bicritical and polycritical points, where at least two or three second-order phase-transition lines intersect, respectively. The results are applied to the superconductivity phase diagram of ${\mathrm{UPt}}_{3}$.

Crossover at a bicritical point: Behavior of the hard susceptibility

We show how to use the minimal subtraction scheme and asymmetric renormalization procedure to calculate the crossover behavior of the hard susceptibility in the case of an anisotropic system near a bicritical point. The effective exponent governing the singular temperature dependence of the hard susceptibility changes from γ N to 1− α M as T → T c , where N is the total number of spin components, M is the number of soft components, γ N is the O( N) susceptibility exponent and α M is the O( M) specific heat exponent. The full crossover function is obtained using a trajectory integral approach.