Real Order Parameter Research Articles

Comparative study on phase transition behaviors of fractional molecular field theory and random-site Ising model

Fractional molecular field theory (FMFT) is a phenomenological theory that describes phase transitions in crystals with randomly distributed components, such as the relaxor-ferroelectrics and spin glasses. In order to verify the feasibility of this theory, this paper fits it to the Monte Carlo simulations of specific heat and susceptibility versus temperature of two-dimensional (2D) random-site Ising model (2D-RSIM). The results indicate that the FMFT deviates from the 2D-RSIM significantly. The main reason for the deviation is that the 2D-RSIM is a typical system of component random distribution, where the real order parameter is spatially heterogeneous and has no symmetry of space translation, but the basic assumption of FMFT means that the parameter is spatially uniform and has symmetry of space translation.

The role of multiplicative noise in critical dynamics

We study the role of multiplicative stochastic processes in the description of the dynamics of an order parameter near a critical point. We study equilibrium as well as out-of-equilibrium properties. By means of a functional formalism, we build the Dynamical Renormalization Group equations for a real scalar order parameter with Z2 symmetry, driven by a class of multiplicative stochastic processes with the same symmetry. We compute the flux diagram using a controlled ϵ-expansion, up to order ϵ2. We find that, for dimensions d=4−ϵ, the additive dynamic fixed point is unstable. The flux runs to a multiplicative fixed point driven by a diffusion function G(ϕ)=1+g∗ϕ2(x)/2, where ϕ is the order parameter and g∗=ϵ2/18 is the fixed point value of the multiplicative noise coupling constant. We show that, even though the position of the fixed point depends on the stochastic prescription, the critical exponents do not. Therefore, different dynamics driven by different stochastic prescriptions (such as Itô, Stratonovich, anti-Itô and so on) are in the same universality class.

On synchronization in Kuramoto models on spheres

We analyze two classes of Kuramoto models on spheres that have been introduced in previous studies. Our analysis is restricted to ensembles of identical oscillators with the global coupling. In such a setup, with an additional assumption that the initial distribution of oscillators is uniform on the sphere, one can derive equations for order parameters in closed form. The rate of synchronization in a real Kuramoto model depends on the dimension of the sphere. Specifically, synchronization is faster on higher-dimensional spheres. On the other side, real order parameter in complex Kuramoto models always satisfies the same ODE, regardless of the dimension. The derivation of equations for real order parameters in Kuramoto models on spheres is based on recently unveiled connections of these models with geometries of unit balls. Simulations of the system with several hundreds of oscillators yield perfect fits with the theoretical predictions, that are obtained by solving equations for the order parameter.

Swift-Hohenberg equation with third-order dispersion for optical fiber resonators

We investigate the dynamics of a ring cavity made of photonic crystal fiber and driven by a coherent beam working near to the resonant frequency of the cavity. By means of a multiple-scale reduction of the Lugiato-Lefever equation with high-order dispersion, we show that the dynamics of this optical device, when operating close to the critical point associated with bistability, is captured by a real order parameter equation in the form of a generalized Swift-Hohenberg equation. A Swift-Hohenberg equation has been derived for several areas of nonlinear science such as chemistry, biology, ecology, optics, and laser physics. However, the peculiarity of the obtained generalized Swift-Hohenberg equation for photonic crystal fiber resonators is that it possesses a third-order dispersion. Based on a weakly nonlinear analysis in the vicinity of the modulational instability threshold, we characterize the motion of dissipative structures by estimating their propagation speed. Finally, we numerically investigate the formation of moving temporal localized structures often called cavity solitons.

Critical behavior of the QED3 -Gross-Neveu-Yukawa model at four loops

We study the universal critical properties of the QED$_3$-Gross-Neveu-Yukawa model with $N$ flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the $\epsilon$ expansion below four dimensions. For $N=1$, the model is conjectured to be infrared dual to the $SU(2)$-symmetric noncompact $\mathbb{C}$P$^1$ model, which describes the deconfined quantum critical point of the N\'eel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For $N=2$, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagom\'e antiferromagnet. For general $N$ we determine the order parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of $SU(N)$ singlet and adjoint fermion mass bilinears at the critical point. We use Pad\'e approximants to obtain estimates of critical properties in 2+1 dimensions.

Superconductivity in quantum wires: A symmetry analysis

We study properties of quantum wires with spin–orbit coupling and time reversal symmetry breaking, in normal and superconducting states. Electronic band structures are classified according to quasi-one-dimensional magnetic point groups, or magnetic classes. The latter belong to one of three distinct types, depending on the way the time reversal operation appears in the group elements. The superconducting gap functions are constructed using antiunitary operations and have different symmetry properties depending on the type of the magnetic point group. We obtain the spectrum of the Andreev boundary modes near the end of the wire in a model-independent way, using the semiclassical approach with the boundary conditions described by a phenomenological scattering matrix. Explicit expressions for the bulk topological invariants controlling the number of the boundary zero modes are presented in the general multiband case for two types of the magnetic point groups with real order parameters, corresponding to DIII and BDI symmetry classes.

Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model.

We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second-order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial Gaussian pumping beam and subjected to time-delayed feedback. The Gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assuming a continuous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use an order parameter approach to derive a normal form of the delay-induced Hopf bifurcation leading to an oscillating solution. Additionally we model the interplay of an attracting inhomogeneity and destabilizing time delay by describing the localized solution as an overdamped particle in a potential well generated by the inhomogeneity. In this case, the time-delayed feedback acts as a driving force. Comparing results from the later approach with the full Swift-Hohenberg model, we show that the approach not only provides an instructive description of the depinning dynamics, but also is numerically accurate throughout most of the parameter regime.

Soliton-induced Majorana fermions in a one-dimensional atomic topological superfluid

We theoretically investigate the behavior of dark solitons in a one-dimensional spin-orbit coupled atomic Fermi gas in harmonic traps by solving self-consistently the Bogoliubov--de Gennes equations. The dark soliton---to be created by phase imprinting in future experiments---is characterized by a real order parameter, which changes sign at a point node and hosts localized Andreev bound states near the node. By considering both cases of a single soliton and multiple solitons, we find that the energy of these bound states decreases to zero when the system is tuned to enter the topological superfluid phase by increasing an external Zeeman field. As a result, two Majorana fermions emerge in the vicinity of each soliton, in addition to the well-known Majorana fermions at the trap edges associated with the nontrivial topology of the superfluid. We propose that the soliton-induced Majorana fermions can be directly observed by using spatially resolved radio-frequency spectroscopy or indirectly probed by measuring the density profile at the point node. For the latter, the deep minimum in the density profile will disappear due to the occupation of the soliton-induced zero-energy Majorana fermion modes. Our prediction could be tested in a resonantly interacting spin-orbit coupled ${}^{40}\text{K}$ Fermi gas confined in a two-dimensional optical lattice.

Fractional models of anomalous relaxation based on the Kilbas and Saigo function

We revisit the Kilbas and Saigo functions of the Mittag-Leffler type of a real variable $$t$$ , with two independent real order-parameters. These functions, subjected to the requirement to be completely monotone for $$t>0$$ , can provide suitable models for the responses and for the corresponding spectral distributions in anomalous (non–Debye) relaxation processes, found e.g. in dielectrics. Our analysis includes as particular cases the classical models referred to as Cole–Cole (the one-parameter Mittag-Leffler function) and to as Kohlrausch (the stretched exponential function). After some remarks on the Kilbas and Saigo functions, we discuss a class of fractional differential equations of order $$\alpha \in (0,1]$$ with a characteristic coefficient varying in time according to a power law of exponent $$\beta $$ , whose solutions will be presented in terms of these functions. We show 2D plots of the solutions and, for a few of them, the corresponding spectral distributions, keeping fixed one of the two order-parameters. The numerical results confirm the complete monotonicity of the solutions via the non-negativity of the spectral distributions, provided that the parameters satisfy the additional condition $$0<\alpha +\beta \le 1$$ , assumed by us.

Dynamical symmetry breaking with optimal control: Reducing the number of pieces

We analyse the production of defects during the dynamical crossing of a mean-field phase transition with a real order parameter. When the parameter that brings the system across the critical point changes in time according to a power-law schedule, we recover the predictions dictated by the well-known Kibble-Zurek theory. For a fixed duration of the evolution, we show that the average number of defects can be drastically reduced for a very large but finite system, by optimising the time dependence of the driving using optimal control techniques. Furthermore, the optimised protocol is robust against small fluctuations.

Quantum phase transition to unconventional multi-orbital superfluidity in optical lattices

Orbital physics plays a significant role for a vast number of important phenomena in complex condensed matter systems such as high-T$_c$ superconductivity and unconventional magnetism. In contrast, phenomena in superfluids -- especially in ultracold quantum gases -- are commonly well described by the lowest orbital and a real order parameter. Here, we report on the observation of a novel multi-orbital superfluid phase with a {\it complex} order parameter in binary spin mixtures. In this unconventional superfluid, the local phase angle of the complex order parameter is continuously twisted between neighboring lattice sites. The nature of this twisted superfluid quantum phase is an interaction-induced admixture of the p-orbital favored by the graphene-like band structure of the hexagonal optical lattice used in the experiment. We observe a second-order quantum phase transition between the normal superfluid (NSF) and the twisted superfluid phase (TSF) which is accompanied by a symmetry breaking in momentum space. The experimental results are consistent with calculated phase diagrams and reveal fundamentally new aspects of orbital superfluidity in quantum gas mixtures. Our studies might bridge the gap between conventional superfluidity and complex phenomena of orbital physics.

Order parameters from image analysis: a honeycomb example

Honeybee combs have aroused interest in the ability of honeybees to form regular hexagonal geometric constructs since ancient times. Here we use a real space technique based on the pair distribution function (PDF) and radial distribution function (RDF), and a reciprocal space method utilizing the Debye-Waller Factor (DWF) to quantify the order for a range of honeycombs made by Apis mellifera ligustica. The PDFs and RDFs are fit with a series of Gaussian curves. We characterize the order in the honeycomb using a real space order parameter, OP ( 3 ), to describe the order in the combs and a two-dimensional Fourier transform from which a Debye-Waller order parameter, u, is derived. Both OP ( 3 ) and u take values from [0, 1] where the value one represents perfect order. The analyzed combs have values of OP ( 3 ) from 0.33 to 0.60 and values of u from 0.59 to 0.69. RDF fits of honeycomb histograms show that naturally made comb can be crystalline in a 2D ordered structural sense, yet is more 'liquid-like' than cells made on 'foundation' wax. We show that with the assistance of man-made foundation wax, honeybees can manufacture highly ordered arrays of hexagonal cells. This is the first description of honeycomb utilizing the Debye-Waller Factor, and provides a complete analysis of the order in comb from a real-space order parameter and a reciprocal space order parameter. It is noted that the techniques used are general in nature and could be applied to any digital photograph of an ordered array.

Phase transitions in low-dimensional ψ 4 systems

The phase transition in a planar array of weakly coupled Ginzburg–Landau chains with real order parameter is studied, using an original variant of the two-level approximation. The results are extended to the quantum phase transition in a chain of quantum double well oscillators, coupled with an elastic interaction, using the matrix transfer method.

Ginzburg–Landau theory beyond the linear-chain approximation: the 2D case

Within the Ginzburg–Landau theory applied to a planar array of chains, characterizedby a real order parameter, the inter-chain fluctuations are taken into accountexactly. So, the ‘linear chain approximation’ is replaced by a rigorous treatment.The single-chain problem is addressed taking advantage of a precise and simplenon-perturbative solution of the Schrödinger equation for the anharmonic oscillator.

Critical dynamics of stochastic models with two conserved densities (modelC′)

We calculate the field-theoretic functions of the generalized dynamical model C(*') , where two conserved secondary densities are coupled to a nonconserved complex order parameter (OP), in two-loop order. A transformation to "orthogonalized" densities can be performed where only one secondary density with nontrivial static coupling to the OP exists while the second one remains Gaussian. The secondary densities remain dynamically coupled by the nondiagonal diffusion coefficent. General relations for the field-theoretic functions allow us to relate the asymptotic critical properties of model C(*') to the simpler model C(*) with only one conserved density coupled to the OP. The nonasymptotic properties, however, differ as can be seen from the flow of the dynamic parameters, which is presented for the case of a real OP with componets n=1,2,3 .

Optical patterns with different wavelengths.

The semiconductor resonator is an example of an optical system where two modulational instabilities with different wave numbers coexist. In the limit of nascent bistability, the dynamics is generically described by a nonvariational real order parameter equation, of which we give a detailed derivation. This considerably simplifies the linear and weakly nonlinear stability analyses. When the two instabilities are close together, we derive normal form equations and put special emphasis on "envelope" branches of solutions. These particular solutions may connect the two instability points or form an isola. On the basis of these rigorous results, we finally discuss the case of distant modulational instabilities, in both one and two transverse dimensions.

Moving localized structures and spatial patterns in quadratic media with a saturable absorber

For near the first lasing threshold, we give a detailed derivation of a real order parameter equation for the degenerate optical parametric oscillator with a saturable absorber. For this regime, we study analytically the role of the quasi-homogeneous neutral mode in the pattern formation process. We show that this effect stabilized the hexagonal patterns below the lasing threshold. More importantly, we find numerically that when Turing and Hopf bifurcations interact, a stable moving asymmetric localized structure with a constant transverse velocity is generated. The formation of the moving localized structures is analysed for both the propagation and the mean field models. A quantitative confrontation of the two models is discussed.

Critical dynamics of stochastic models with energy conservation (model C).

We calculate the field-theoretic functions of the generalized dynamical model C*, in which a conserved secondary density is coupled to a nonconserved complex order parameter, in two-loop order. We show that the fixed points in this extended model are equal to the fixed points obtained in model C with a real order parameter, which has been introduced by Halperin, Hohenberg, and Ma. Our results correct long-standing errors in the field-theoretic functions in model C published by several authors leading also to different fixed point values w* for the ratio of the two time scales involved. The stability regions of the fixed points, which remained partially unclear, considered in a "phase diagram"--whose axes are the spatial dimension d and number of order parameter components n--are now clarified. Especially an anomalous region found by previous authors, in which the scaling properties remained unsolved, does not exist. There are only two regions: one with a finite fixed point w* where the dynamical exponent z of the order parameter is z=2+alpha/nu and another region where w*=0 and z is equal to the model A value.

Excitation of phase patterns and spatial solitons via two-frequency forcing of a 1:1 resonance.

We show that a self-oscillatory system, driven at two frequencies close to that of the unforced system (resonance 1:1), becomes phase locked and exhibits two equivalent stable states of opposite phases. For spatially extended systems this phase bistability results in patterns characteristic for real order parameter systems, such as phase domains, labyrinths, and phase spatial solitons. In variational cases, the phase-locking mechanism is interpreted as a result of the periodic "rocking" of the system potential. Rocking could be tested experimentally in lasers and in oscillatory chemical reactions.

Transition to Fulde-Ferrel-Larkin-Ovchinnikov phases near the tricritical point: an analytical study

We explore analytically the nature of the transition to the Fulde-Ferrel-Larkin-Ovchinnikov superfluid phases in the vicinity of the tricritical point, where these phases begin to appear. We make use of an expansion of the free energy up to an overall sixth order, both in order parameter amplitude and in wavevector. We first explore the minimization of this free energy within a subspace, made of arbitrary superpositions of plane waves with wavevectors of different orientations but same modulus. We show that the standard second order FFLO phase transition is unstable and that a first order transition occurs at higher temperature. Within this subspace we prove that it is favorable to have a real order parameter and that, among these states, those with the smallest number of plane waves are preferred. This leads to an order parameter with a cos( . ) dependence, in agreement with preceding work. Finally we show that the order parameter at the transition is only very slightly modified by higher harmonics contributions when the constraint of working within the above subspace is released.