The Ginzburg–Landau theory and the surface energy of a colour superconductor

We apply the Ginzburg–Landau theory to the colour superconducting phase of a lump of dense quark matter. We calculate the surface energy of a domain wall separating the normal phase from the super phase with the bulk equilibrium maintained by a critical external magnetic field. Because of the symmetry of the problem, we are able to simplify the Ginzburg–Landau equations and express them in terms of two components of the di-quark condensate and one component of the gauge potential. The equations also contain two dimensionless parameters: the Ginzburg–Landau parameter κ and ρ. The main result of this paper is a set of inequalities obeyed by the critical value of the Ginzburg–Landau parameter—the value of κ for which the surface energy changes sign—and its derivative with respect to ρ. In addition we prove a number of inequalities of the functional dependence of the surface energy on the parameters of the problem and obtain a numerical solution of the Ginzburg–Landau equations. Finally a criterion for the types of colour superconductivity (type I or type II) is established in the weak coupling approximation.     

Ginzburg–Landau equations and boundary conditions for superconductors in static magnetic fields

We derive the Ginzburg–Landau equations for superconductors in static magnetic fields. Instead of the square of the gauge‐invariant gradient of the order‐parameter wave function, we consider the quantum‐mechanical expression for the kinetic energy in the Ginzburg–Landau energy functional. We introduce a new surface term in the free energy functional which results in the de Gennes interface conditions. The phenomenological Ginzburg–Landau theory thus contains three length scales which must be determined from microscopic theory: the Ginzburg–Landau coherence length, the London penetration depth, and the de Gennes length.     

Phase transitions in shape memory alloys: A non-isothermal Ginzburg–Landau model

We propose a model to describe non-isothermal transitions from the austenite to the martensite phase occurring in shape memory materials. The phenomenon is set in the context of the Ginzburg–Landau theory of phase transitions, postulating a free energy depending on the temperature, the stress and the order parameter. In the one-dimensional case, when only two martensitic variants are involved and stress and deformation have a fixed direction, our choice of free energy allows us to deduce a phase diagram describing the main features of a typical SMA. The Ginzburg–Landau equation ruling the evolution of the order parameter is coupled with the equations of thermoelasticity by assuming a constitutive equation relating stress, strain and order parameter. The consistency of the model with the second law of Thermodynamics in the form of the Clausius–Duhem inequality is proved. Finally a possible generalization to a three-dimensional model is proposed, by introducing a tensor-valued order parameter.     

Properties of parallel upper critical field within continuous Ginzburg–Landau model

In this paper, we employ a continuous Ginzburg–Landau model to study the behaviors of the parallel upper critical field of an intrinsically layered superconductor. Near T_c where the order parameter is nearly homogeneous, the parallel upper critical field is found to vary as (1−T/T_c)^1/2. With a well-localized order parameter, the same field temperature dependence holds over the whole temperature range. The profile of the order parameter at the parallel upper critical field is of a Gaussian type, which is consistent with the usual Ginzburg–Landau theory. In addition, the influences of the unit cell dimension and the average effective masses on the parallel upper critical field and the associated order parameter are also addressed.     

On the Ginzburg–Landau Functional in the Surface Superconductivity Regime

We present new estimates on the two-dimensional Ginzburg–Landau energy of a type-II superconductor in an applied magnetic field varying between the second and third critical fields. In this regime, superconductivity is restricted to a thin layer along the boundary of the sample. We provide new energy lower bounds, proving that the Ginzburg–Landau energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. Estimates relating the density of the Ginzburg–Landau order parameter to that of the 1D problem follow. In the particular case of a disc sample, a refinement of our method leads to a pointwise estimate on the Ginzburg–Landau order parameter, thereby proving a strong form of uniformity of the surface superconductivity layer, which is related to a conjecture by Xing-Bin Pan.     

QCD with two colors at finite baryon density at next-to-leading order

We study QCD with two colors and quarks in the fundamental representation at finite baryon density in the limit of light-quark masses. In this limit the free energy of this theory reduces to the free energy of a chiral Lagrangian which is based on the symmetries of the microscopic theory. In earlier work this Lagrangian was analyzed at the mean-field level and a phase transition to a phase of condensed diquarks was found at a chemical potential of half the diquark mass (which is equal to the pion mass). In this article we analyze this theory at next-to-leading order in chiral perturbation theory. We show that the theory is renormalizable and calculate the next-to-leading order free energy in both phases of the theory. By deriving a Landau–Ginzburg theory for the order parameter we show that the finite one-loop contribution and the next-to-leading order terms in the chiral Lagrangian do not qualitatively change the phase transition. In particular, the critical chemical potential is equal to half the next-to-leading order pion mass, and the phase transition is of second order.     

On the Effect of Boundary Conditions in the Ginzburg–Landau Theory on the Results of Calculations of the Critical State of Layered Structures

The effect of boundary conditions in the Ginzburg–Landau theory on the critical state of superconducting layered structures is studied. The method is based on the numerical solution of the Ginzburg–Landau nonlinear equations describing the behavior of a superconducting plate carrying a transport current in a magnetic field, provided the absence of vortices in it. The use of the general boundary condition for the Ginzburg–Landau system of equations leads to a change in the order parameter over the thickness of thin superconducting plates. The calculated dependences of the critical current of plates on the magnetic field applied in parallel to layers are used to determine the critical current of multilayered structures. It is assumed that the mutual influence of superconducting layers occurs only through the magnetic field induced by them.     

Minimizers of the Lawrence–Doniach Functional with Oblique Magnetic Fields

We study minimizers of the Lawrence–Doniach energy, which describes equilibrium states of superconductors with layered structure, assuming Floquet-periodic boundary conditions. Specifically, we consider the effect of a constant magnetic field applied obliquely to the superconducting planes in the limit as both the layer spacing s → 0 and the Ginzburg–Landau parameter \({\kappa = \epsilon^{-1} \to \infty}\), under the hypotheses that \({s=\epsilon^\alpha}\) with 0 < α < 1. By deriving sharp matching upper and lower bounds on the energy of minimizers, we determine the lower critical field and the orientation of the flux lattice, to leading order in the parameter \({\epsilon}\). To leading order, the induced field is characterized by a convex minimization problem in \({\mathbb {R}^3}\). We observe a “flux lock-in transition”, in which flux lines are pinned to the horizontal direction for applied fields of small inclination, and which is not present in minimizers of the anisotropic Ginzburg–Landau model. The energy profile we obtain suggests the presence of “staircase vortices”, which have been described qualitatively in the physics literature.     

Interfaces and ripple states in ferroelastic crystals—A simple model

A simple atomistic model with two harmonic potentials between layers of atoms describes surface relaxations equivalent to those found as kink solitary waves in ferroelastic crystals. The parameters of the model are expressed in terms of the entropy term in a Landau potential and the Ginzburg gradient energy. We discuss the possibility that metastable ripple states exist in which a modulation is superimposed to an underlying uniform order parameter. Such ripple states can occur at temperatures close to the transition point between a ferroelastic phase and an incommensurate phase. In the ripple state, domain walls consist of kinks with modulations on either side of the kink.     

The density of superconductivity in domains with corners

We compute the \(L^2\)-norm of any minimizer of the Ginzburg–Landau functional in a planar domain with a finite number of corners. Our computations are valid for a uniform applied magnetic field, large Ginzburg–Landau parameter and in the regime where superconductivity is confined near the corners of the domain.     

Flux confinement by regular arrays and clusters of antidots in Pb/Cu bilayers

We have studied the magnetoresistance and superconducting–normal phase boundary of superconducting films with an antidot array and mesoscopic antidot clusters of 2 × 2 μm²with only four antidots. For both systems characteristic minima have been observed in the magnetoresistance which are caused by the formation of certain vortex configurations minimizing the free energy. By comparing experimental data with calculations carried out in the London limit of the Ginzburg Landau theory, these vortex configurations have been identified.     

Critical magnetic field ratio of anisotropic magnetic superconductors

The upper critical field, the lower critical field and the critical magnetic field ratio of anisotropic magnetic superconductors are calculated by Ginzburg–Landau theory analytically. The effect of the Ginzburg–Landau parameter (κ₀), magnetic susceptibility (χ) and magnetic-to-anisotropic parameter ratio (θ) on the critical field ratio are considered. We find that the value of critical field ratio increases with increasing κ₀ and θ, and decreases with increasing χ. The highest and the lowest value of critical field ratio is found in the diamagnetic superconductors and the ferromagnetic superconductors, respectively.     

Relaxation model of the orientational phase transition in fullerite C60

The model of thermal behavior of a thermoelastic medium is developed in the context of the Landau theory of phase transitions. In the framework of this model, two different problems are considered with allowance for order parameter relaxation: the problem of relatively slow uniform heating (cooling) of the medium under external hydrostatic pressure and the problem of order parameter relaxation at thermal isolation. A finite value of the relaxation constant τ of the order parameter is demonstrated to bring about the heating (cooling) rate dependence of the physical quantities, such as specific heat. The relaxation time of the order parameter is shown to be twice larger than the temperature relaxation time, as a consequence of the Landau expansion of the free energy.     

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.     

Relationship between crystalline order and melting mechanisms of solids

The mechanism for homogeneous nucleation of the liquid phase in Lennard-Jones solids is studied by combining the Landau free energy approach with some of the methodology developed to characterise transition path ensembles. The second-order bond orientational order parameter, Q ₆ which indexes the overall degree of crystalline order, is shown to provide a dynamically significant collective coordinate describing the melting process. Trajectories generated from configurations sampled in the vicinity of the maximum in the Landau free energy curve, F(Q ₆), are shown to have equal likelihood of teminating in either the solid or liquid-like free energy minima. It is also demonstrated that Q ₆ is necessary but not sufficient as a dynamical coordinate to describe melting and it is necessary to explore possiblities for additional coordinates which are critical for initiating melting. Our sudy suggests that the additional coordinates for describing the melting process would be some type of localised defect, much smaller in spatial extent than the size of the critical nucleus predicted by classical nucleation theory.      

Tricritical point in quantum phase transitions of the Coleman–Weinberg model at Higgs mass

The tricritical point, which separates first and second order phase transitions in three-dimensional superconductors, is studied in the four-dimensional Coleman–Weinberg model, and the similarities as well as the differences with respect to the three-dimensional result are exhibited. The position of the tricritical point in the Coleman–Weinberg model is derived and found to be in agreement with the Thomas–Fermi approximation in the three-dimensional Ginzburg–Landau theory. From this we deduce a special role of the tricritical point for the Standard Model Higgs sector in the scope of the latest experimental results, which suggests the unexpected relevance of tricritical behavior in the electroweak interactions.     

Unconventional critical behaviour of fermions hybridized with bosons

A phase transition into the condensed state of fermions hybridized with immobile bosons is examined beyond the ordinary mean-field approximation (MFA) in two and three dimensions. The hybridization interaction does not provide the Cooper pairing of fermions and the Bose condensation in two dimensions. In the three-dimensional boson–fermion model (BFM), an expansion in the strength of the order parameter near the transition yields no linear homogeneous term in the Ginzburg–Landau–Gor’kov equation. This indicates that previous mean-field discussions of the model are flawed in any dimension. In particular, the conventional (MFA) upper critical field is zero in any-dimensional BFM.     

The Effect of Nonisothermality on the Early Stages of Spinodal Decomposition

Journal of Experimental and Theoretical Physics - A nonisothertmal model of spinodal decomposition is proposed for a binary system described by the Ginzburg–Landau energy. The initial stages...