Ginzburg-Landau theory

This are some notes on Statistical mechanics.
The topics are
-Recap of thermodynamics                                                     
-Thermodynamic of phase transitions                                                   
-recap of theory of ensamble                                                                               
-Statistical mechanics and phase transition                                                             
-Models (Ising in particular)                                                                                         
-Role of symmetry (symmetry breaking), dimension and range of interaction 
-Mean field theory (and Vann der Walls)                                                                   
-Landau Theory                     
-Ginzburg criterium
-Scaling theory (Windom)
-Renormalization group
-Symmetry breaking in other parts of physics (particles physics)

The paper provides the basics of Landau's Theory applied to structural Phase Transitions in general, and in particular to ferroleastic and ferroelastoelectric transitions. Examples investigated in detail are: LiKSO4, Rb4LiH3(SO4)4, (NH4)2CuCl4

Thermodynamics of type II superconductors in electromagnetic field based on the Ginzburg - Landau theory is presented. The Abrikosov flux lattice solution is derived using an expansion in a parameter characterizing the "distance" to the superconductor - normal phase transition line. The expansion allows a systematic improvement of the solution. The phase diagram of the vortex matter in magnetic field is determined in detail. In the presence of significant thermal fluctuations on the mesoscopic scale (for example in high $T_{c}$ materials) the vortex crystal melts into a vortex liquid. A quantitative theory of thermal fluctuations using the lowest Landau level approximation is given. It allows to determine the melting line and discontinuities at melt, as well as important characteristics of the vortex liquid state. In the presence of quenched disorder (pinning) the vortex matter acquires certain "glassy" properties. The irreversibility line and static properties of the vortex glass state are studied using the "replica" method. Most of the analytical methods are introduced and presented in some detail. Various quantitative and qualitative features are compared to experiments in type II superconductors, although the use of a rather universal Ginzburg - Landau theory is not restricted to superconductivity and can be applied with certain adjustments to other physical systems, for example rotating Bose - Einstein condensate.

In this article the Ginzburg-Landau theory ideas are considered in their application to the description of fluctuations influence on the superfluid density in superconductor. The conclusion about the availability of two incompatible mathematical definitions of the superfluid density is made. That is why, it is suggested not to consider the fluctuating part of order parameter, while calculating the superconductor thermodynamical characteristics in the Mean Field Approximation. Comment: 5 pages, 0 figures

We use the weakly nonlocal hydrodynamics approach to obtain a dynamical equation for the peculiar velocity field in which the viscosity term is physically motivated. Based on some properties of the Ginzburg-Landau equation and the wave mechanics analog of hydrodynamics we find the nonlocal adhesion approximation taking into account the internal structures of the Zeldovich pancakes. If the internal structures correspond to significant mesoscopic fluctuations, viscosity is probably driven by a stochastic force and dynamics is given by the noisy Burgers equation.

The upper critical field in MgB2 is investigated in the framework of the two-gap Ginzburg-Landau theory. A variational solution of linearized Ginzburg-Landau equations agrees well with the Landau level expansion and demonstrates that spatial distributions of the gap functions are different in the two bands and change with temperature. The temperature variation of the ratio of two gaps is responsible for the upward temperature dependence of in-plane Hc2 as well as for the deviation of its out-of-plane behavior from the standard angular dependence. The hexagonal in-plane modulations of Hc2 can change sign with decreasing temperature.

Several field theoretical approaches to the superconducting phase transition are discussed. Emphasis is given to theories of scaling and renormalization group in the context of the Ginzburg-Landau theory and its variants. Also discussed is the duality approach, which allows to access the strong-coupling limit of the Ginzburg-Landau theory.

The Ginzburg-Landau functional for a two-gap superconductor is derived within the weak-coupling BCS model. The two-gap Ginzburg-Landau theory is, then, applied to investigate various magnetic properties of MgB2 including an upturn temperature dependence of the transverse upper critical field and a core structure of an isolated vortex. Orientation of vortex lattice relative to crystallographic axes is studied for magnetic fields parallel to the c-axis. A peculiar 30-degree rotation of the vortex lattice with increasing strength of an applied field observed by neutron scattering is attributed to the multi-gap nature of superconductivity in MgB2.

A classical SU(2) Einstein-Yang-Mills theory in (3+1)-dimensional anti-de Sitter spacetime is believed to be dual to a p-wave superconductor in (2+1)-dimensional flat spacetime. In order to calculate the superconducting coherence length ξ of the holographic superconductor near the superconducting phase transition point, we make a perturbative study of the gravity theory analytically. The superconducting coherence length ξ is found to be proportional to (1-T/Tc)-1/2 near the critical temperature Tc. We also obtain the magnetic penetration depth λ∝(Tc-T)1/2 by adding a small external homogeneous magnetic field. The results agree with the Ginzburg-Landau theory.

The thermal conductivity κ of BiâSr{sub 1.8}Ca{sub 1.2}CuâOâ and DyBaâCuâO{sub 7-x} polycrystals is analyzed near the critical temperature (T{sub c}=79.5 and 83.8 K, respectively) in order to extract the superconducting fluctuation contribution κ{sub fl}. The fluctuationless background is calculated in a formal way, taking into account both an electronic and a phonon contribution above and below the critical temperature. The fluctuation contribution to the electronic thermal conductivity is then extracted. A crossover from a critical to a Gaussian fluctuation regime ({vert_bar}T{sub G}-T{sub c}{vert_bar}â2.2K) is observed on the BiâSr{sub 1.8}Ca{sub 1.2}CuâOâ sample both above and below T{sub c}, followed by a crossover from a three-dimensional (3D) to a 2D Gaussian behavior ({vert_bar}T{sub VL}-T{sub c}{vert_bar}â3.7K) as the sample temperature is further moved away from T{sub c}. On the other hand, a 3D behavior only is observed on the DyBaâCuâO{sub 7-x} material in the normal and superconducting states. In both systems, the critical exponents are found to be those theoretically predicted. Moreover, the crossover temperatures are consistent with those expected from the differently anisotropic structures of these compounds. From the crossover temperatures and the thermal conductivity fluctuation contribution amplitudes, quite realistic values for the Ginzburg-Landau parameter k{sub GL}, the interlayer coupling energy J{sub c}, and the electronic mean free path l{sub e} near T{sub c} are found in those materials. {copyright} {ital 1997} {ital The American Physical Society}

We study the effect of critical fluctuations on the $(B,T)$ phase diagram in extreme type-II superconductors in zero and finite magnetic field using large-scale Monte Carlo simulations on the Ginzburg-Landau model in a frozen gauge approximation. We show that a vortex-loop unbinding gives a correct picture of the zero field superconducting-normal transition even in the presence of amplitude fluctuations, which are far from being critical at $T_c$. We extract critical exponents of the dual model by studying the topological excitations of the original model. From the vortex-loop distribution function we extract the anomalous dimension of the dual field $\eta \simeq -0.18$, and conclude that the charged Ginzburg-Landau model and the neutral 3DXY model belong to different universality classes. We find are two distinct scaling regimes for the vortex-line lattice melting line: a high-field scaling regime and a distinct low-field 3DXY critical scaling regime. We also find indications of an abrupt change in the connectivity of the vortex-tangle in the vortex liquid along a line $T_L \geq T_M$. This is the finite field counter-part of the zero-field vortex-loop blowout. Which at low enough fields appears to coincide with $T_M$. Here, a description of the vortex system only in terms of field induced vortex lines is inadequate at and above the VLL melting temperature.

We describe a semianalytic approach to the two-band Ginzburg-Landau theory, which predicts the behavior of vortices in two-band superconductors. We show that the character of the short-range vortex-vortex interaction is determined by the sign of the normal domain-superconductor interface energy, in analogy with the conventional differentiation between type I and type II superconductors. However, we also show that the long-range interaction is determined by a modified Ginzburg-Landau parameter κ*, different from the standard κ of a bulk superconductor. This opens the possibility for nonmonotonic vortex-vortex interaction, which is temperature dependent, and can be further tuned by alterations of the material on the microscopic scale.

In this work we explore the interplay between superconductivity and nematicity in the framework of a Ginzburg Landau theory with a nematic order parameter coupled to the superconductor order parameter, often used in the description of superconductivity of Fe based materials. In particular, we focus on the study of the vortex-vortex interaction in order to determine the way nematicity affects its attractive or repulsive character. To do so, we use a dynamical method based on the solutions of the Time Dependent Ginzburg Landau equations in a bulk superconductor. An important contribution of our work is the implementation of a pseudo-spectral method to solve the dynamics, known to be highly efficient and of very high order in comparison to the usual finite differences/elements methods. The coupling between the superconductor and the (real) nematic order parameters is represented by two terms in the free energy: a biquadratic term and a coupling of the nematic order parameter to the covariant derivatives of the superconductor order parameter. Our results show that there is a competing effect: while the former independently of its competitive or cooperative character generates an attractive vortex-vortex interaction, the latter always generates a repulsive interaction.

The properties of many electronic materials are governed by defect type, distribution, and concentration. In this paper, we demonstrate the influence of equilibrated nonstoichiometry on the ferroelectric properties of the perovskite BaTiO3. It is shown from measurements of the transition temperature, thermal hysteresis, latent heat, and transformation strain that the concentration of partial and/or full Schottky defects significantly alters the

A detailed theoretical interpretation of the Josephson interference experiment between Sr$_2$RuO$_4$ and Pb reported by Kidwingira \textit{et al} is given. Assuming chiral p-wave pairing symmetry a Ginzburg-Landau theory is derived in order to investigate the structure of domain walls between chiral domains. It turns out that anisotropy effects of the Fermi surface and the orientation of the domain walls are essential for their internal structure. Introducing a simple model for a Josephson junction the effect of domain walls intersecting the interface between Sr$_2$RuO$_4$ and Pb is discussed. It is shown that characteristic deviations of the Fraunhofer interference pattern for the critical Josephson current as a function of the magnetic field occurs in qualitative agreement with the experimental finding. Moreover the model is able also to account for peculiar hysteresis effects observed in the experiment.

Recent results for the coexistence of ferromagnetism and unconventional superconductivity with spin-triplet Cooper pairing are reviewed on the basis of the quasi-phenomenological Ginzburg-Landau theory. New results are reported. The results are discussed in view of applications to metallic compounds as UGe2, URhGe, ZrZn2.