Generalized Mean Field Research Articles

Dynamical analysis of the low-temperature phase diagram of a cuprate superconductor

For a cuprate superconductor, specifically the yttrium barium copper oxide system, we analyze the low-temperature phase diagram consisting of the superconducting, pseudogap, and normal phases by using a phonon-mediated attraction and Coulomb repulsion mechanism. Through this analysis, we show this mechanism in the cuprate is quite important for its key dynamics. For the pairing mechanism, a variational method is developed and applied, and a generalized mean field theory is used to calculate the condensate energy and the energy gap of the quasiparticle. We obtain a self-consistent system of equations for these quantities. The superconducting and pseudogap phases, which involve pair-condensates, are assumed to be equilibrium systems of the condensate and normal subphases. The condition for the phase transition point is given by the equal total electronic energy gains. Assuming that the difference in the pairing range of the pair condensate is small between just above and below the transition temperature, computations are performed to solve the self-consistent system of equations. We find solutions compatible with experimental data. Plausible values of the energy gap in the superconducting phase and in the pseudogap phase are obtained. The present analysis shows that the phonon-mediated attraction and Coulomb repulsion mechanism itself is compatible with the experimental data. The importance of the screening effect through one-phonon exchange in the pairing mechanism and of the structure of the valence band is discussed.

A generalized mean field type flow on a closed Riemann surface

AbstractLet be a closed Riemann surface. Let ψ, h be two smooth functions on Σ with and . In this paper, using the method of flow due to Castras (Pacific J. Math. 276(2015), no. 2, 321–345) and Sun–Zhu (Calc. Var. Partial Differential Equations 60(2021), no. 1, 26), we prove that the solution of the equation exists under given conditions. This gives a new proof of the main results of Zhu (Nonlinear Anal. 169(2018), 38–58).

Wetting Behavior of a Three-Phase System in Contact with a Surface.

We extend the Cahn-Landau-de Gennes mean field theory of wetting in binary mixtures to understand the wetting thermodynamics of a three phase system (e.g., polymer dispersed liquid crystals or polymer-colloid mixtures) that is in contact with an external surface, which prefers one of the phases. Using a model free-energy, which has three minima in its landscape, we show that as the central minimum becomes more stable compared to the remaining ones, the bulk phase diagram encounters a triple point and then bifurcates and we observe a novel non-monotonic dependence of the surface tension as a function of the stability of the central minimum. We show that this non-monotonicity in surface tension is associated with a complete to partial wetting transition. We obtain the complete wetting phase behavior as a function of phase stability and the surface interaction parameters when the system is close to the bulk triple point. The model free-energy that we use is qualitatively similar to that of a renormalized free energy, which arises in the context of polymer-liquid crystal mixtures. Finally, we study the thermodynamics of wetting for an explicit polymer-liquid crystal mixture and show that its thermodynamics is similar to that of our model free-energy.

Interplay between Microscopic and Macroscopic Properties of Charged Hydrogels

We have investigated aqueous charged poly(acrylamide-co-acrylate) gels by varying polymer concentrations at fixed concentrations of added sodium chloride salt. Using a combination of static and dynamic light scattering, rheology, and water permeation experiments, we have made independent measurements of shear modulus, friction coefficient, diffusion coefficient, and mesh size of the gel. We have also derived a generalized mean field theory for charged gels by including electrostatic interactions between polymer segments, in addition to the contributions from free energy of mixing, elasticity, and the Donnan equilibrium. The measured dependencies of modulus, friction coefficient, and gel diffusion coefficient on polymer concentration are in accordance with the predictions of the generalized mean field theory, enabling a reliable bridge between the macroscopic and microscopic properties of polyelectrolyte gels. When the contributions to free energy of the gel from fluctuations in polymer concentration and c...

Homogeneous cooling of mixtures of particle shapes

In this work, we examine theoretically the cooling dynamics of binary mixtures of spheres and rods. To this end, we introduce a generalized mean field analytical theory, which describes the free cooling behavior of the mixture. The relevant characteristic time scale for the cooling process is derived, depending on the mixture composition and the aspect ratio of the rods. We simulate mixtures of spherocylinders and spheres using a molecular dynamics algorithm implemented on graphics processing unit (GPU) architecture. We systematically study mixtures composed of spheres and rods with several aspect ratios and varying the mixture composition. A homogeneous cooling state, where the time dependence of the system’s intensive variables occurs only through a global granular temperature, is identified. We find cooling dynamics in excellent agreement with Haff’s law, when using an adequate time scale. Using the scaling properties of the homogeneous cooling dynamics, we estimated numerically the efficiency of the energy interchange between rotational and translational degrees of freedom for collisions between spheres and rods.

Variational principle of Bogoliubov and generalized mean fields in many-particle interacting systems

The approach to the theory of many-particle interacting systems from a unified standpoint, based on the variational principle for free energy is reviewed. A systematic discussion is given of the approximate free energies of complex statistical systems. The analysis is centered around the variational principle of Bogoliubov for free energy in the context of its applications to various problems of statistical mechanics. The review presents a terse discussion of selected works carried out over the past few decades on the theory of many-particle interacting systems in terms of the variational inequalities. It is the purpose of this paper to discuss some of the general principles which form the mathematical background to this approach and to establish a connection of the variational technique with other methods, such as the method of the mean (or self-consistent) field in the many-body problem. The method is illustrated by applying it to various systems of many-particle interacting systems, such as Ising, Heisenberg and Hubbard models, superconducting (SC) and superfluid systems, etc. This work proposes a new, general and pedagogical presentation, intended both for those who are interested in basic aspects and for those who are interested in concrete applications.

Suppression of s-wave superconductivity by kinetic disorder in a two-dimensional attractive Hubbard model

We investigate the influence of diagonal and off-diagonal disorder potentials on superconductivity in an attractive Hubbard model. The study is motivated by recent experimental and theoretical interest in understanding the microscopic mechanism by which impurities destroy superconductivity. In order to capture the spatial correlations accurately, we make use of the real-space Bogoliubov-de Gennes mean field method. We find that the response of a superconductor to disorder crucially depends, even qualitatively, on the type of disorder considered. Superconductivity is suppressed spatially homogeneously by off-diagonal (kinetic) disorder in comparison to the suppression by diagonal (potential) disorder which proceeds via the formation of strongly superconducting islands. Moreover, the non-superconducting phase is gapless in the case of kinetic disorder, suggesting a fermionic superconductor-insulator transition (SIT). This is in sharp contrast to the SIT tuned by diagonal disorder, which is understood to be bosonic in nature. A qualitatively distinct mechanism that allows for a BCS-like suppression of superconductivity with increasing disorder is, in fact, consistent with recent experiments on amorphous Bi films.

Generalized mean field approximation for parallel dynamics of the Ising model

.The dynamics of the non-equilibrium Ising model with parallel updates is investigated using a generalized mean field approximation that incorporates multiple two-site correlations at any two time steps, which can be obtained recursively. The proposed method shows significant improvement in predicting local system properties compared to other mean field approximation techniques, particularly in systems with symmetric interactions. Results are also evaluated against those obtained from Monte Carlo simulations. The method is also employed to obtain parameter values for the kinetic inverse Ising modeling problem, where couplings and local field values of a fully connected spin system are inferred from data.

Mean field model for synchronization of coupled two-state units and the effect of memory

A prototypical model for a mean field second order transition is presented, which is based on an ensemble of coupled two-states units. This system is used as a basic model to study the effect of memory. To wit, we distinguish two types of memories: weak and strong, depending on the feasibility of linearizing the generalized mean field master equation. For weak memory we find static solutions that behave much like those of the memoryless (Markovian) system. The latter exhibits a pitchfork bifurcation as the control parameter is increased, with two stable and one unstable solution. The former exhibits an imperfect pitchfork bifurcation to states with the same behaviors. In both cases, the stability of the static solutions is analyzed via the usual linearization around the equilibrium solution. For strong memories we again find an imperfect pitchfork bifurcation, with two stable and one unstable branch. However, it is no longer possible to analyze these behaviors via the usual linearization, which is local in time, because a strong memory requires knowledge of the system for its entire past. Finally, we are pleased to dedicate this publication to Helmut Brand on the occasion of his 60th birthday.

Equivalent topological invariants for one-dimensional Majorana wires in symmetry classD

Topological superconductors in one spatial dimension exhibiting a single Majorana bound state at each end are distinguished from trivial gapped systems by a Z_2 topological invariant. Originally, this invariant was calculated by Kitaev in terms of the Pfaffian of the Majorana representation of the Hamiltonian: The sign of this Pfaffian divides the set of all gapped quadratic forms of Majorana fermions into two inequivalent classes. In the more familiar Bogoliubov de Gennes mean field description of superconductivity, an emergent particle hole symmetry gives rise to a quantized Zak-Berry phase the value of which is also a topological invariant. In this work, we explicitly show the equivalence of these two formulations by relating both of them to the phase winding of the transformation matrix that brings the Majorana representation matrix of the Hamiltonian into its Jordan normal form.

Pairing Correlation in Quark Matter

The superconductivity in quark matter is investigated by a generalized mean field approximation. If we assume that two quarks interact each other via the one-gluon exchange potential, attractive force appears in the color-triplet and spin-parallel state. Applying a BCS-like theory to the system, it is shown that we obtain a new superconducting state which is more stable than the normal (Fermi gas) state. We also give the explicit expression of the superconducting state.

Gapless inhomogeneous superfluid phase with spin-dependent disorder

We show that the presence of a spin-dependent random potential in a superconductor or a superfluid atomic gas leads to distinct transitions at which the energy gap and average order parameter vanish, generating an intermediate gapless superfluid phase, in marked contrast to the case of spin-symmetric randomness where no such gapless superfluid phase is seen. By allowing the pairing amplitude to become inhomogeneous, the gapless superconducting phase persists up to considerably higher disorder compared with the prediction of Abrikosov–Gorkov. The low-lying excited states are located predominantly in regions where the pairing amplitude vanishes and coexist with the superfluid regions with a finite pairing. Our results are based on inhomogeneous Bogoliubov–de Gennes mean field theory for a two-dimensional attractive Hubbard model with spin-dependent disorder.

Interplay of superconductivity and spin-dependent disorder

The finite temperature phase diagram for the 2D attractive fermion Hubbard model with spin-dependent disorder is considered within Bogoliubov-de Gennes mean field theory. Three types of disorder are studied. In the first, only one species is coupled to a random site energy; in the second, the two species both move in random site energy landscapes which are of the same amplitude, but different realizations; and finally, in the third, the disorder is in the hopping rather than the site energy. For all three cases we find that, unlike the case of spin-symmetric randomness, where the energy gap and average order parameter do not vanish as the disorder strength increases, a critical disorder strength exists separating distinct phases. In fact, the energy gap and the average order parameter vanish at distinct transitions, $V_{c}^{\rm gap}$ and $V_{c}^{\rm op}$, allowing for a gapless superconducting (gSC) phase. The gSC phase becomes smaller with increasing temperature, until it vanishes at a temperature $T^{\ast}$.

Dependence of the critical temperature of high-temperature cuprate superconductors on hoppings and spin correlations between CuO2 planes

The influence of interlayer hoppings on the superconducting transition temperature (T c) in bilayer cuprates has been studied. The parameter of hopping between layers is expressed as t ⊥(k) = t ⊥(cos(k x ) − cos(k y ))2 and treated as a small perturbation for the states of two CuO2 planes described by the t-t′-t″-J* model. In the generalized mean field approximation for \({d_{{x^2} - {y^2}}}\) symmetry of the superconducting gap, neither the interlayer hopping or exchange interaction, nor the pair hopping between CuO2 layers provides an additional mechanism of Cooper pair formation or an increase in T c. In the concentration dependence of T c, the bilayer splitting of the upper Hubbard band of quasi-holes is manifested as two peaks with temperatures slightly lower than the maximum T c for a single-layer cuprate. Interlayer antiferromagnetic spin correlations suppress bilayer splitting.

A generalized mean field theory of coarse-graining

A general mean field theory is presented for the construction of equilibrium coarse-grained models. Inverse methods that reconstruct microscopic models from low resolution experimental data can be derived as particular implementations of this theory. The theory also applies to the opposite problem of reduction, where relevant information is extracted from available equilibrium ensemble data. Additionally, a complementary approach is presented and problems of representability in coarse-grained modeling analyzed using information theoretic arguments. These problems are central to the construction of coarse-grained representations of complex systems, and commonly used coarse-graining methods and variational principles for coarse-graining are derived as particular cases of the general theory.

Spatial structure of superconducting correlations of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa8hzam % aaBaaaleaacaWF4bWaaWbaaWqabeaacaaIYaaaaSGaeyOeI0IaamyE % amaaCaaameqabaGaaGOmaaaaaSqabaaaaa!3BD7! $$ d_{x^2 - y^2 } $$ symmetry in high-temperature superconductors

The spatial structure of the correlation function of the order parameter of the \( d_{x^2 - y^2 } \) symmetry has been investigated within the t-J model in the generalized mean field approximation taking into account the “no-double occupancy” constraint due to strong electron correlations. A slow decay of the correlation function to the thirtieth coordination sphere has been found. The limitation by pairing of only the nearest neighbors makes it possible to take into account less than 17% of the superconducting correlations.

Brownian particles with long- and short-range interactions

We develop the kinetic theory of Brownian particles with long- and short-range interactions. Since the particles are in contact with a thermal bath fixing the temperature T, they are described by the canonical ensemble. We consider both overdamped and inertial models. In the overdamped limit, the evolution of the spatial density is governed by the generalized mean field Smoluchowski equation including a mean field potential due to long-range interactions and a generically nonlinear barotropic pressure due to short-range interactions. This equation describes various physical systems such as self-gravitating Brownian particles (Smoluchowski–Poisson system), bacterial populations experiencing chemotaxis (Keller–Segel model) and colloidal particles with capillary interactions. We also take into account the inertia of the particles and derive corresponding kinetic and hydrodynamic equations generalizing the usual Kramers, Jeans, Euler and Cattaneo equations. For each model, we provide the corresponding form of free energy and establish the H-theorem and the virial theorem. Finally, we show that the same hydrodynamic equations are obtained in the context of nonlinear mean field Fokker–Planck equations associated with generalized thermodynamics. However, in that case, the nonlinear pressure is due to the bias in the transition probabilities from one state to the other leading to non-Boltzmannian distributions while in the former case the distribution is Boltzmannian but the nonlinear pressure arises from the two-body correlation function induced by the short-range potential of interaction. As a whole, our paper develops connections between the topics of long-range interactions, short-range interactions, nonlinear mean field Fokker–Planck equations and generalized thermodynamics. It also justifies from a kinetic theory based on microscopic processes, the basic equations that were introduced phenomenologically to describe self-gravitating Brownian particles, chemotaxis and colloidal suspensions with attractive interactions.

Quasiaverages, symmetry breaking and irreducible Green functions method

The development and applications of the method of quasiaverages to quantum statistical physics and to quantum solid state theory and, in particular, to quantum theory of magnetism, were considered. It was shown that the role of symmetry (and the breaking of symmetries) in combination with the degeneracy of the system was reanalyzed and essentially clarified within the framework of the method of quasiaverages. The problem of finding the ferromagnetic, antiferromagnetic and superconducting “symmetry broken” solutions of the correlated lattice fermion models was discussed within the irreducible Green functions method. A unified scheme for the construction of generalized mean fields (elastic scattering corrections) and self-energy (inelastic scattering) in terms of the equations of motion and Dyson equation was generalized in order to include the “source fields”. This approach complements previous studies of microscopic theory of antiferromagnetism and clarifies the concepts of Neel sublattices for localized and itinerant antiferromagnetism and “spin-aligning fields” of correlated lattice fermions.

Wetting transition of a nematic liquid crystal on a periodic wedge-structured substrate

It is known that the wetting behaviour of a fluid is deeply altered by the presence of rough or structured substrates. We first review some simple considerations about isotropic fluids and rough substrates, and then we generalize Wenzel's law, which assigns an effective contact angle to a droplet on a rough substrate, when the wetting layer has an ordered phase, like a nematic. We estimate the conditions for which the wetting behavior of an ordered fluid can be qualitatively different from that usually found in a simple fluid. To support our general considerations, we use the Landau-de Gennes mean field approach to investigate theoretically and numerically the wetting transition of a nematic phase on a periodic triangular structured substrate.

Quasi-two-dimensional diluted magnetic semiconductors with arbitrary carrier degeneracy

In the framework of the generalized mean field theory, conditions for arising the ferromagnetic state in a two-dimensional diluted magnetic semiconductor and the features of that state are defined. RKKY-interaction of magnetic impurities is supposed. The spatial disorder of their arrangement and temperature alteration of the carrier degeneracy are taken into account.