Landau Ginzburg Research Articles

Noncommutative homological mirror symmetry of elliptic curves

We prove an equivalence of two A∞-functors, via Orlov’s Landau–Ginzburg/ Calabi–Yau (LG/CY) correspondence. One is the Polishchuk–Zaslow mirror symmetry functor of elliptic curves, and the other is a localized mirror functor from the Fukaya category of T2 to a category of noncommutative matrix factorizations. As a corollary, we prove that the noncommutative mirror functor LMgrLt realizes homological mirror symmetry for any t.

Mahler measure of 3D Landau–Ginzburg potentials

Abstract We express the Mahler measures of 23 families of Laurent polynomials in terms of Eisenstein–Kronecker series. These Laurent polynomials arise as Landau–Ginzburg potentials on Fano 3-folds, sixteen of which define K ⁢ 3 {K3} hypersurfaces of generic Picard rank 19, and the rest are of generic Picard rank less than 19. We relate the Mahler measure at each rational singular moduli to the value at 3 of the L-function of some weight-3 newform. Moreover, we find ten exotic relations among the Mahler measures of these families.

Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory

This paper presents our study of the presence of the unstable critical point in spontaneous symmetry breaking (SSB) in the framework of Ginzburg–Landau (G-L) free energy. Through a 3D Ising spin lattice simulation, we found a zone of hysteresis where the unstable critical point continued to exist, despite the system having entered the broken symmetry phase. Within the hysteresis zone, the presence of the kink–antikink SSB solitons expands and, therefore, these can be observed. In scalar field theories, such as Higgs fields, the mass of this soliton inside the hysteresis zone could behave as a tachyon mass, namely as an imaginary quantity. Due to the fact that groups Ζ(2) and SU(2) belong to the same universality class, one expects that, in future experiments of ultra-relativistic nuclear collisions, in addition to the expected bosons condensations, structures of tachyon fields could appear.

Optimizing K0.5Na0.5NbO3 Single Crystal by Engineering Piezoelectric Anisotropy.

K0.5Na0.5NbO3 is considered as one of the most promising lead-free piezoelectric ceramics in the field of wearable electronics because of its excellent piezoelectric properties and environmental friendliness. In this work, the temperature-dependent longitudinal piezoelectric coefficient was investigated in K0.5Na0.5NbO3 single crystals via the Landau–Ginzburg–Devonshire theory. Results show that the piezoelectric anisotropy varies with the temperature and the maximum of deviates from the polar direction of the ferroelectric phase. In the tetragonal phase, parallels with cubic polarization direction near the tetragonal-cubic transition region, and then gradually switches toward the nonpolar direction with decreasing temperatures. The maximum of in the orthorhombic phase reveals a distinct varying trend in different crystal planes. As for the rhombohedral phase, slight fluctuation of the maximum of was observed and delivered a more stable temperature-dependent maximum and its corresponding angle θmax in comparison with tetragonal and orthorhombic phases. This work not only sheds some light on the temperature-dependent phase transitions, but also paves the way for the optimization of piezoelectric properties in piezoelectric materials and devices.

Control of Domain States in Rhombohedral Lead Zirconate Titanate Films via Misfit Strains and Surface Charges

AbstractUsing the Landau–Ginzburg–Devonshire approach and numerical modeling, an influence of the misfit strain and surface screening charges as well as the role of the flexoelectric effect have been studied in the case of a rhombohedral lead zirconate titanate ferroelectric/ferroelastic thin film with an anisotropic misfit caused by a substrate. The magnitude and sign of the misfit strain influence the domain structure, providing misfit‐dependent phases with different favorable polarization components. Whilst strong enough compressive misfit strains favor a phase with orthorhombic‐like polarization directions, strong tensile misfits only yield in‐plane polarization components. The strength of surface screening conditions the existence of closure domain structures and, by increasing, supports the single‐domain state. The flexoelectric effect exhibits a weak influence on the phase diagram of multi‐domain states when compared with the phase diagram of single‐domain states. Its effect, however, becomes significant in the case of skyrmion topological states spontaneously formed near the film surface when compressive misfit strains are applied. Cooperative influence of the misfit strain, surface screening charges, and temperature can set a thin rhombohedral ferroelectric film into a number of different polar and structural states, whereby the role of the flexoelectric effect is pronounced for topologically nontrivial structures.

Reservoirs, Fick law, and the Darken effect

We study the stationary measures of Ginzburg–Landau (GL) stochastic processes, which describe the magnetization flux induced by the interaction with reservoirs. To privilege simplicity to generality, we restrict to quadratic Hamiltonians where almost explicit formulas can be derived. We discuss the case where reservoirs are represented by boundary generators (mathematical reservoirs) and compare with more physical reservoirs made by large-infinite systems. We prove the validity of the Fick law away from the boundaries. We also obtain in the context of the GL models a mathematical proof of the Darken effect, which shows uphill diffusion of carbon in a specimen partly doped with the addition of Si.

Dimer models and Hochschild cohomology

Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau–Ginzburg models dual to punctured Riemann surfaces. For a consistent dimer embedded in a torus, we explicitly compute the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. This includes a full characterization of the Batalin–Vilkovisky structure induced by the Calabi–Yau structure of the Jacobi algebra. We then compute the compactly supported Hochschild cohomology of the category of matrix factorizations for the Jacobi algebra with its canonical potential.

Microscopic derivation of superconductor-insulator boundary conditions for Ginzburg-Landau theory revisited: Enhanced superconductivity at boundaries with and without magnetic field

Using the standard Bardeen-Cooper-Schrieffer (BCS) theory, we revise microscopic derivation of the superconductor-insulator boundary conditions for the Ginzburg-Landau (GL) model. We obtain a negative contribution to free energy in the form of surface integral. Boundary conditions for the conventional superconductor have the form $\textbf{n} \cdot \nabla \psi = \text{const} \psi$. These are shown to follow from considering the order parameter reflected in the boundary. The boundary conditions are also derived for more general GL models with higher-order derivatives and pair-density-wave states. It shows that the boundary states with higher critical temperature and the boundary gap enhancement, found recently in BCS theory, are also present in microscopically-derived GL theory. In the case of an applied external field, we show that the third critical magnetic-field value $H_{c3}$ is higher than what follows from the de Gennes boundary conditions and is also significant in type-I regime.

Exponential reduction in critical current density in Eu1-xCexBa2Cu3O7-δsuperconductors near critical temperature

We have studied how the current-voltage (IV) characteristic below the critical temperature gets affected by electron doping in Eu1-xCexBa2Cu3O7-δ (ECBCO) superconducting systems. The critical current density as a function of T has been extracted in the vicinity but below the critical temperature. Variation of the low critical current density with temperature in the limited range of T has been explained by using an exponential dependence on T. Possibility of the Ambegaokar-Baratoff (AB) to Ginzburg-Landau (GL) crossover has been explored. An exponential nature of the variation in all electron doped superconductors with large angular separation between grain boundaries is attributed to the existence of the vortex fluid state in a narrow range of T.

WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

AbstractMotivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

Landau–Ginzburg models of complete intersections in Lagrangian Grassmannians

Landau–Ginzburg models of complete intersections in Lagrangian Grassmannians

A critical point analysis of Landau–Ginzburg potentials with bulk in Gelfand–Cetlin systems

Using the bulk deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya–Oh–Ohta–Ono’s bulk-deformed potential function, we prove that every complete flag manifold Fl(n) (n≥3) with a monotone Kirillov–Kostant–Souriau (KKS) symplectic form carries a continuum of nondisplaceable Lagrangian tori which degenerates to a nontorus fiber in the Hausdorff limit. In particular, the Lagrangian S3-fiber in Fl(3) is nondisplaceable, answering a question raised by Nohara and Ueda who computed its Floer cohomology to be vanishing.

Intrinsic switching in Si-doped HfO2: A study of Curie–Weiss law and its implications for negative capacitance field-effect transistor

HfO2-based ferroelectrics are considered a promising class of materials for logic and memory applications due to their CMOS compatibility and ferroelectric figures of merit. A steep-slope field-effect-transistor (FET) switch is a device for logic applications in which a ferroelectric gate stack exploits a stabilized negative capacitance regime capable to differentially amplify the surface potential in a metal–oxide–semiconductor FET structure, resulting in the improvement of the subthreshold swing and overdrive. In a number of relevant studies of negative capacitance, intrinsic (thermodynamic) switching is assumed, since alternative switching scenarios predict undesirable hysteretic responses in logic devices. However, there is little support from the experimental data showing that the polarization reversal in HfO2-based ferroelectrics is really driven by the intrinsic switching mechanism. In this work, polarization hysteresis loops are measured over wide temperature ranges on polycrystalline Si-doped HfO2 (Si:HfO2) capacitors. The analysis herein, which is based on the classic Landau–Ginzburg–Devonshire theory, yields the temperature-dependent dielectric susceptibility values, which fit the Curie–Weiss law. The extrapolated Curie temperature value is in line with the data obtained for other HfO2-based ferroelectrics using different techniques. The work also illustrates a method to evaluate the ferroelectric equivalent negative capacitance value and range of voltages, aiming at study and optimization of a stabilized negative capacitance FET. This study indicates that the intrinsic switching provides an adequate description of the polarization hysteresis in Si:HfO2 films. This confirms the usability of hafnia-based ferroelectrics for negative capacitance logic devices, and the important role that the intrinsic mechanism plays in the dielectric response of these materials.

Dynamics of spatiotemporal modulated damped signals in a nonlinear RLC transmission network

The dynamics of spatiotemporal modulated damped signals in a nonlinear LC transmission network with dissipative elements are investigated analytically. The complex cubic Ginzburg–Landau (GL) equation governing slowly modulated wave propagation is presented. Considering linear wave propagating in the network, we derive in terms of the propagating frequency the spatial decreasing rate (linear dissipation parameter) and show that its must important contribution comes from the dissipative element of the shunt branch. The modulational instability (MI) criterion of modulated Stokes wave propagating in the network is investigated and the analytical expression of the MI growth rate is derived; we show that in the case of weak dissipation, there are no significant changes for the bandwidth frequency where the network may exhibit MI. Exact and approximative envelope soliton-like solutions of the derived GL equation are presented and used to investigate the dynamics of spatiotemporal modulated damped signals along the network. We show that the solution parameters can be used for managing the evolution of the envelope soliton signals along the network. Our investigation shows that the amplitude decays in both space (cell number n) and time t, while the velocity remains constant when the envelope soliton signal propagates along the dissipative network.

Phase transition properties of ferroelectric and antiferroelectric liquid crystals

ABSTRACT Thermodynamic models based on the Landau free-energy density are popular and well suited to study the phase transition properties of ferroelectric and antiferroelectric liquid crystals. Ferroelectric and antiferroelectric liquid crystals have molecular chirality which play an important role in the technology of liquid crystals for the application of devices performance. The structural properties of ferroelectric phase transitions which give rise to the macroscopic tensors coupled to the macroscopic fields, the parameters describing the antiferroelectric phase transitions are still under investigation due to the lack of precise symmetrical properties characterising their existence. The study of ferroelectric and antiferroelectric liquid crystals has been substantial experimental strides. In theoretical approach, there has been relatively inappreciable work on this fascinating class of material. The present review paper focused on a comprehensive overview of the current scenario of the phase transition properties of ferroelectric and antiferroelectric liquid crystals for the fundamental and technological applications. The Landau Ginzburg theory of the phase transitions between ferroelectric and antiferroelectric liquid mesophases is investigated. The important experimental and theoretical developments of ferroelectric and antiferroelectric mesogens are also summarised.

Study of Carrier Transfer Mechanism When Substituting Strontium at Barium Sites in CuTl-1223 Superconducting Phase

Strontium-doped CuTl-1223 phase has been obtained by employing a nonstoichiometric composition of the form Tl1/2Cu1/2(Ba1−xSrx)Ca1Cu4Oy (x = 0, 0.1, 0.2, 0.3). The superconducting properties of the samples were investigated by x-ray diffraction analysis and four-probe resistivity, alternating-current (AC) susceptibility, and Fourier-transform infrared (FTIR) absorption measurements. The effect of doping strontium atoms on the intrinsic superconductor parameters of these samples was studied by excess conductivity analysis. All the samples showed orthorhombic crystal structure in space group Pmmm, and the cell parameters were determined by using all the planar reflections. The c-axis length and the unit cell volume decreased with Sr doping in the final compound. Suppression of the room-temperature resistivity followed by metallic variations in resistivity versus temperature measurements are typical features for these samples. The zero-resistivity critical temperature and the onset of diamagnetism were suppressed with increasing Sr doping in the final compound. The apical oxygen phonon mode of type Cu(1)–OA–Cu(2) observed at around 548 cm−1 hardened with increasing Sr doping. The excess conductivity analysis revealed that the coherence length along the c-axis, the interlayer coupling, and the Fermi velocity of the carriers increased for the Sr doping levels of x = 0.1 and 0.2 but decreased for the sample with x = 0.3. The values of Bc0(T), Bc1(T), and Jc0(0) increased with increasing Sr doping in the final compound. It is proposed that this effect arises due to an increase in the superconducting volume fraction. A decrease in the value of the London penetration depth λp.d. and the Ginzburg–Landau (GL) parameter shows that the flux-pinning characteristics of the samples were improved by Sr doping. An increase in the mean free time of the carriers and a decrease in the energy required to break apart Cooper pairs result from a decrease in the remanent field scattering induced by the increased population of pinning centers in Sr-doped samples.

Pressure effect on the formation kinetics of ferroelectric domain structure under first order phase transitions

Within the framework of Landau–Ginzburg theory the kinetics of domain structure formation in ferroelectrics that undergo first order phase transition was investigated under the influence of hydrostatic pressure. It was established that mechanical action increases the tendency of nonequilibrium system in the formation of stable polydomain structure. Numerical analysis showed that the process of domain ordering can proceed both directly and with formation of short-living metastable polydomain phase. It was found that the incubation period of domain structure formation can be observed at the initial stage of relaxation. Using the KNO3 crystal as an example, it was shown that the time of evolution of the system to the thermodynamic equilibrium is inversely proportional to the pressure value. It was established that in KDP crystals near phase transition point (TC–T < 10 K) the hydrostatic pressure induces the destruction of ferroelectric ordering. It was shown that the value of critical pressure pcr increases with the growth of quenching temperature.

Landau–Ginzburg/Calabi–Yau Correspondence for a Complete Intersection via Matrix Factorizations

Abstract By generalizing the Landau–Ginzburg/Calabi–Yau correspondence for hypersurfaces, we can relate a Calabi–Yau complete intersection to a hybrid Landau–Ginzburg model: a family of isolated singularities fibered over a projective line. In recent years Fan, Jarvis, and Ruan have defined quantum invariants for singularities of this type, and Clader and Clader–Ross have provided an equivalence between these invariants and Gromov–Witten invariants of complete intersections, in this way quantum cohomology yields an identification of the cohomology groups of the Calabi–Yau and of the hybrid Landau–Ginzburg model. It is not clear how to relate this to the known isomorphism descending from derived equivalences (due to Segal and Shipman, and Orlov and Isik). We answer this question for Calabi–Yau complete intersections of two cubics.

Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics

In this paper, the first in a series, we study the deformed Hermitian–Yang–Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas’ GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold \({\mathcal {H}}\) closely related to Solomon’s space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with \(C^{1,\alpha }\) regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. As an application of our techniques we give a simplified proof of Chen’s theorem on the existence of \(C^{1,\alpha }\) geodesics in the space of Kähler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM [26] and special Lagrangians in Landau–Ginzburg models [27].

A linearized element-free Galerkin method for the complex Ginzburg–Landau equation

In this paper, an effective linearized element-free Galerkin (EFG) method is developed for the numerical solution of the complex Ginzburg–Landau (GL) equation. To deal with the time derivative and the nonlinear term of the GL equation, an explicit linearized procedure is presented. The unconditional stability and the error estimate of the procedure are analyzed. Then, a stabilized EFG method is proposed to establish linear algebraic systems. In the method, the penalty technique is used to facilitate the satisfaction of boundary conditions, and the stabilized moving least squares approximation is used to enhance the stability and performance. The linearized EFG method is a meshless method and possesses high precision and convergence rate in both space and time. Theoretical error and convergence of the linearized EFG method are analyzed. Finally, some numerical results are provided to demonstrate the efficiency of the method and confirm the theoretical results.