Landau-Ginzburg Potential Research Articles

A new description of the E6 singularity

We discuss a new type of Landau-Ginzburg potential for the E 6 singularity of the form W = const+( Q 1(x)+P 1(x) P 2(x) x 3 which featured in a recent study of heterotic/typeII string duality. Here Q 1, P 1 and P 2 are polynomials of degree 15,10 and 10, respectively. We study the properties of the potential in detail and show that it gives a new and consistent description of the E 6 singularity.

Mirror symmetry of elliptic curves and Ising model

We study the differential equations governing mirror symmetry of elliptic curves, and obtain a characterization of the ODEs which give rise to the integral q-expansion of mirror maps. Through theta function representation of the defining equation, we express the mirror correspondence in terms of theta constants. By investigating the elliptic curves in X 9-family, the identification of the Landau-Ginzburg potential with the spectral curve of Ising model is obtained. Through the Jacobi elliptic function parametrization of Boltzmann weights in the statistical model, an exact Jacobi form-like formula of mirror map is described.

A computational model for nucleation of solid-solid phase transformations

A computational methodology for nucleation of phase transformations in a class of grade 2, non-linearly elastic materials is presented. Nucleation is treated as an energy extremum problem. The material is assumed to be governed by a non-linear, non-local elastic constitutive relation represented by a Landau-Ginzburg potential. The extremum problem is solved using the element-free Galerkin (EFG) method and a perturbed Lagrangian technique. The EFG method is used because of its ability to handle continuity of displacement gradients required in the weak form. Applications to homogeneous nucleation in two dimensions are presented which illustrate the accuracy of the method and its suitability for problems of this type.

Refining the elliptic genus

We show how special forms an N = 2 Landau-Ginzburg potential directly imply the presence of an N = 2 super- W algebra. If the Landau-Ginzburg model has a super- W algebra, we show how the elliptic genus can be refined so as to give much more complete information about the structure of the model. We study the super- W 3 model in some detail, and present some results and conjectures about more general models.

Generalized two-dimensional QCD

We study two-dimensional gauge theories with fundamental fermions and a general first-order gauge-field lagrangian. For the case of U(1) we show how standard bosonization of the Schwinger model generalizes to give mesons interacting through a general Landau-Ginzburg potential. We then show how for the case of SU( N) 't Hooft's solution of large- N two-dimensional QCD can be generalized in a consistent and natural manner. We finally point out the possible relevance of studying these theories to the string formulation of two-dimensional QCD as well as to understanding QCD in higher dimensions.

Martensitic phase transition with two-component order parameter in a stressed cubic crystal

Abstract Martensitic phase transitions in a stressed cubic crystal are studied by means of Landau theory with a two-component order parameter. The stress-temperature phase diagram is analyzed theoretically and is shown to consist of six essentially different regions isolated by the liability lines of austenite, tetragonal martensitic and orthorhombic martensitic phases. The triple point is also present in the phase diagram. The role of the cubic term in the Landau-Ginzburg potential in the description of the phase transition properties is emphasized.

Tunneling from nothing toward induced gravity inflation.

We tackle the problem of the nucleation of the universe for the case in which the underlying gravity is an induced gravity with a Ginzburg-Landau potential for the scalar field. In order to make use of Vilenkin's wave function [Phys. Rev. D 33, 3560 (1986)], we cast the theory into its canonical Einstein form through a suitable conformal transformation: by using Vilenkin's boundary conditions we show then that the semiclassical tunneling from nothing solves the problem of the initial conditions in the induced gravity inflation and then selects the present observed values of the gravitational and the cosmological constants (${G}_{\mathrm{eff}}={G}_{N}$, ${\ensuremath{\Lambda}}_{\mathrm{eff}}=0$). Therefore our result is an improvement with respect to those obtained by other authors either (i) with the assumption of no-boundary conditions for the cosmic wave function, or (ii) with the same Ginzburg-Landau potential, but in the standard Einstein gravity.

Nonlinear noise in cosmology.

This paper derives and analyzes exact, nonlocal Langevin equations appropriate in a cosmological setting to describe the interaction of some collective degree of freedom with a surrounding ``environment.'' Formally, these equations are much more general, involving as they do a more or less arbitrary ``system,'' characterized by some time-dependent potential, which is coupled via a nonlinear, time-dependent interaction to a ``bath'' of oscillators with time-dependent frequencies. The analysis reveals that, even in a Markov limit, which can often be justified, the time dependences and nonlinearities can induce new and potentially significant effects, such as systematic and stochastic mass renormalizations and state-dependent ``memory'' functions, aside from the standard ``friction'' of a heuristic Langevin description. One specific example is discussed in detail, namely the case of an inflaton field, characterized by a Landau-Ginsburg potential, that is coupled quadratically to a bath of scalar ``radiation.'' The principal conclusion derived from this example is that nonlinearities and time-dependent couplings do {\em not} preclude the possibility of deriving a fluctuation-dissipation theorem, and do {\em not} change the form of the late-time steady state solution for the system, but {\em can} significantly shorten the time scale for the approach towards the steady state.

Intermittency in second-order phase transitions.

It is shown that self-similar behavior in multiplicity fluctuations exists in the Ginzburg-Landau description of second-order phase transitions. Furthermore, there exists a numerical exponent that characterizes the intermittency properties in the hadronic phase and is independent of the specific values of the coefficients in the Ginzburg-Landau potential. Current data on intermittency are only 2\ensuremath{\sigma} away from the critical exponent.

Topological Landau-Ginzburg matter at c=3

The topological correlation functions, their prepotential, and the Landau-Ginzburg potential are computed for the N=2 supersymmetric, c=3, matter model that is a tensor product of three c=1 models.

The dynamics of unstable-state decay

We consider a Fokker-Planck equation in the “order parameter” space and present an asymptotic method to obtain in the low-noise limit the time-dependent probability density, its moments and the first-passage time distribution. We apply the general results to the Ginzburg-Landau potential and its extention for a laser model with saturation. Special attention is paid to the stable equilibrium point and the macroscopically unaccessible region beyond it not studied before. We demonstrate that some of our results (e.g. mean first-passage time) coincide with those yielded by exact expressions in the appropriate limit.

Mean field theory, topological field theory, and multi-matrix models

We show that the genus zero correlation functions of an arbitrary topological field theory coupled to two-dimensional topological gravity are determined by an appropriate Landau-Ginzburg potential. We determine the potentials that arise for topological sigma models with CP 1 or a Calabi-Yau manifold for target space. We present substantial evidence that the multi-matrix models that have been studied recently are equivalent to certain topological field theories coupled to topological gravity. We also describe a topological version of the general “string equation”.

An exactly soluble model for interfacial kinetics

We study an exactly soluble model for planar interfacial kinetics for both locally conserved and nonconserved order parameters, where the Landau–Ginzburg potential is approximated by two parabolas. Long-range interactions with arbitrary interaction kernels are included. In particular, the relaxation of the initial order parameter interfacial profile to its equilibrium profile is considered. Furthermore, the steady-state solution of planar interfacial growth for a nonconserved order parameter is determined. Here, the influence of a second nonconserved order parameter is also discussed.

Remarks on the Landau-Ginzburg potential and RG flow for SU(2)-coset models

The existence of a Landau-Ginzburg (LG) field for the SU (2)-coset models is motivated and conjectured. The general form of the LG potential for the A series is found, and the RG flow pattern suggested by this is shown to agree with that found by other authors, thereby further supporting the conjecture.

Nucleation theory for a model bistable chemical reaction

The theory of homogeneous nucleation is developed for a model nonlinear bistable chemical reaction driven far from equilibrium (trimolecular Schlogl model). The theory is restricted to the vicinity of the stable/unstable transition, where the nucleation barrier is small but nonvanishing. The nucleation rates are derived for two types of fluctuations: first, fluctuations due to a homogeneous external white noise source, and second, internal chemical fluctuations, described by a reastion-diffusion multivariate master equation. In the white noise case, a Landau-Ginzburg potential can be defined, and the standard nucleation formalism can be applied; this is not true for the internal case and a new result is used. The inhomogeneous chemical fluctuations, due to the coupling between the nonlinear reaction and diffusion, are shown to have an influence on the nucleation rate. Quantitative conditions are also given to evaluate the possibility of homogeneous nucleation in nonlinear chemical systems.

Inflation with a non-minimally coupled scalar field

Cosmological inflation is frequently associated with a scalar field φ, which could also generate G N , via a coupling term 1 2 ϵ φ 2 R in the lagrangian, as a result of symmetry breaking. We derive a formula for density perturbations δϱ/ ϱ in the theory L = [(16π G 0) −1 + 1 2 ϵφ 2] R + 1 2 ϵ; kφ ;k −V(φ) . When G 0 −1 = 0, our results largely coincide with those of Accetta, Zoller and Turner, who assumed a Ginzburg-Landau potential V = 1 4 λ(φ 2 − φ 0 2) 2 and the formula δϱ/ ϱ = αH 2/ φ . We verify that formula and find the constant of proportionality to be α ≈ 2 × 10 −2.

Electron-hole droplet condensation — A phase transition in an open system

We derive from hydrodynamical equations a generalized Ginzburg-Landau potential for the amplitude of the u stable mode, which describes a first order phase transition in the electron-hole plasma.

Method for evaluating one-dimensional path integrals

A practical method is developed for calculating numerically one-dimensional path integrals for an arbitrary potential $V(x)$. For the oscillator potential $V(x)=\frac{k{x}^{2}}{2}$ the well-known analytic solution is obtained. To illustrate the numerical convergence of this method the path integral for the Ginzburg-Landau potential, $V(x)=\ensuremath{-}\frac{A{x}^{2}}{2}+\frac{B{x}^{4}}{4}$, is calculated over a range of the positive constants $A$ and $B$ and compared with numerical solutions of the Schr\"odinger equation.