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Register a new userVortex-Line Percolation in the Three-Dimensional Complex Ginzburg-Landau Model
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We study the phase transition of the three-dimensional complex |psi|^4 theory by considering the geometrically defined vortex-loop network as well as the magnetic properties of the system in the vicinity of the critical point. Using high-precision Monte Carlo techniques we examine an alternative formulation of the geometrical excitations in relation to the global O(2)-symmetry breaking, and check if both of them exhibit the same critical behavior leading to the same critical exponents and therefore to a consistent description of the phase transition. Different percolation observables are taken into account and compared with each other. We find that different definitions of constructing the vortex-loop network lead to different results in the thermodynamic limit, and the percolation thresholds do not coincide with the thermodynamic phase transition point.
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In discussing the phase transition of the three-dimensional complex |psi|^4 theory, we study the geometrically defined vortex-loop network as well as the magnetic properties of the system in the vicinity of the critical point. Using high-precision Monte Carlo techniques we investigate if both of them exhibit the same critical behavior leading to the same critical exponents and hence to a consistent description of the phase transition. Different percolation observables are taken into account and compared with each other. We find that different connectivity definitions for constructing the vortex-loop network lead to different results in the thermodynamic limit, and the percolation thresholds do not coincide with the thermodynamic phase transition point.
We study the critical behaviour of the three-dimensional U(1) gauge+Higgs theory (Ginzburg-Landau model) at large scalar self-coupling lambda (``type II region) by measuring various correlation lengths as well as the Abrikosov-Nielsen-Olesen vortex tension. We identify different scaling regions as the transition is approached from below, and carry out detailed comparisons with the criticality of the 3d O(2) symmetric scalar theory.
It is believed that the two-dimensional massless $mathcal{N}=2$ Wess--Zumino model becomes the $mathcal{N}=2$ superconformal field theory (SCFT) in the infrared (IR) limit. We examine this theoretical conjecture of the Landau--Ginzburg (LG) description of the $mathcal{N}=2$ SCFT by numerical simulations on the basis of a supersymmetric-invariant momentum-cutoff regularization. We study a single supermultiplet with cubic and quartic superpotentials. From two-point correlation functions in the IR region, we measure the scaling dimension and the central charge, which are consistent with the conjectured LG description of the $A_2$ and $A_3$ minimal models, respectively. Our result supports the theoretical conjecture and, at the same time, indicates a possible computational method of correlation functions in the $mathcal{N}=2$ SCFT from the LG description.
For each given $ngeq 2$, we construct a family of entire solutions $u_varepsilon (z,t)$, $varepsilon>0$, with helical symmetry to the 3-dimensional complex-valued Ginzburg-Landau equation begin{equation*} onumber Delta u+(1-|u|^2)u=0, quad (z,t) in mathbb{R}^2times mathbb{R} simeq mathbb{R}^3. end{equation*} These solutions are $2pi/varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior as $varepsilonto 0$ $$ u_varepsilon (z,t) approx prod_{j=1}^n Wleft( z- varepsilon^{-1} f_j(varepsilon t) right), $$ where $W(z) =w(r) e^{itheta} $, $z= re^{itheta},$ is the standard degree $+1$ vortex solution of the planar Ginzburg-Landau equation $ Delta W+(1-|W|^2)W=0 text{ in } mathbb{R}^2 $ and $$ f_j(t) = frac { sqrt{n-1} e^{it}e^{2 i (j-1)pi/ n }}{ sqrt{|logvarepsilon|}}, quad j=1,ldots, n. $$ Existence of these solutions was previously conjectured, being ${bf f}(t) = (f_1(t),ldots, f_n(t))$ a rotating equilibrium point for the renormalized energy of vortex filaments there derived, $$ mathcal W_varepsilon ( {bf f} ) :=pi int_0^{2pi} Big ( , frac{|log varepsilon|} 2 sum_{k=1}^n|f_k(t)|^2-sum_{j eq k}log |f_j(t)-f_k(t)| , Big ) mathrm{d} t, $$ corresponding to that of a planar logarithmic $n$-body problem. These solutions satisfy $$ lim_{|z| to +infty } |u_varepsilon (z,t)| = 1 quad hbox{uniformly in $t$} $$ and have nontrivial dependence on $t$, thus negatively answering the Ginzburg-Landau analogue of the Gibbons conjecture for the Allen-Cahn equation, a question originally formulated by H. Brezis.
After a brief introduction to the complex Ginzburg-Landau equation, some of its important features in two space dimensions are reviewed. A comprehensive study of the various phases observed numerically in large systems over the whole parameter space is then presented. The nature of the transitions between these phases is investigated and some theoretical problems linked to the phase diagram are discussed.
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