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تسجيل مستخدم جديدVortex-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 Mo
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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 m
athbb{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.
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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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