[Paper Review] Fractal vortex structure in the lattice model of superconductors

Serguei Ktitorov|arXiv (Cornell University)|Dec 31, 2002

Theoretical and Computational Physics3 citations

TL;DR

This paper proposes a nonlinear Harper-Ginzburg-Landau (HGL) equation to describe vortex structures in lattice superconductors under strong magnetic fields, showing that fractal vortex configurations emerge from a two-dimensional nonlinear map on the complex field. The key result is that the vortex lattice becomes a fractal structure governed by dynamical maps, with the order parameter distribution generalizing elliptic theta functions to include fractal effects via discrete maps such as the Chirikov standard map.

ABSTRACT

The problem of the vortex structure in the lattice modele of superconductors has been reduced to the nonlinear map problem characteristic for the fractal theory.

Motivation & Objective

• To investigate vortex structure formation in lattice superconductors at strong magnetic fields where the flux quantum is not small.
• To reformulate the lattice Ginzburg-Landau equation in a way that does not assume small magnetic flux parameter β = Φ/Φ₀.
• To explore the emergence of fractal vortex configurations due to nonlinear dynamics and quasiperiodic potential effects.
• To establish a connection between vortex lattice geometry and nonlinear dynamical maps on the complex field.

Proposed method

• Derive the lattice Ginzburg-Landau (LGL) equation from a mean-field free energy functional with nearest-neighbor tunneling and vector potential coupling.
• Apply the Landau gauge and tight-binding approximation to reduce the LGL equation to a nonlinear Harper equation (NHE) in terms of complex order parameter amplitudes.
• Introduce a transformation φₘₙ = uₘ exp(iκm) to decouple the phase dependence and obtain the nonlinear Harper-Ginzburg-Landau (HGL) equation.
• Reformulate the HGL equation as a two-dimensional nonlinear map on the complex field, enabling analysis via dynamical systems theory.
• Use the Chirikov standard map and its generalized form to model vortex nucleation and interaction, linking to the Frenkel-Kontorova model.
• Generalize the Gaussian-Wannier-type wavefunction ansatz to include fractal effects through discrete maps, leading to a theta-function-like expression with fractal parameters.

Experimental results

Research questions

• RQ1 How does the vortex structure in a lattice superconductor deviate from periodicity when the magnetic flux per plaquette is irrational?
• RQ2 What dynamical mechanism leads to the formation of self-similar, fractal vortex configurations in the absence of small-β approximations?
• RQ3 Can the nonlinear Ginzburg-Landau equation on a lattice be mapped to a known class of nonlinear dynamical systems?
• RQ4 How do fractal eigenfunctions of the Harper operator influence the spatial distribution of the superconducting order parameter?
• RQ5 In what way do nonlinear maps on the complex field describe the nucleation and spatial organization of vortices in a crystal lattice?

Key findings

• The vortex structure in the lattice superconductor is governed by a nonlinear map on the complex field, leading to fractal configurations that are not periodic in general.
• The nonlinear Harper-Ginzburg-Landau (HGL) equation emerges as a two-dimensional map, allowing the description of vortex lattices via dynamical systems theory.
• At irrational flux ratios β, the spectrum of the linear Harper operator forms a Cantor set, enabling the absence of dimensional crossover and supporting fractal behavior.
• The order parameter distribution is generalized to a sum involving fractal sequences uₙ, which reduce to an elliptic theta function when uₘ = 0.
• The Chirikov standard map and its generalized form are identified as viable models for vortex dynamics, particularly when vortex interactions are included.
• The absence of a small-β assumption allows the study of fractal vortex structures even at strong magnetic fields, where traditional perturbative methods fail.