Вихревая решетка в системе периодических и случайных центров пиннинга В. Погосов, Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, Russia In collaboration with V. R. Misko, H. J. Zhao, and F. M. Peeters, Departement Fysica, Universiteit Antwerpen, Antwerpen, Belgium W. V. Pogosov, V. R. Misko, H. J. Zhao, and F. M. Peeters "Collective vortex phases in periodic plus random pinning potential", Phys. Rev. B 79, (2009).

Introduction Model Defects of vortex lattice Random pinning Phase diagram Summary Outline

Introduction Vortex lattice in regular square array of pins strong pinningintermediate pinningweak pinning Competition between the symmetries of pinning and vortex arrays: versus Δ W. V. Pogosov, A. L. Rakhmanov, and V. V. Moshchalkov, Phys. Rev. B 67, (2003).

Vortices in ultracold gases (theory + experiment) Submillimeter-size charged balls (experiment)

Why random pinning? - it can always be found in real systems; what is its influence on critical current? - interesting physics with interdisciplinary importance - other dimensionalities: FLL in superconductors with layers or twin boundaries; spin, charge, mass, polarization density waves, etc. Vortex lattice structure is determined by a competition of three factors: (i)a square array of regular pinning sites (ii)vortex-vortex interaction, which favors a triangular symmetry, (iii)a random potential, which is going to destroy the regularity in vortex positions ?

Vortices as point-like objects in 2D Model

Pinning potential Energy of the system Molecular-dynamic simulations

Why defects? They play a crucial role in the process of disordering Our strategy is: -to identify and classify possible defects -to estimate their typical energies and sizes -to find random pinning strength, required for their generation Defects of vortex lattice

sine-Gordon for 2D elastic media u is a deformation field Smooth elastic kinks are elementary defects destroying the order J.-P. Bouchaud and A. Georges, PRL (1992); T. Emig and T. Natterman, EPJB (1999) Is our system of sine-Gordon type?

Our system is NOT of a sine- Gordon type! Square array of free interacting vortices is locally unstable with respect to deformations -elastic constant is negative -kink is not smooth

Smooth kinks versus sharp defects Структура бесконечного кинка энергия взаимодействия ряда вихрей с другими вихрями энергия кинка (межвихревое взаимодействие): энергия кинка (взаимодействие с центрами пиннинга): полная энергия кинка:

1.These strings are elastic, i.e., vortices behave collectively 2. Strings can be considered as nuclei of the half-pinned phase (memory of this phase)

Elementary defect in the half-pinned phase Nuclei of pinned phase

Another defect: domains of a deformed triangular lattice inside square pinned and half-pinned phases These domains are pieces of elastic media, vortices behave collectively All defects considered so far are two-dimensional! domain

Quasi-1D defects at higher values of regular pinning strength

Effective 1D sine-Gordon system elasticity theory энергия взаимодействия с рядами невозмущенных вихрей энергия пиннинга

Quasi-1D defects length and energy - Quasi-1D defects are elastic objects, vortices behave collectively - Smooth transition to the single-pinning regime with quasi-0D defects энергия внутренней деформации дефекта (растяжение + сжатие)

Random pinning n r is a concentration of pinning sites Effective pinning potential

Phase diagram We know now energies and sizes of various kinds of defects

Molecular-dynamics simulations

Fractal-like defects with quasi-self-similarity appear in the system on the first step of disordering Fractals dimensionality evolves smoothly from 2 to 0, i.e., from the collective pinning regime towards a single- pinning regime Domains of totally depinned vortices appear on the second step The width of this region on the phase diagram shrinks when approaching to the single-pinning regime, where these domains become quasi-0D defects matching with fractals

weak pinningintermediatestrong pinning regular pinning strength Random vsregularRandom vsregular

- The disordering of vortex lattices was studied both analytically and by molecular-dynamics simulations - The U r - U 0 phase diagram was constructed and various regimes of one- or two-step disordering (via chain defects and domain-like defects) were identified with a lot of nontrivial features. - We discovered a unified scenario of disordering of square lattice by fractal-like defects with smoothly varying dimensionality from 2 to 0, in the whole region of pinning strengths – from very weak to very strong ones Analytics + molecular-dynamics simulations is a nice approach! Summary

Дальнейшие планы: -критические токи: влияние беспорядка, динамические фазы, перколяционные явления -температурные флуктуации в отсутствии случайного потенциала и в его присутствии -трехмерный случай