1. VORTEX PHASES IN PERIODIC PLUS RANDOM PINNING POTENTIAL Walter Pogosov, 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

2. Introduction Model Defects of vortex lattice Phase diagram Summary Outline

3. Introduction Vortex lattice in regular square array of pins strong pinningintermediate pinningweak pinning Competition between the symmetries of the pinning and vortex arrays: □ versus Δ

4. London approximation Linearized Ginzburg-Landau equations Full system of Ginzburg-Landau equations Molecular-dynamic simulations Vortices in ultracold gases (theory + experiment) Submillimeter-size charged balls (experiment)

5. Why random pinning? - it can always be found in real systems - 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 ?

6. Vortices as point-like objects in 2D Pinning potential Energy of the system Model

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

8. 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 values of random pinning strength, required for their generation Defects of vortex lattice

9. sine-Gordon for 2D elastic media u is a deformation field Smooth elastic kinks are elementary defects destroying the order Is our system of well-studied sine- Gordon type?

10. 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

11. Smooth kinks versus sharp defects Elastic energy of interaction of vortex rows Energy of depinning Total energy

12. Schematic plot of defects in square lattice 1.These strings are elastic, i.e., vortices behave collectively 2. Strings can be considered as nuclei of half-pinned phase (memory of this phase)

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

14. Another defect: domains of 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

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

16. Effective 1D sine-Gordon system theory of elasticity is applicable

17. Quasi-1D defect’s length and energy - Quasi-1D defects are elastic objects, vortices behave collectively - Smooth transition to the single-pinning regime with quasi-0D defects!

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

20. Molecular-dynamics simulations

24. Fractal-like defects with quasi-self-similarity appear in the system on the first step of disordering Fractal’s 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

25. weak pinningintermediatestrong pinning regular pinning strength Random vsregularRandom vsregular

26. - 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