### Article

## Anisotropic superconductors between types I and II

Self-duality or matching between the magnetic and the condensate coherence lengths is a fundamental property

of isotropic superconductors at the critical Bogomolnyi point (B point). The self-dual state of the condensate

is infinitely degenerate, which is the core reason for the sharp transition between the superconductivity types in

the nearest vicinity of the critical temperature Tc. Below Tc nonlocal interactions in the condensate remove

the degeneracy, which leads to the appearance of a finite intertype (IT) domain between types I and II. This

domain exhibits the mixed state with exotic field-condensate configurations and nonstandard magnetic response,

which cannot be understood within the dichotomy of the conventional superconductivity types. At a first glance,

this picture does not apply to an anisotropic system because no spatial matching between the condensate and

magnetic field can be generally expected for direction-dependent characteristic lengths. However, contrary

to these expectations, here we demonstrate that anisotropic superconductors follow the same scenario of the

interchange between types I and II. In anisotropic materials the IT domain is governed by the B point of the

effective isotropic model obtained by the appropriate scaling transformation of the initial anisotropic formalism.

This transformation depends on the direction of the applied magnetic field, and thus the superconductivity type

of strongly anisotropic materials can be dependent on this direction.

The Bogomolnyi point separates superconductivity types I and II while itself hiding infinitely degenerate magnetic flux configurations, including very exotic states (referred to here as flux “monsters”). When the degeneracy is removed, the Bogomolnyi point unfolds into a finite, intertype domain in the phase diagram between types I and II. One can expect that in this case the flux monsters can escape their “prison” at the Bogomolnyi point, occupying the intertype domain and shaping its internal structure. Our calculations reveal that such exotic flux distributions are indeed stable in the intertype regime of thin superconductors made of a type-I material, where the Bogomolnyi degeneracy is removed by stray magnetic fields. They can be classified into three typical patterns that are qualitatively different from those in types I and II: superconducting islands separated by vortex chains; stripes/worms/labyrinths patterns; and mixtures of giant vortices and vortex clusters. Our findings shed light on the problem of the interchange between types I and II, raising important questions on the completeness of the textbook classification of the superconductivity types.

It is well known that superconducting films made of a type-I material can demonstrate a type-II magnetic response, developing stable vortex configurations in a perpendicular magnetic field. Here we show that the superconducting state of a type-I nanowire undergoes more complex transformations, depending on the nanowire thickness. Sufficiently thin nanowires deviate from type I and develop multiquantum vortices and vortex clusters similar to intertype (IT) vortex states in bulk superconductors between conventional superconductivity types I and II. When the nanowire thickness decreases further, the quasi-one-dimensional vortex matter evolves towards type II so that the IT vortex configurations gradually disappear in favor of the standard Abrikosov lattice (chain) of single-quantum vortices. However, type II is not reached. Instead, an ultrathin nanowire re-enters abruptly the type-I regime while vortices tend to be suppressed by the boundaries, eventually becoming one-dimensional phase-slip centers. Our results demonstrate that arrays of nanowires can be used to construct composite superconducting materials with a widely tunable magnetic response.

Superconducting films are usually regarded as type II superconductors even when they are made of a type I material. The reason is the presence of stray magnetic fields that stabilize the vortex matter by inducing long-range repulsive interactions between vortices. While very thin films indeed reach this limit, there is a large interval of thicknesses where magnetic properties of superconducting films cannot be classified as either of the two conventional superconductivity types. Recent calculations revealed that in this interval the system exhibits spontaneous formation of magnetic flux-condensate patterns and superstructures appearing due to the interplay between the long-range stray field effects and proximity to the Bogomolnyi selfduality point. These calculations were based on the periodic in-plane boundary conditions which, as is well known from classical electrodynamics, for systems with long-range interactions can lead to field distortions and considerable discrepancies between results of different calculation methods. Here we demonstrate that similar spontaneous patterns are obtained for superconducting films with open in-plane boundary conditions (vanishing in-plane currents perpendicular to the edges of the finite film) and thus the phenomenon is not an artefact of chosen boundary conditions.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

Let G be a semisimple algebraic group whose decomposition into the product of simple components does not contain simple groups of type A, and P⊆G be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples (G, P, n) such that (a) there exists an open G-orbit on the multiple flag variety G/P × G/P × . . . × G/P (n factors), (b) the number of G-orbits on the multiple flag variety is finite.

I give the explicit formula for the (set-theoretical) system of Resultants of m+1 homogeneous polynomials in n+1 variables