### Article

## Topology of the planar phase of superfluid 3He and bulk-boundary correspondence for three-dimensional topological superconductors

We provide topological classification of possible phases with the symmetry of the planar phase of superfluid $\mathrm{He3.\; Compared\; to\; the\; B\; phase\; [class\; DIII\; in\; classification\; of\; A.\; Altland\; and\; M.\; R.\; Zirnbauer,Phys.\; Rev.\; B55,\; 1142\; (1997)],\; it\; has\; an\; additional\; symmetry,\; which\; modifies\; the\; topology.\; We\; analyze\; the\; topology\; in\; terms\; of\; explicit\; mappings\; from\; the\; momentum\; space\; and\; also\; discuss\; explicitly\; topological\; invariants\; for\; the\; B\; phase.\; We\; further\; show\; how\; the\; bulk-boundary\; correspondence\; for\; the\; three-dimensional\; (3D)\; B\; phase\; can\; be\; inferred\; from\; that\; for\; the\; 2D\; planar\; phase.\; A\; general\; condition\; is\; derived\; for\; the\; existence\; of\; topologically\; stable\; zero\; modes\; at\; the\; surfaces\; of\; 3D\; superconductors\; with\; class-DIII\; symmetries.}$

We analyze the entanglement spectrum of superfluid phases of $^3$He, the 3D B-phase and the planar phase in two dimensions. We find explicitly the wave functions of the low-lying eigenstates, including Majorana zero modes, as well as the corresponding part of the spectrum of the entanglement Hamiltonian.

The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal’s Γ-spaces. The formalism of topological operads generalises well to different categories yielding such notions as (Formula presented.)-algebras in chain complexes, while the Γ-space approach faces difficulties. In this paper we discuss how, by attempting to extend the Segal approach to arbitrary categoires, one arrives to the problem of understanding “weak” sections of a homotopical Grothendieck fibration. We propose a model for such sections, called derived sections, and study the behaviour of homotopical categories of derived sections under the base change functors. The technology developed for the base-change situation is then applied to a specific class of “resolution” base functors, which are inspired by cellular decompositions of classifying spaces. For resolutions, we prove that the inverse image functor on derived sections is homotopically full and faithful.

Collective plasmon excitations in a helical electron liquid on the surface of strong three-dimensional topological insulator are considered. The properties and internal structure of these excitations are studied. Due to spin-momentum locking in helical liquid on a surface of topological insulator, the collective excitations should manifest themselves as coupled charge- and spin-density waves.

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.