Topological Classification of Correlations in 2D Electron Systems in Magnetic or Berry Fields

Recent topology classification of 2D electron states induced by different homotopy classes of mappings of the planar Brillouin zone into Bloch space can be supplemented by a homotopy classification of various phases of multi-electron homotopy patterns induced by Coulomb interaction between electrons. The general classification of such type is presented. It explains the topologically protected correlations responsible for integer and fractional Hall effects in 2D multi-electron systems in the presence of perpendicular quantizing magnetic field or Berry field, the latter in topological Chern insulators. The long-range quantum entanglement is essential for homotopy correlated phases in contrast to local binary entanglement for conventional phases with local order parameters. The classification of homotopy long-range correlated phases induced by the Coulomb interaction of electrons has been derived in terms of homotopy invariants and illustrated by experimental observations in GaAs 2DES, graphene monolayer, and bilayer and in Chern topological insulators. The homotopy phases are demonstrated to be topologically protected and immune to the local crystal field, local disorder, and variation of the electron interaction strength. The nonzero interaction between electrons is shown, however, to be essential for the definition of the homotopy invariants, which disappear in gaseous systems.

Tree invariants and Milnor linking numbers with indeterminacy

This paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak, which are closely related to the classical Milnor linking numbers also known as [Formula: see text]-invariants. We prove that, analogously as for [Formula: see text]-invariants, certain residue classes of tree invariants yield link-homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor’s indeterminacy. For practical purposes, we also provide a recursive procedure for computing arrow polynomials of tree invariants.

The homology groups of the space $\Omega_n(m)$

The spaces $\Omega_n(m)$ that generalize the spaces $\Omega_n$ are introduced. In order to investigate the homotopy invariants of the space $\Omega_n(m)$ the CW-structure of the space $\Omega_n(m)$ is built. Using exact homology sequence the homology groups of the space $\Omega_n(m)$ are calculated.

CATEGORY AND TOPOLOGICAL COMPLEXITY OF THE CONFIGURATION SPACE

The Lusternik–Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik–Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product $G\times \mathbb{R}^{n}$ and apply the results to the planar and spatial motion of two rigid bodies in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ respectively.

-homotopy invariants of topological Fukaya categories of surfaces

We provide an explicit formula for localizing$\mathbb{A}^{1}$-homotopy invariants of topological Fukaya categories of marked surfaces. Following a proposal of Kontsevich, this differential$\mathbb{Z}$-graded category is defined as global sections of a constructible cosheaf of dg categories on any spine of the surface. Our theorem utilizes this sheaf-theoretic description to reduce the calculation of invariants to the local case when the surface is a boundary-marked disk. At the heart of the proof lies a theory of localization for topological Fukaya categories which is a combinatorial analog of Thomason–Trobaugh’s theory of localization in the context of algebraic$K$-theory for schemes.