Inspired by the human brain, there is a strong effort to find alternative models of information processing capable of imitating the high energy efficiency of neuromorphic information processing. One possible realization of cognitive computing are reservoir computing networks. These networks are built out of non-linear resistive elements which are recursively connected. We propose that a skyrmion network embedded in frustrated magnetic films may provide a suitable physical implementation for reservoir computing applications.The significant key ingredient of such a network is a two-terminal device with non-linear voltage characteristics originating from single-layer magnetoresistive effects, like the anisotropic magnetoresistance or the recently discovered non-collinear magnetoresistance. The most basic element for a reservoir computing network built from "skyrmion fabrics" is a single skyrmion embedded in a ferromagnetic ribbon. In order to pave the way towards reservoir computing systems based on skyrmion fabrics, here we simulate and analyze i) the current flow through a single magnetic skyrmion due to the anisotropic magneto-resistive effect and ii) the combined physics of local pinning and the anisotropic magneto-resistive effect. 1 arXiv:1702.04298v2 [cond-mat.dis-nn]

We investigate how a magnetic field induces one-dimensional edge channels when the twodimensional surface states of three-dimensional topological insulators become gapped. The Hall effect, measured by contacting those channels, remains quantized even in situations, where the θ-term in the bulk and the associated surface Hall conductivities, σ S xy , are not quantized due to the breaking of time-reversal symmetry. The quantization arises as the θ-term changes by ±2πn along a loop around n edge channels. Model calculations show how an interplay of orbital and Zeeman effects leads to quantum Hall transitions, where channels get redistributed along the edges of the crystal. The network of edges opens new possibilities to investigate the coupling of edge channels.In topological insulators (TIs) the topological properties of the band structure leads to the formation of protected metallic states at the surface. Soon after their theoretical prediction 1,2 as a generalization of Haldane's Chern insulator 3 their two-dimensional (2d) variantthe quantum spin Hall insulator -was realized in quantum well heterostructures by König et al.4 . Later, threedimensional (3d) varieties of this novel form of insulator were predicted 5-7 and again realized afterwards 8,9 . 3d TIs come as strong and weak TIs. Strong TIs (STIs) have a metallic surface which is protected against localization due to its helical nature. Band topology enforces that the surface metal can be described as an effective 2d Dirac theory of massless electrons. Soon after the realization of 3d STIs it was understood that the surfaces of these materials could give way to an unconventional Hall response whenever the surface is gapped by perturbations which break time-reversal symmetry (TRS). This is related to the axion quantum electrodynamics 10-12 and the variation of the θ-angle at an interface between a bulk material (θ = 0) and vacuum (θ = 0). STIs are characterized by the quantized value θ = π. When a gap is opened, a dissipationless surface conductivity with a half-integer Hall conductivity 11,12 , σ S xy = e 2 2h , arises. Importantly, this value is not quantized if both TRS and inversion symmetry (IS) are broken in the bulk.Recently, Brüne et al.9 have experimentally observed a quantized Hall effect for films of HgTe which can be considered 3d STIs, with the magnetic field applied perpendicular to the film. Both in the experiment and in two theoretical studies 13-15 a field configuration was used, where only two faces of the crystal were gapped.In this paper we investigate the Hall response of a 3d STI in the presence of a bulk magnetic field oriented such that all surface excitations are gapped. We first study how the θ-term characterizing the 3d bulk, the surface Hall conductivities, the edge channels, and the quantum Hall effect observed by contacting those edge channels are related to each other. We then study a concrete lattice model (inspired by the HgTe band structure) to investigate how the location and number of edge channels can be controlled.We co...

Berry phases occur when a system adiabatically evolves along a closed curve in parameter space. This tutoriallike article focuses on Berry phases accumulated in real space. In particular, we consider the situation where an electron traverses a smooth magnetic structure, while its magnetic moment adjusts to the local magnetization direction. Mapping the adiabatic physics to an effective problem in terms of emergent fields reveals that certain magnetic textures -skyrmions -are tailormade to study these Berry phase effects.Over the last decades physicists have realized that a seemingly harmless phase -the Berry phase -leads to a lot of interesting physical consequences like quantum, This paper is structured as follows: First, we review the general concept of Berry phases and introduce the main formulas. In Sec. II, we focus on real-space Berry phase effects. Here we discuss the example of a free electron traversing a spatially or temporally inhomogeneous, smooth magnetic structure. In the adiabatic limit, the magnetic moment of the electron adjusts constantly to the local magnetization direction and thereby the electron picks up a Berry phase. To interpret the physical consequences of these Berry phases, we review the mapping onto a problem, where the electron moves in a uniform Zeeman magnetic field, but instead "feels" an emergent electric field and an emergent orbital magnetic field, leading, for example, to the topological Hall effect 4 . In Sec. III, we first introduce skyrmions in a formal way and then later focussing on skyrmions in magnetic systems, in particular those in a certain material class denoted as B20 materials. We discuss that those magnetic textures are tailormade to observe the emergent electric fields introduced in Sec. II A. In Sec. III C we switch the perspective and discuss the back reaction of the electronic Berry phase effects on the skyrmion lattice. I. BERRY PHASESBerry phases occur in all parts of physics and arise whenever a system adiabatically evolves along a cyclic process C in some parameter space X. A simple classical example is the parallel transport of a vector along a closed curve on a sphere (see Fig. 1). Although the system returns to its initial state it aquires a geometric phase characterized by the enclosed area on the sphere (Gauss-Bonnet theorem).A quantum-mechanical example is a system whose states are non-degenerate and which evolves adiabatically in the parameter space X = X(t). In that case, the solution of the time-dependent Schrödinger equation i ∂ t |ψ(t) = H(X(t)) |ψ(t) is given by 11 |ψ(t) = n a n (t 0 )e iγn(C) e − i t 0 dt n (X(t )) |ψ n (t) , (1) where |ψ n (t) and n (X(t)) are the time-evolved eigenstates and energies of the Hamiltonian H(X(t)), |ψ(t 0 ) = n a n (t 0 ) |ψ n (t 0

Reservoir Computing is a type of recursive neural network commonly used for recognizing and predicting spatio-temporal events relying on a complex hierarchy of nested feedback loops to generate a memory functionality. The Reservoir Computing paradigm does not require any knowledge of the reservoir topology or node weights for training purposes and can therefore utilize naturally existing networks formed by a wide variety of physical processes. Most efforts prior to this have focused on utilizing memristor techniques to implement recursive neural networks. This paper examines the potential of skyrmion fabrics formed in magnets with broken inversion symmetry that may provide an attractive physical instantiation for Reservoir Computing.

We consider a quantum wire with two subbands of spin-polarized electrons in the presence of strong interactions. We focus on the quantum phase transition when the second subband starts to get filled as a function of gate voltage. Performing a one-loop renormalization group analysis of the effective Hamiltonian, we identify the critical fixed-point theory as a conformal field theory having an enhanced SU(2) symmetry and central charge 3/2. While the fixed point is Lorentz invariant, the effective "speed of light" nevertheless vanishes at low energies due to marginally irrelevant operators leading to a diverging critical specific heat coefficient.