Fabian R. Lux

Understanding the interaction of real- and reciprocal space topology in skyrmion crystals is an open problem. We approach it from the viewpoint of C^\ast C * -algebras and calculate all admissible Chern numbers of a strongly coupled tight-binding skyrmion system on a triangular lattice as a function of Fermi energy and texture parameters. Our analysis reveals the topological complexity of electronic states coupled to spin textures, and the failure of the adiabatic picture to account for it in terms of emergent electromagnetism. On the contrary, we explain the discontinuous jumps in the real-space winding number in terms of collective evolution in real-, reciprocal, and mixed space Chern numbers. Our work sets the stage for further research on topological dynamics in complex dynamic spin textures coupled to external fields.

Frieze groups are discrete subgroups of the full group of isometries of a flat strip. We investigate here the dynamics of specific architected materials generated by acting with a frieze group on a collection of self-coupling seed resonators. We demonstrate that, under unrestricted reconfigurations of the internal structures of the seed resonators, the dynamical matrices of the materials generate the full self-adjoint sector of the stabilized group C*-algebra of the frieze group. As a consequence, in applications where the positions, orientations and internal structures of the seed resonators are adiabatically modified, the spectral bands of the dynamical matrices carry a complete set of topological invariants that are fully accounted by the K-theory of the mentioned algebra. By resolving the generators of the K-theory, we produce the model dynamical matrices that carry the elementary topological charges, which we implement with systems of plate resonators to showcase several applications in spectral engineering. The paper is written in an expository style.

The anomalous Hall effect has been front and center in solid state research and material science for over a century now, and the complex transport phenomena in nontrivial magnetic textures have gained an increasing amount of attention, both in theoretical and experimental studies. However, a clear path forward to capturing the influence of magnetization dynamics on anomalous Hall effect even in smallest frustrated magnets or spatially extended magnetic textures is still intensively sought after. In this work, we present an expansion of the anomalous Hall tensor into symmetrically invariant objects, encoding the magnetic configuration up to arbitrary power of spin. We show that these symmetric invariants can be utilized in conjunction with advanced regularization techniques in order to build models for the electric transport in magnetic textures which are, on one hand, complete with respect to the point group symmetry of the underlying lattice, and on the other hand, depend on a minimal number of order parameters only. Here, using a four-band tight-binding model on a honeycomb lattice, we demonstrate that the developed method can be used to address the importance and properties of higher-order contributions to transverse transport. The efficiency and breadth enabled by this method provides an ideal systematic approach to tackle the inherent complexity of response properties of noncollinear magnets, paving the way to the exploration of electric transport in intrinsically frustrated magnets as well as large-scale magnetic textures.

The nontrivial topology of spin systems such as skyrmions in real space can promote complex electronic states. Here, we provide a general viewpoint at the emergence of topological electronic states in spin systems based on the methods of noncommutative K-theory. By realizing that the structure of the observable algebra of spin textures is determined by the algebraic properties of the noncommutative hypertorus, we arrive at a unified understanding of topological electronic states which we predict to arise in various noncollinear setups. The power of our approach lies in an ability to categorize emergent topological states algebraically without referring to smooth real- or reciprocal-space quantities. This opens a way towards an educated design of topological phases in aperiodic, disordered, or non-smooth textures of spins and charges containing topological defects.

Owing to their interesting spectral properties, the synthetic crystals over lattices other than regular Euclidean lattices, such as hyperbolic and fractal ones, have attracted renewed attention, especially from materials and meta-materials research communities. They can be studied under the umbrella of quantum dynamics over Cayley graphs of finitely generated groups. In this work, we investigate numerical aspects related to the quantum dynamics over such Cayley graphs. Using an algebraic formulation of the "periodic boundary condition" due to Lueck [Geom. Funct. Anal. 4, 455-481 (1994)], we devise a practical and converging numerical method that resolves the true bulk spectrum of the Hamiltonians. Exact results on the matrix elements of the resolvent, derived from the combinatorics of the Cayley graphs, give us the means to validate our algorithms and also to obtain new combinatorial statements. Our results open the systematic research of quantum dynamics over Cayley graphs of a very large family of finitely generated groups, which includes the free and Fuchsian groups.

Among antiferromagnetic materials, Mn2Au is one of the most intensively studied, and it serves as a very popular platform for testing various ideas related to antiferromagnetic magnetotransport and dynamics. Since recently, this material has also attracted considerable interest in the context of optical properties and optically-driven antiferromagnetic switching. In this work, we use first principles methods to explore the physics of charge photocurrents, spin photocurrents, and the inverse Faraday effect in antiferromagnetic Mn2Au. We predict the symmetry and magnitude of these effects and speculate that they can be used for tracking the dynamics of staggered moments during switching. Our calculations reveal the emergence of large photocurrents of spin in collinear Mn2Au, whose properties can be understood as a result of a non-linear optical version of the spin Hall effect, which we refer to as the photospin Hall effect, encoded into the relation between the driving charge and resulting spin photocurrents. Moreover, we suggest that even a very small canting in Mn2Au can give rise to colossal spin photocurrents that are chiral in flavor. We conclude that the combination of staggered magnetization with the structural and electronic properties of this material results in a unique blend of prominent photocurrents, which makes Mn2Au a unique platform for advanced optospintronics applications.

Tessellations of the hyperbolic spaces by regular polygons are becoming popular because they support discrete quantum and classical models displaying unique spectral and topological characteristics. Resolving the true bulk spectra and the thermodynamic response functions of these models requires converging periodic boundary conditions and our work delivers a practical solution for this open problem on generic {p,q}-tessellations. This enables us to identify the true spectral gaps of bulk Hamiltonians and, as an application, we construct all but one topological models that deliver the topological gaps predicted by the K-theory of the lattices. We demonstrate the emergence of the expected topological spectral flows whenever two such bulk models are deformed into each other and, additionally, we prove the emergence of topological channels whenever a soft physical interface is created between different topological classes of Hamiltonians.