The long fascination that antiferromagnetic materials has exerted on the scientific community over about a century has been entirely renewed recently with the discovery of several unexpected phenomena, including various classes of anomalous spin and charge Hall effects and unconventional magnonic transport, and also homochiral magnetic entities such as skyrmions. With these breakthroughs, antiferromagnets stand out as a rich playground for the investigation of novel topological behavior, and as promising candidate materials for disruptive low-power microelectronic applications. Remarkably, the newly discovered phenomena are all related to the topology of the magnetic, electronic or magnonic ground state of the antiferromagnets. This review exposes how non-trivial topology emerges at different levels in antiferromagnets and explores the novel mechanisms that have been discovered recently. We also discuss how novel classes of quantum magnets could enrich the currently expanding field of antiferromagnetic spintronics and how spin transport can in turn favor a better understanding of exotic quantum excitations.

No abstract

This article is the second of a trilogy that addresses the perturbative response of general quantum systems, with possibly nontrivial ground state geometry, beyond linear order. Here, we establish concise, general formulae for second order response to a spatially uniform, time-varying electric field in the velocity gauge that are manifestly free of static limit spurious divergences. We first discuss general quantum evolution in a curved space, then detail how such a situation is a natural byproduct of Hilbert space truncation, and point out crucial subtleties associated with the resulting finite curvatures. We then present a geometric perspective of the two popular gauges often used in quantum transport theories, the velocity gauge and the length gauge, and discuss how they, taking truncation-induced curvature effects into account, naturally lead to the same results in spite of the truncation. We highlight subtle formal discrepancies in the literature. Finally, we provide a general scheme for removing static limit spurious divergences in the velocity gauge without frequency expansions and present concise and comprehensive Green's function formulae for responses up to second order. As an application of specific aspects of our theory, second order charge current responses in selected cases are analyzed in Refs. [3] and [63].

D. The retarded correlator as a combination of plain correlators 40 1. The 3-point retarded correlator 41 2. The retarded 4-point correlator 41 3. Time-translation invariance 42 4. Correlators in the frequency-domain 42 E. Spectral representation of stripped and retarded correlators 42 1. Green's operators 43 2. Spectral representation of the 2-point stripped correlator 43 3. Spectral representation of the 3-point stripped correlator 44 4. Spectral representation of the 2-point retarded correlator 45 5. Spectral representation of the 3-point retarded correlator 45 6. Spectral representation of the 4-point retarded correlator 46 7. Spectral representation of advanced correlators 47 References 47