Topological spin torque emerging in classical-spin systems with different time scales

Abstract

In classical spin systems with two largely different inherent time scales, the configuration of the fast spins almost instantaneously follows the slow-spin dynamics. We develop the emergent effective theory for the slow-spin degrees of freedom and demonstrate that this generally includes a topological spin torque. This torque gives rise to anomalous real-time dynamics. It derives from the holonomic constraints defining the fast-spin configuration space and is given in terms of a topological charge density which becomes a quantized homotopy invariant when integrated.

Nonequilibrium dynamics in a spin valve with non-collinear magnetization

“…Despite these differences, the methods that combine classical localized spins with quantum conduction electrons are in a rather good qualitative agreement with the full quantum mechanical treatments [44,45,51,57] and capture most of its details. As such, these hybrid techniques proved to be extremely useful in the investigation of various phenomena not described by classical or adiabatic LLG based approaches, e.g., geometrical torque [51], magnetic inertia [15], chiral spin and charge pumping [49], formation of some nontrivial magnetic textures [53,64] or resonant dependence of the spin damping on voltage [56,65]. In addition, they are generalizable to realistic band structures [50].…”

“…A natural compromise between completely classical and fully quantum-mechanical approaches present hybrid methods which combine both classical and quantum degrees of freedom [15,[39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. In the case of magnetic systems, these methods consider classical localized magnetic moments interacting with quantum conduction electrons.…”

“…Under this assumption, electrons respond instantaneously to the slow time-dependent potential of the classical degrees of freedom and, therefore, can be described by steady-state approaches, e.g., via nonequilibrium Green functions (NEGF) [40,41,[46][47][48]55]. However, several recent studies have shown that this approach is often invalid [15,[43][44][45][49][50][51][52][53] because the two timescales can not be strictly separated in general.…”

“…These approaches reveal a time-dependent misalignment between the localized magnetic moments and the local nonequilibrium spin density of conduction electrons. Its most important consequences materialize in the form of additional torques and time-retarded damping effects [15,[43][44][45][49][50][51][52][53].…”