Abstract
A clear separation of the timescales governing the dynamics of “slow” and “fast” degrees of freedom often serves as a prerequisite for the emergence of an independent low-energy theory. Here, we consider (slow) classical spins exchange coupled to a tight-binding system of (fast) conduction electrons. The effective equations of motion are derived under the constraint that the quantum state of the electron system at any instant of time lies in the -dimensional low-energy subspace for the corresponding spin configuration at . The effective low-energy theory unfolds itself straightforwardly and takes the form of a non-Abelian gauge theory with the gauge freedom given by the arbitrariness of the basis spanning the instantaneous low-energy sector. The holonomic constraint generates a gauge-covariant spin-Berry curvature tensor in the equations of motion for the classical spins. In the non-Abelian theory for , opposed to the adiabatic spin dynamics theory, the spin-Berry curvature is generically nonzero, even for time-reversal-symmetric systems. Its expectation value with the representation of the electron state is gauge invariant and gives rise to an additional geometrical spin torque. Aside from anomalous precession, the theory also captures the spin nutational motion, which is usually considered as a retardation effect. This is demonstrated by proof-of-principle numerical calculations for a minimal model with a single classical spin. Already for and in parameter regimes where the adiabatic theory breaks down, we find good agreement with results obtained from the full (unconstrained) theory.
5 More- Received 9 February 2022
- Revised 28 June 2022
- Accepted 12 September 2022
DOI:https://doi.org/10.1103/PhysRevB.106.094433
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