Collective charge and spin excitations in solids can sometimes be studied through a simplified bosonic model, e.g., in the description of spin via Heisenberg-like models, and of charge ordering in alloys via the Ising model. Such models are derived by estimating the exchange interactions and writing an effective classical Hamiltonian. Alternatively, one maps the electronic model onto an effective bosonic problem, most commonly using a Schrieffer-Wolff transformation [1]. However, spin degrees of freedom are mapped into composite fermions, and not into physical bosonic fields, which would require artificial constraints conserving the spin length.
All these derivations do not answer a fundamental question: what is the correct equation of motion for spin? Firstly, classical spin Hamiltonians can only describe a uniform precession of the local magnetic moment, because Gilbert damping requires coupling classical spins to electrons [2]. Secondly, Higgs fluctuations of the modulus of the local magnetic moment cannot be addressed in classical spin problems. Finally, the above discussion relies on the average local magnetization being nonzero. Consequently these state-of-the-art methods cannot be used in the most challenging case of paramagnetism.
I will address these long-standing problems and provide a consistent description of spin dynamics of electronic systems [3]. First, from the initial interacting electronic problem I will derive an effective quantum bosonic action in terms of physical charge and spin variables. This derivation does not require a nonzero average magnetization, nor any artificial constraints. The corresponding equation of motion correctly describes the rotational dynamics of the local magnetic moment via the topological Berry phase, and also accounts for Higgs fluctuations of the magnetic moment. Finally, I will introduce a physical criterion for the formation of the local magnetic moment in the system and show that our approach is applicable even in the paramagnetic regime.
[1] Phys. Lett. A 64,163('77); J.Phys.C: Solid State Phys. 10,L271('77); Phys.Rev.B 37,9753 ('88).
[2] New J. Phys. 17, 113058 (2015); Phys. Rev. Lett. 117, 127201 (2016).
[3] E. A. Stepanov et al.: Phys. Rev. Lett. 121, 037204 (2018); Phys. Rev. B 99, 115124 (2019); Phys. Rev. B 100, 205115 (2019); arXiv:2106.12462 (2021).
