Geometrical contributions to the exchange constants: Free electrons with spin-orbit interaction

Abstract

Using thermal quantum field theory we derive an expression for the exchange constant that resembles Fukuyama's formula for the orbital magnetic susceptibility (OMS). Guided by this formal analogy between the exchange constant and OMS we identify a contribution to the exchange constant that arises from the geometrical properties of the band structure in mixed phase space. We compute the exchange constants for free electrons and show that the geometrical contribution is generally important. Our formalism allows us to study the exchange constants in the presence of spin-orbit interaction (SOI). Thereby, we find sizable differences between the exchange constants of helical and cycloidal spin spirals. Furthermore, we discuss how to calculate the exchange constants based on a gauge-field approach in the case of the Rashba model with an additional exchange splitting and show that the exchange constants obtained from this gauge-field approach are in perfect agreement with those obtained from the quantum field theoretical method.

References in corpus (4)

• Geometric orbital susceptibility: quantum metric without Berry curvature
• Real-Space and Reciprocal-Space Berry Phases in the Hall Effect of Mn${}_{1-\mathrm{x}}$Fe${}_{\mathrm{x}}$Si
• Doppler shift picture of the Dzyaloshinskii--Moriya interaction
• Orbital magnetism of coupled bands models

Cited by in corpus (12)

• Relations between topology and the quantum metric for Chern insulators
• Extracting the quantum metric tensor through periodic driving
• Semi-classical wave packet dynamics in non-uniform electric fields
• Berry Curvature and Quantum Metric in $N$-band systems -- an Eigenprojector Approach
• Light-matter coupling and quantum geometry in moiré materials
• Nontrivial quantum geometry of degenerate flat bands
• Steady-state Hall response and quantum geometry of driven-dissipative lattices
• The relation of the Dzyaloshinskii-Moriya interaction to spin currents and to the spin-orbit field
• Ground-state quantum geometry in superconductor-quantum dot chains
• Wave packet dynamics of Bogoliubov quasiparticles: quantum metric effects
• Chiral damping, chiral gyromagnetism and current-induced torques in textured one-dimensional Rashba ferromagnets
• Orbital Longitudinal Magneto-electric Coupling in Multilayer Graphene