  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.
