  We study the unidirectional magnetoresistance (UMR) and the nonlinear Hall
effect (NLHE) in the ferromagnetic Rashba model. For this purpose we derive
expressions to describe the response of the electric current quadratic in the
applied electric field. We compare two different formalisms, namely the
standard Keldysh nonequilibrium formalism and the Moyal-Keldysh formalism, to
derive the nonlinear conductivities of UMR and NLHE. We find that both
formalisms lead to identical numerical results when applied to the
ferromagnetic Rashba model. The UMR and the NLHE nonlinear conductivities tend
to be comparable in magnitude according to our calculations. Additionally,
their dependencies on the Rashba parameter and on the quasiparticle broadening
are similar. The nonlinear zero-frequency response considered here is several
orders of magnitude higher than the one at optical frequencies that describes
the photocurrent generation in the ferromagnetic Rashba model. Additionally, we
compare our Keldysh nonequilibrium expression in the independent-particle
approximation to literature expressions of the UMR that have been obtained
within the constant relaxation time approximation of the Boltzmann formalism.
We find that both formalisms converge to the same analytical formula in the
limit of infinite relaxation time. However, remarkably, we find that the
Boltzmann result does not correspond to the intraband term of the Keldysh
expression. Instead, the Boltzmann result corresponds to the sum of the
intraband term and an interband term that can be brought into the form of an
effective intraband term due to the f-sum rule.
