  The auxiliary field quantum Monte Carlo (AFQMC) method has been a workhorse
in the field of strongly correlated electrons for a long time and has found its
most recent implementation in the ALF package
(https://alf.physik.uni-wuerzburg.de). The utilization of the Trotter
decomposition to decouple the interaction from the non-interacting Hamiltonian
makes this method inherently second order in terms of the imaginary time slice.
We show that due to the use of the Hubbard-Stratonovich transformation (HST) a
semigroup structure on the time evolution is imposed that necessitates the
introduction of a new family of complex-hermitian splitting methods for the
purpose of reaching higher order. We will give examples of these new methods
and study their efficiency, as well as perform comparisons with other
established second and higher order methods in the realm of the AFQMC method.
