  Phase-space representations are a family of methods for dynamics of both
bosonic and fermionic systems, that work by mapping the system's density matrix
to a quasi-probability density and the Liouville-von Neumann equation of the
Hamiltonian to a corresponding density differential equation for the
probability. We investigate here the accuracy and the computational efficiency
of one approximate phase-space representation, called the fermionic Truncated
Wigner Approximation (fTWA), applied to the Fermi-Hubbard model. On a many-body
2D system, with hopping strength and Coulomb $U$ tuned to represent the
electronic structure of graphene, the method is found to be able to capture the
time evolution of first-order (site occupation) and second-order (correlation
functions) moments significantly better than the mean-field, Hartree-Fock
method. The fTWA was also compared to results from the exact diagonalization
method for smaller systems, and in general the agreement was found to be good.
The fully parallel computational requirement of fTWA scales in the same order
as the Hartree-Fock method, and the largest system considered here contained
198 lattice sites.
