Grassmann phase space dynamics of strongly-correlated fermion

Abstract: We discuss the numerical implementation of two related representations of fermionic density matrices which have been introduced in Annals of Physics 370, 12 (2016). In both of them, the density matrix is expanded in a basis of Bargmann coherent states with weights given by the two phase space distributions. We derive the equations of motion for the distributions when imaginary time evolution is generated by the Hubbard Hamiltonian. One of them is a Grassmann Fokker-Planck equation that can be re-cast into a remarkably simple It^{o} form involving solely complex variables. In spite of this simple form, we demonstrate that complications arise in numerically computing the expectation value of any observable. These are due to exponential growth in the matrix elements of the stochastic propagator, delicate numerical sensitivity in performing primitive linear algebra operations, and the re-appearance of a sign problem.