Electron conductance in a solid state is usually driven by electric field - making some direction of jumps more likely.
It makes (e.g. Hubbard's) Hamiltonian no longer self-adjoint, how to simulate QM with non-Hermitian Hamiltonian? How electron conductance in electric field is being modeled?
PS. I have just found non-Hermitian QM Wikipedia article, PI lecture by Carl Bender, where he says that PT symmetric Hamiltonian should still have real spectrum. However, I have checked MERW-based conductance simulator which uses Hubbard-like Hamiltonian, and turns out its spectrum has not only real eigenvalues if turning on the potential.