Inhomogeneous spectral moment sum rules for the retarded Green function
and self-energy of strongly correlated electrons or ultracold fermionic atoms
in optical lattices
Spectral moment sum rules are presented for the inhomogeneous many-body
problem described by the fermionic Falicov-Kimball or Hubbard models. These
local sum rules allow for arbitrary hoppings, site energies, and interactions.
They can be employed to quantify the accuracy of numerical solutions to the
inhomogeneous many-body problem like strongly correlated multilayered devices,
ultracold atoms in an optical lattice with a trap potential, strongly
correlated systems that are disordered, or systems with nontrivial spatial
ordering like a charge density wave or a spin density wave. We also show how
the spectral moment sum rules determine the asymptotic behavior of the Green
function, self-energy, and dynamical mean field, when applied to the dynamical
mean-field theory solution of the many body problem. In particular, we
illustrate in detail how one can dramatically reduce the number of Matsubara
frequencies needed to solve the Falicov-Kimball model, while still retaining
high precision, and we sketch how one can incorporate these results into
Hirsch-Fye quantum Monte Carlo solvers for the Hubbard (or more complicated)
models. Since the solution of inhomogeneous problems is significantly more time
consuming than periodic systems, efficient use of these sum rules can provide a
dramatic speed up in the computational time required to solve the many-body
problem. We also discuss how these sum rules behave in nonequilibrium
situations as well, where the Hamiltonian has explicit time dependence due to a
driving field or due to the time-dependent change of a parameter like the
interaction strength or the origin of the trap potential.Comment: (28 pages, 6 figures, ReVTeX) Paper updated to correct equations 11,
24, and 2
