Inhomogeneous spectral moment sum rules for the retarded Green function and self-energy of strongly correlated electrons or ultracold fermionic atoms in optical lattices

Abstract

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.

(28 pages, 6 figures, ReVTeX) Paper updated to correct equations 11, 24, and 28

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• Quenching Bloch oscillations in a strongly correlated material
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• Dynamical mean-field theory for light fermion--heavy boson mixtures on optical lattices
• Minimal Effective Gibbs Ansatz (MEGA): A simple protocol for extracting an accurate thermal representation for quantum simulation
• Theoretical description of pump/probe experiments in electron mediated charge-density-wave insulators
• Exact solution for Bloch oscillations of a simple charge-density-wave insulator
• Nonadiabatic Time-Dependent Spin-Density Functional Theory for strongly correlated systems
• Spectral moment sum rules for the retarded Green's function and self-energy of the inhomogeneous Bose-Hubbard model in equilibrium and nonequilibrium
• Nonequilibrium sum rules for the Holstein model
• Nonequilibrium spectral moment sum rules of the Holstein-Hubbard model