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

Spectral moment sum rules for strongly correlated electrons in time-dependent electric fields

We derive exact operator average expressions for the first two spectral moments of nonequilibrium Green's functions for the Falicov-Kimball model and the Hubbard model in the presence of a spatially uniform, time-dependent electric field. The moments are similar to the well-known moments in equilibrium, but we extend those results to systems in arbitrary time-dependent electric fields. Moment sum rules can be employed to estimate the accuracy of numerical calculations; we compare our theoretical results to numerical calculations for the nonequilibrium dynamical mean-field theory solution of the Falicov-Kimball model at half-filling.Comment: (16 pages, submitted to Phys. Rev. B

Nonequilibrium sum rules for the retarded self-energy of strongly correlated electrons

We derive the first two moment sum rules of the conduction electron retarded self-energy for both the Falicov-Kimball model and the Hubbard model coupled to an external spatially uniform and time-dependent electric field (this derivation also extends the known nonequilibrium moment sum rules for the Green's functions to the third moment). These sum rules are used to further test the accuracy of nonequilibrium solutions to the many-body problem; for example, we illustrate how well the self-energy sum rules are satisfied for the Falicov-Kimball model in infinite dimensions and placed in a uniform electric field turned on at time t=0. In general, the self-energy sum rules are satisfied to a significantly higher accuracy than the Green's functions sum rules

F-electron spectral function of the Falicov-Kimball model in infinite dimensions: the half-filled case

The f-electron spectral function of the Falicov-Kimball model is calculated via a Keldysh-based many-body formalism originally developed by Brandt and Urbanek. We provide results for both the Bethe lattice and the hypercubic lattice at half filling. Since the numerical computations are quite sensitive to the discretization along the Kadanoff-Baym contour and to the maximum cutoff in time that is employed, we analyze the accuracy of the results using a variety of different moment sum-rules and spectral formulas. We find that the f-electron spectral function has interesting temperature dependence becoming a narrow single-peaked function for small U and developing a gap, with two broader peaks for large U.Comment: (13 pages, 11 figures, typeset in RevTex 4

Nonequilibrium perturbation theory of the spinless Falicov-Kimball model

We perform a perturbative analysis for the nonequilibrium Green functions of the spinless Falicov-Kimball model in the presence of an arbitrary external time-dependent but spatially uniform electric field. The conduction electron self-energy is found from a strictly truncated second-order perturbative expansion in the local electron-electron repulsion U. We examine the current at half-filling, and compare to both the semiclassical Boltzmann equation and exact numerical solutions for the contour-ordered Green functions from a transient-response formalism (in infinite dimensions) on the Kadanoff-Baym-Keldysh contour. We find a strictly truncated perturbation theory in the two-time formalism cannot reach the long-time limit of the steady state; instead it illustrates pathological behavior for times larger than approximately 2/U

Time-dependent density-functional theory for ultrafast interband excitations

We formulate a time-dependent density functional theory (TDDFT) in terms of the density matrix to study ultrafast phenomena in semiconductor structures. A system of equations for the density matrix components, which is equivalent to the time-dependent Kohn-Sham equation, is derived. From this we obtain a TDDFT version of the semiconductor Bloch equations, where the electronic many-body effects are taken into account in principle exactly. As an example, we study the optical response of a three-dimensional two-band insulator to an external short-time pulsed laser field. We show that the optical absorption spectrum acquires excitonic features when the exchange-correlation potential contains a $1/q^{2}$ Coulomb singularity. A qualitative comparison of the TDDFT optical absorption spectra with the corresponding results obtained within the Hartree-Fock approximation is made

Steady-state nonequilibrium dynamical mean-field theory and the quantum Boltzman

We derive the formalism for steady state nonequilibrium dynamical mean-field theory in a real-time formalism along the Kadanoff-Baym contour. The resulting equations of motion are first transformed to Wigner coordinates (average and relative time), and then re-expressed in terms of differential operators. Finally, we perform a Fourier transform with respect to the relative time, and take the first-order limit in the electric field to produce the quantum Boltzmann equation for dynamical mean-field theory. We next discuss the structure of the equations and their solutions, describing how these equations reduce to the Drude result in the limit of a constant relaxation time. We also explicitly demonstrate the equivalence between the Kubo and nonequilibrium approaches to linear response. There are a number of interesting modifications of the conventional quantum Boltzmann equation that arise due to the underlying bandstructure of the lattice.Comment: (14 pages, proceedings of the Workshop on Progress in Nonequilibrium Green's Functions III, Kiel Germany

Steady-state nonequilibrium density of states of driven strongly correlated lattice models in infinite dimensions

The formalism for exactly calculating the retarded and advanced Green's functions of strongly correlated lattice models in a uniform electric field is derived within dynamical mean-field theory. To illustrate the method, we solve for the nonequilibrium density of states of the Hubbard model in both the metallic and Mott insulating phases at half-filling (with an arbitrary strength electric field) by employing the numerical renormalization group as the impurity solver. This general approach can be applied to any strongly correlated lattice model in the limit of large dimensions.Comment: (5 pages, 2 figures, RevTeX

Electron Thermalization and Relaxation in Laser-Heated Nickel by Few-Femtosecond Core-Level Transient Absorption Spectroscopy

Direct measurements of photoexcited carrier dynamics in nickel are made using few-femtosecond extreme ultraviolet (XUV) transient absorption spectroscopy at the nickel M$_{2,3}$ edge. It is observed that the core-level absorption lineshape of photoexcited nickel can be described by a Gaussian broadening ($\sigma$) and a red shift ($\omega_{s}$) of the ground state absorption spectrum. Theory predicts, and the experimental results verify that after initial rapid carrier thermalization, the electron temperature increase ($\Delta T$) is linearly proportional to the Gaussian broadening factor $\sigma$, providing quantitative real-time tracking of the relaxation of the electron temperature. Measurements reveal an electron cooling time for 50 nm thick polycrystalline nickel films of 640$\pm$80 fs. With hot thermalized carriers, the spectral red shift exhibits a power-law relationship with the change in electron temperature of $\omega_{s}\propto\Delta T^{1.5}$. Rapid electron thermalization via carrier-carrier scattering accompanies and follows the nominal 4 fs photoexcitation pulse until the carriers reach a quasi-thermal equilibrium. Entwined with a <6 fs instrument response function, carrier thermalization times ranging from 34 fs to 13 fs are estimated from experimental data acquired at different pump fluences and it is observed that the electron thermalization time decreases with increasing pump fluence. The study provides an initial example of measuring electron temperature and thermalization in metals in real time with XUV light, and it lays a foundation for further investigation of photoinduced phase transitions and carrier transport in metals with core-level absorption spectroscopy.Comment: 20 pages, 8 figure