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arXiv:0805.2304 · Condensed Matter Physics

F-electron spectral function of the Falicov-Kimball model and the Wiener-Hopf sum equation approach

Andrij M. Shvaika, James K. Freericks

Published Sep 1, 2008• 6 citations
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Abstract

We derive an alternative representation for the f -electron spectral function of the Falicov-Kimball model from the original one proposed by Brandt and Urbanek.In the new representation, all calculations are restricted to the real time axis, allowing us to go to arbitrarily low temperatures.The general formula for the retarded Green's function involves two determinants of continuous matrix operators that have the Toeplitz form.By employing the Wiener-Hopf sum equation approach and Szeg ö's theorem, we can derive exact analytic formulas for the large-time limits of the Green's function; we illustrate this for cases when the logarithm of characteristic function (which defines the continuous Toeplitz matrix) does and does not wind around the origin.We show how accurate these asymptotic formulas are to the exact solutions found from extrapolating matrix calculations to the zero discretization size limit.

DOI
10.5488/cmp.11.3.425

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We derive an alternative representation for the f -electron spectral function of the Falicov-Kimball model from the original one proposed by Brandt and Urbanek.In the new representation, all calculations are restricted to the real time axis, allowing us to go to arbitrarily low temperatures.The general formula for the retarded Green's function involves two determinants of continuous matrix operators that have the Toeplitz form.By employing the Wiener-Hopf sum equation approach and Szeg ö's theorem, we can derive exact analytic formulas for the large-time limits of the Green's function; we illustrate this for cases when the logarithm of characteristic function (which defines the continuous Toeplitz matrix) does and does not wind around the origin.We show how accurate these asymptotic formulas are to the exact solutions found from extrapolating matrix calculations to the zero discretization size limit.
The fff-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 fff-electron spectral function has interesting temperature dependence, becoming a narrow single-peaked function for small UUU and developing a gap, with two broader peaks for large UUU.
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 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=0t=0t=0. In general, the self-energy sum rules are satisfied to a significantly higher accuracy than the Green function sum rules.
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 such as 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 such as 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 in a parameter such as the interaction strength or the origin of the trap potential.
The fff-electron spectral function of the Falicov-Kimball model is calculated within the dynamical mean-field theory using the numerical renormalization group method as the impurity solver. Both the Bethe lattice and the hypercubic lattice are considered at half filling. For small UUU we obtain a single-peaked fff-electron spectral function, which---for zero temperature---exhibits an algebraic (x-ray) singularity $({\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{\ensuremath{-}\ensuremath{\alpha}})$ for $\ensuremath{\omega}\ensuremath{\rightarrow}0$. The characteristic exponent $\ensuremath{\alpha}$ depends on the Coulomb (Hubbard) correlation UUU. This x-ray singularity cannot be observed when using alternative (Keldysh-based) many-body approaches. With increasing $U,\ensuremath{\alpha}$ decreases and vanishes for sufficiently large UUU when the fff-electron spectral function develops a gap and a two-peak structure (metal-insulator transition).
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.