Spectral Moment Sum Rules Research Articles

Nonequilibrium spectral moment sum rules of the Holstein–Hubbard model

We derive a general procedure for evaluating the nth derivative of a time-dependent operator in the Heisenberg representation and employ this approach to calculate the zeroth to third spectral moment sum rules of the retarded electronic Green’s function and self-energy for a system described by the Holstein–Hubbard model allowing for arbitrary spatial and time variation of all parameters (including spatially homogeneous electric fields and parameter quenches). For a translationally invariant (but time-dependent) Hamiltonian, we also provide sum rules in momentum space. The sum rules can be applied to various different phenomena like time-resolved angle-resolved photoemission spectroscopy and benchmarking the accuracy of numerical many-body calculations. This work also corrects some errors found in earlier work on simpler models.

Spectral moment sum rules for the retarded Green's function and self-energy of the inhomogeneous Bose-Hubbard model in equilibrium and nonequilibrium

We derive expressions for the zeroth and the first three spectral moment sum rules for the retarded Green's function and for the zeroth and the first spectral moment sum rules for the retarded self-energy of the inhomogeneous Bose-Hubbard model in nonequilibrium, when the local on-site repulsion and the chemical potential are time-dependent, and in the presence of an external time-dependent electromagnetic field. We also evaluate these moments for equilibrium, where all time dependence and external fields vanish. Unlike similar sum rules for the Fermi-Hubbard model, in the Bose-Hubbard model case, the sum rules often depend on expectation values that cannot be determined simply from parameters in the Hamiltonian like the interaction strength and chemical potential, but require knowledge of equal time many-body expectation values from some other source. We show how one can approximately evaluate these expectation values for the Mott-insulating phase in a systematic strong-coupling expansion in powers of the hopping divided by the interaction. We compare the exact moments to moments of spectral functions calculated from a variety of different approximations and use them to benchmark their accuracy.

Erratum: Inhomogeneous spectral moment sum rules for the retarded Green function and self-energy of strongly correlated electrons or ultracold fermionic atoms in optical lattices [Phys. Rev. B80, 115119 (2009)

Erratum: Inhomogeneous spectral moment sum rules for the retarded Green function and self-energy of strongly correlated electrons or ultracold fermionic atoms in optical lattices [Phys. Rev. B<b>80</b>, 115119 (2009)

Dynamical mean-field theory for light-fermion–heavy-boson mixtures on optical lattices

We theoretically analyze Fermi-Bose mixtures consisting of light fermions and heavy bosons that are loaded into optical lattices (ignoring the trapping potential). To describe such mixtures, we consider the Fermi-Bose version of the Falicov-Kimball model on a periodic lattice. This model can be exactly mapped onto the spinless Fermi-Fermi Falicov-Kimball model at zero temperature for all parameter space as long as the mixture is thermodynamically stable. We employ dynamical mean-field theory to investigate the evolution of the Fermi-Bose Falicov-Kimball model at higher temperatures. We calculate spectral moment sum rules for the retarded Green's function and self-energy, and use them to benchmark the accuracy of our numerical calculations, as well as to reduce the computational cost by exactly including the tails of infinite summations or products. We show how the occupancy of the bosons, single-particle many-body density of states for the fermions, momentum distribution, and the average kinetic energy evolve with temperature. We end by briefly discussing how to experimentally realize the Fermi-Bose Falicov-Kimball model in ultracold atomic systems.

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.

Erratum: Spectral moment sum rules for strongly correlated electrons in time-dependent electric fields [Phys. Rev. B73, 075108 (2006)

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.

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.

Check of QCD based on theτ-decay data analysis in the complexq2plane

The thorough analysis of the ALEPH data on hadronic $\ensuremath{\tau}$ decay is performed in the framework of QCD. The perturbative calculations are performed in three- and four-loop approximations. The terms of the operator product expansion (OPE) are accounted for up to the dimension $D=8.$ The value of the QCD coupling constant ${\ensuremath{\alpha}}_{s}{(m}_{\ensuremath{\tau}}^{2})=0.355\ifmmode\pm\else\textpm\fi{}0.025$ is found from the hadronic branching ratio ${R}_{\ensuremath{\tau}}.$ The $V+A$ and V spectral functions are analyzed using the analytical properties of the polarization operators in the whole complex ${q}^{2}$ plane. Borel sum rules in the complex ${q}^{2}$ plane along the rays, starting from the origin, are used. It is demonstrated that QCD with OPE terms is in agreement with the data for a coupling constant close to the lower error edge ${\ensuremath{\alpha}}_{s}{(m}_{\ensuremath{\tau}}^{2})=0.330.$ The restriction on the value of the gluonic condensate was found to be $〈({\ensuremath{\alpha}}_{s}/\ensuremath{\pi}{)G}^{2}〉=0.006\ifmmode\pm\else\textpm\fi{}0.012 {\mathrm{GeV}}^{2}.$ The analytical perturbative QCD is compared with the data. It is demonstrated to be in strong contradiction with experiment. The restrictions on the renormalon contribution are found. The instanton contributions to the polarization operator are analyzed in various sum rules. In the Borel transformation they appear to be small, but not in the spectral moment sum rules.

The [formula omitted] sum rules and the operator product expansion in complex [formula omitted]-plane from [formula omitted]-decay data

The operator product expansion (OPE) for the difference of vector and axial current correlators is analyzed for complex values of momentum q 2 . The vector and axial spectral functions, taken from hadronic τ-decay data, are treated with the help of Borel, gaussian and spectral moments sum rules. The range of applicability, advantages and disadvantages of each type are discussed. The general features of OPE are confirmed by the data. The vacuum expectation values of dimension-6 and -8 operators are found to be O 6=−(6.8±2.1)×10 −3 GeV 6 , O 8=(7±4)×10 −3 GeV 8 .

The strong coupling constant in full QCD

To leading order in , the leading and non-leading corrections to the strong coupling constant of the excited meson are calculated in the framework of QCD spectral moment sum rules in the full theory. Our prediction is in good agreement with the light-cone QCD sum rule result.

Formula omitted] couplings and [formula omitted] within a [formula omitted] in full QCD

To leading order in α s , we evaluate the leading and non-leading 1 M b corrections to the B ∗Bπ and B ∗Bγ couplings using QCD spectral moment sum rules in the full theory. We find that, for large M b and contrary to the heavy-to-light B → π(ϱ)l v form factors, which are dominated by the soft light quark vacuum condensate, these couplings are governed by the hard perturbative graph, like other heavy-to-heavy transitions. We also find that for the B ∗ → Bγ , the 1 M b correction is mainly due to the perturbative and light quark condensate contributions originating from the graphs involving the heavy quark part of the electromagnetic current, which are essential for explaining the large charge dependence in the observed D ∗− → D −γ and D ∗0 → D 0γ decays. Our best numerical predictions without any free parameters for the B ∗- meson are: g B ∗−B 0π − ⋍ 14±4 , Γ B ∗−→B −γ ⋍ (0.10±0.03) keV and the large charge dependence of the ratio: Γ B ∗−→B −γ Γ B ∗0→B 0γ ⋍2.5 . For the D ∗- meson , we find: Γ D ∗−→D 0π − ⋍ 1.54Γ D ∗0→D 0π 0 ⋍ (8±5) keV, Γ D ∗− → D −γ ⋍ (0.09 −0.07 +0.40) keV aand Γ D ∗0→D 0γ ⋍ (3.7±1.4) keV, where the branching ratios agree within the errors with the present data, while the total widths Γ D ∗0→all ⋍ (11±4) keV and Γ D ∗− →all ⋍ (12±7) keV are much smaller than the present experimental upper limits.