Partial Sum Rules Research Articles

Shining light on transition-metal oxides: unveiling the hidden Fermi liquid.

We use low energy optical spectroscopy and first principles local density approximation plus dynamical mean field theory calculations to test the hypothesis that the anomalous transport properties of strongly correlated metals originate in the strong temperature dependence of their underlying resilient quasiparticles. We express the resistivity in terms of an effective plasma frequency ω(p)* and an effective scattering rate 1/τ(tr)*. We show that in the archetypal correlated material V₂O₃, ω(p)* increases with increasing temperature, while the plasma frequency from the partial sum rule exhibits the opposite trend. 1/τ(tr)* has a more pronounced temperature dependence than the scattering rate obtained from the extended Drude analysis. The theoretical calculations of these quantities are in quantitative agreement with experiment. We conjecture that these are robust properties of all strongly correlated metals, and test the conjecture by carrying out a similar analysis on thin film NdNiO₃ on a LaAlO₃ substrate.

Optical constants determination of samarium, holmium, and erbium in the 15-850 eV spectral range using a transmittance method

The optical constants β and δ of the complex refractive index ñ=1-δ+iβ of Sm, Ho, and Er were obtained in the 1.5-850eV energy range using a transmittance method. Thin films of Sm, Ho, and Er were deposited by magnetron sputtering, and transmittance was measured using synchrotron radiation under a high vacuum condition. All films were directly coated on Si photodiodes, which were used as coating substrates, as well as photon detectors. Si was used as capping layer while a thin W layer was used as barrier against interface diffusion between Si and the highly reactive rare earth elements. The constants β were extracted from transmittance results, and the constants δ were calculated based on measured β values using the Kramers-Kronig formalism. Small deficiencies determined from the present data using the partial sum rules were partly attributed to the sputtered film densities that could be slightly lower than the bulk values.

Serber symmetry, largeNc, and Yukawa-like one-boson exchange potentials

The Serber force has relative orbital parity symmetry and requires vanishing $\mathit{NN}$ interactions in partial waves with odd angular momentum. We illustrate how this property is well fulfilled for spin triplet states with odd angular momentum and violated for odd singlet states for realistic potentials but fails for chiral potentials. The analysis is carried out in terms of partial wave sum rules for $\mathit{NN}$ phase shifts, $r$-space potentials at long distances, and ${V}_{\mathrm{low} k}$ potentials. We analyze how Serber symmetry can be accommodated within a large-${N}_{c}$ perspective when interpreted as a long-distance symmetry. A prerequisite for this is the numerical similarity of the scalar and vector meson resonance masses. The conditions under which the resonance exchange potential can be approximated by a Yukawa form are also discussed. Although these masses arise as poles on the second Riemann in $\ensuremath{\pi}\ensuremath{\pi}$ scattering, we find that within the large-${N}_{c}$ expansion the corresponding Yukawa masses correspond instead to a well-defined large-${N}_{c}$ approximation to the pole that cannot be distinguished from their location as Breit-Wigner resonances.

Measurement of dysprosium optical constants in the 2-830 eV spectral range using a transmittance method, and compilation of the revised optical constants of lanthanum, terbium, neodymium, and gadolinium

The optical constants beta, delta of the complex refractive index (ñ = 1-delta+ibeta) of Dy were obtained in the 2-830 eV energy range using a novel transmittance method. Si/W/Dy/W films were deposited by dc-magnetron sputtering on Si photodiode substrates, and the transmittance was characterized using synchrotron radiation. The extinction coefficients beta of Dy and the transmittance of a Si capping layer and two W interface barrier layers as functions of energy were solved simultaneously using a nonlinear optimization routine. The measured transmittances of the capping and barrier layers were primarily used as indicators for any flaws in the transmittance results. The dispersion coefficients delta of Dy were calculated using the Kramers-Kronig integral, and a complete set of beta values required for this integral was obtained by combining the present data with data from the literature. Sum rule tests on Dy show some deficiencies in the present data, which may be attributed to lower film density compared with the bulk value. Similar procedures were applied to previously measured transmittances of B4C/La, Si/Tb, Si/Nd, and Si/Gd films, where B4C or Si were used as capping layers on those reactive rare-earth films. The improved sets of transmittance values of B(4)C and Si capping layers were used as input in the optimization routine to solve for more accurate beta values of La, Tb, Nd, and Gd. The revised optical constants of these materials, tested for consistency with partial sum rules, are also reported.

Punctured plane partitions and the q-deformed Knizhnik–Zamolodchikov and Hirota equations

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley–Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ 2 -weighted punctured cyclically symmetric transpose complement plane partitions where τ = − ( q + q − 1 ) . In the cases of no or minimal punctures, we prove that these generating functions coincide with τ 2 -enumerations of vertically symmetric alternating sign matrices and modifications thereof.

Model-independent sum rule analysis based on limited-range spectral data

Partial sum rules are widely used in physics to separate low- and high-energy degrees of freedom of complex dynamical systems. Their application, though, is challenged in practice by the always finite spectrometer bandwidth and is often performed using risky model-dependent extrapolations. We show that, given spectra of the real and imaginary parts of any causal frequency-dependent response function (for example, optical conductivity, magnetic susceptibility, acoustical impedance etc) in a limited range, the sum-rule integral from zero to a certain cutoff frequency inside this range can be safely derived using only the Kramers–Kronig dispersion relations without any extra model assumptions. This implies that experimental techniques providing both active and reactive response components independently, such as spectroscopic ellipsometry in optics, allow an extrapolation-independent determination of spectral weight ‘hidden’ below the lowest accessible frequency.

Dispersion interactions within the Piris natural orbital functional theory: The helium dimer

The authors have investigated the description of the dispersion interaction within the Piris natural orbital functional (PNOF) theory. The PNOF arises from an explicit antisymmetric approach for the two-particle cumulant in terms of two symmetric matrices, Delta and Lambda. The functional forms of these matrices are obtained from the generalization of the two-particle system expressions, except for the off-diagonal elements of Delta. The mean value theorem and the partial sum rule obtained for the off-diagonal elements of Delta provide a prescription for deriving practical functionals. In particular, the previous employed approximation {Jpp/2} for the mean values {Jp*} affords several molecular properties but it is incapable to account for dispersion effects. In this work, the authors analyze a new approach for Jp* obtained by factorization of the matrix Delta within the bounds on its off-diagonal elements imposed by the positivity conditions of the two-particle reduced density matrix. Additional terms for the matrix elements of Lambda proportional to the square root of the holes are again introduced to describe properly the occupation numbers of the lowest occupied levels. The authors have found that the cross products between weakly occupied orbitals must be removed from the functional form of Lambda to obtain a correct long-range asymptotic behavior. The PNOF is used to predict the binding energy as well as the equilibrium distance of the helium dimer. The results are compared with the full configuration-interaction calculations and the corresponding experimental data.

Fingerprints of spin-orbital physics in cubic Mott insulators: Magnetic exchange interactions and optical spectral weights

The temperature dependence and anisotropy of optical spectral weights associated with different multiplet transitions is determined by the spin and orbital correlations. To provide a systematic basis to exploit this close relationship between magnetism and optical spectra, we present and analyze the spin-orbital superexchange models for a series of representative orbital-degenerate transition metal oxides with different multiplet structure. For each case we derive the magnetic exchange constants, which determine the spin wave dispersions, as well as the partial optical sum rules. The magnetic and optical properties of early transition metal oxides with degenerate ${t}_{2g}$ orbitals (titanates and vanadates with perovskite structure) are shown to depend only on two parameters, viz. the superexchange energy $J$ and the ratio $\ensuremath{\eta}$ of Hund's exchange to the intraorbital Coulomb interaction, and on the actual orbital state. In ${e}_{g}$ systems important corrections follow from charge transfer excitations, and we show that ${\mathrm{KCuF}}_{3}$ can be classified as a charge transfer insulator, while ${\mathrm{LaMnO}}_{3}$ is a Mott insulator with moderate charge transfer contributions. In some cases orbital fluctuations are quenched and decoupling of spin and orbital degrees of freedom with static orbital order gives satisfactory results for the optical weights. On the example of cubic vanadates we describe a case where the full quantum spin-orbital physics must be considered. Thus information on optical excitations, their energies, temperature dependence, and anisotropy, combined with the results of magnetic neutron scattering experiments, provides an important consistency test of the spin-orbital models, and indicates whether orbital and/or spin fluctuations are important in a given compound.

A new approach for the two‐electron cumulant in natural orbital functional theory

AbstractThe cumulant expansion gives rise to a useful decomposition of the two‐matrix D in which the pair correlated matrix (cumulant) is disconnected from the antisymmetric product of the one‐matrix Γ. A new explicit antisymmetric approach for the two‐particle cumulant matrix in terms of two symmetric matrices, Δ and Λ, as functionals of the occupation numbers is proposed for singlet ground states of closed‐shell systems. It produces a natural orbital functional that reduces to the exact expression for the total energy in two‐electron systems. The functional form of matrix Λ is readily generalized to any system with an even number of electrons. The diagonal elements of Δ equal the square of the occupation numbers, and the N‐representability positivity necessary conditions of the two‐matrix impose several bounds on the off‐diagonal elements of matrix Δ. The well‐known mean value theorem and the partial sum rule obtained for the off‐diagonal elements of Δ provide a prescription for deriving a practical functional. In particular, when the mean values {J} of the Coulomb interactions {Jij} for a given orbital i taking over all orbitals j ≠ i are assumed to be equal {Kii/2}, a functional close to self‐interaction‐corrected GU functional is obtained, but the two‐matrix fermionic antisymmetric holds. An additional term for the matrix elements of Λ between HF occupied orbitals is proposed to ensure a correct description of the occupation numbers for the lowest occupied levels. The functional is tested in fully variational finite basis set calculations of 57 molecules. It gives reasonable molecular energies at the equilibrium geometries. The calculated values of dipole moments are in good agreement with the available experimental data. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Magnetic interactions and optical spectral weights in Mott insulators with orbital degrees of freedom

AbstractWe outline a unified approach to magnetic and optical properties of Mott insulators at orbital degeneracy, which provides deep insight into the complex spin–orbital physics. It introduces partial sum rules for individual optical excitations which follow from the multiplet structure of transition metal ions. We show that the observed anisotropies and the temperature dependence of the optical spectral weights in LaVO3 and LaMnO3 follow from spin and orbital correlations and are well understood within this approach. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Theory of optical spectral weights in Mott insulators with orbital degrees of freedom

Introducing partial sum rules for the optical multiplet transitions, we outline a unified approach to magnetic and optical properties of strongly correlated transition metal oxides. On the example of ${\mathrm{LaVO}}_{3}$ we demonstrate how the temperature and polarization dependences of different components of the optical multiplet are determined by the underlying spin and orbital correlations dictated by the low-energy superexchange Hamiltonian. Thereby the optical data provide deep insight into the complex spin-orbital physics and the role played by orbital fluctuations.

Optical conductivity of the Bechgaard salts: the sum rules revisited

The validity of the optical sum rules has been addressed eversince and was always matter of debate. Particularly controversial is the proof that the partial sum rules can be extended to both optical conductivity and energy loss function. We show in this paper that for both transverse (optical conductivity) and longitudinal (energy loss function) absorption processes the corresponding sum rule can be theoretically established and through appropriate conditions for the integration limits exactly verified. We also focus our attention on the one-dimensional case within the microscopic Hubbard model. An application of these concepts to the quasi one-dimensional systems, for which we have chosen the organic (TMTSF)2PF6 material, will also be presented.

Absolute photoabsorption measurements of molybdenum in the range 60–930 eV for optical constant determination

Transmission measurements for the optical constants delta, beta of thecomplex refractive index n = 1 - delta + ibeta of molybdenum are performed in the energy range 60-930eV. Free-standing C/Mo/C foils of five different thicknesses are used, and the results are normalized for the presence of the carbon layers in the samples. These absorption results are combined with previous experimental data in the lower energy range and values fromthe atomic tables to obtain the imaginary (absorptive) part of the refractive index for molybdenum in the range 1-30,000 eV. The real(dispersive) part of n was calculated from Kramers-Kronig analysis with the above absorption data. An evaluation with the partial sum rules demonstrates that this new compilation provides an improved set of values for n covering a wider energy range compared with the current tabulated values. The new results are applied so as to calculate thenormal-incidence reflectivities of Mo/Si and Mo/Be multilayer mirrors.

Spectral properties of the large-negative-U Hubbard model

Using the formalism introduced by Harris and Lange [Phys. Rev. 157, 295 (1967)] explicit expressions for partial sum rules of the individual Hubbard bands in the large-negative-U limit are derived. The one-particle spectrum, optical spectrum, as well as charge- and spin-response functions are considered. The approach gives a transparent description of the main features of these spectra. The main sum rules depend on only two independent nearest-neighbor expectation values which are estimated in the ground state using numerical calculations, exact results, and linear-spin-wave theory. Simple expressions for the intensities of the upper and lower Hubbard bands in the optical conductivity show that this spectrum is extremely sensitive on details of the ground state. The charge-density wave and superconducting phases are clearly distinguishable and even the transition to the broken-symmetry state may have a detectable influence on the conductivity.

On the E1 partial wave sum rule and the WKB approximation

The electric dipole (E1) partial wave sum rule, relevant to Coulomb scattering, is derived semiclassically within the WKB theory. Limiting expressions are presented according to the values of the angular momentum l of the colliding particle, the Sommerfeld number and the adiabaticity parameter ξ = ν ƒ - ν i. A comparison with the quantal results, of the WKB sum rule and the classical orbital integrals, shows that the WKB approximation is extremely accurate even in the case of light and/or fast particles (ν ≪ 1). An algorithm is presented for the calculation of the modified Bessel function of the second kind with imaginary argument K iξ(x) and its derivative dK iξ(x)/dx.

Optical properties and electronic structures of alpha - and gamma -Ce.

The optical properties of metallis \ensuremath{\alpha}- and \ensuremath{\gamma}-Ce were investigated by spectroscopic ellipsometry in the energy range 1.5--5.4 eV. The samples were thin films deposited in situ at room temperature and at 25 K for \ensuremath{\gamma}- and \ensuremath{\alpha}-Ce, respectively. The measured conductivities decrease in magnitude as the thickness increases, which can be explained with a microscopic surface roughness model. The optical conductivity increases upon entering the \ensuremath{\alpha} phase because of the increased number of electrons per unit volume due to the volume collapse. The energy band structures and the optical conductivities were calculated using the linearized-augmented-plane-wave (LAPW) method within density-functional theory, treating both the 4f and 5p electrons as bandlike. The application of the partial sum rule to angular-momentum-decomposed optical conductivities shows that the 4f electron contributions in the measured energy range is small.

Optical properties and electronic structures of CeSn3 and LaSn3.

The dielectric functions of ${\mathrm{CeSn}}_{3}$ and ${\mathrm{LaSn}}_{3}$ were remeasured to correct errors in previous measurements. The partial-sum rule for the optical conductivity shows that ${\mathrm{CeSn}}_{3}$ has about 0.54 electrons per formula unit more than ${\mathrm{LaSn}}_{3}$, contributing to the conductivity in the 1.5--5.4-eV region.

Interband Electronic Structure of α‐Alumina up to 2167 K

The electronic structure and interband transitions of α‐Al2O3 have been studied using temperature‐dependent vacuum ultraviolet spectroscopy from 5 to 43 eV, at temperatures ranging from 293 up to 2167 K, which is approaching the melting point of 2327 K. The energy range of the spectra spans the full range of interband electronic transitions. Kramers–Krönig analysis has been employed to recover the phase information, and the interband transition strength (Jcv) of the valence to conduction band transitions. Critical point (CP) analysis of Jcv, a modeling technique based on band structure topology, is then applied to the study of temperature‐induced changes in the interband electronic structure. This approach offers newinsights into the nature of the electronic structure of α‐Al2O3—by allowing us to decompose the interband transitions into subsets associated with states of O 2p nonbonding and Al = O bonding character, and by allowing us to apply partial optical sum rules to these subsets to determine changes in their electron occupancy as a function of temperature. Up to 1700 K the temperature dependence of the electronic structure is linear, corresponding to the linear behavior of the thermal lattice expansion and the vibrational Debye–Waller factors. Below 1700 K the absorption edge shifts at −1.1 meV/K while the exciton and band gap, decomposed through CP modelling, shift at 0.93 and 0.85 meV/K with an exciton binding energy of 0.13 eV. The electron occupancy of the O 2p nonbonding CP set decreases and the occupancy of the Al = O bonding CP set increases. Above 1700 K the temperature dependence of the electronic structure is nonlinear, reflecting the interaction of electrons with phonons in the anharmonic regime, and related to the nonlinearity observed in the vibrational Debye–Waller factors. Also above 1700 K, the O 2p nonbonding and Al = O bonding CP sets merge and this is discussed in the context of temperature‐induced changes in the interatomic bonding.

Finite nucleus Dirac mean field theory and random phase approximation using finite B splines.

We calculate the finite nucleus Dirac mean field spectrum in a Galerkin approach using finite basis splines. We review the method and present results for the relativistic \ensuremath{\sigma}-\ensuremath{\omega} model for the closed-shell nuclei $^{16}\mathrm{O}$ and $^{40}\mathrm{Ca}$. We study the convergence of the method as a function of the size of the basis and the closure properties of the spectrum using an energy-weighted dipole sum rule. We apply the method to the Dirac random-phase-approximation response and present results for the isoscalar ${1}^{\mathrm{\ensuremath{-}}}$ and ${3}^{\mathrm{\ensuremath{-}}}$ longitudinal form factors of $^{16}\mathrm{O}$ and $^{40}\mathrm{Ca}$. We also use a B-spline spectral representation of the positive-energy projector to evaluate partial energy-weighted sum rules and compare with nonrelativistic sum rule results.

New sum rules for the generalized oscillator strength

Partial sum rules for the generalized oscillator strength are reported. The summation runs here over all final target states having a fixed value of the conserved total angular momentum. Total and partial sum rules have been derived provided the initial target state is aligned. In particular it is shown that an alignment destroys the Bethe sum rule.