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arXiv:cond-mat/9912007 · Physical review. B, Condensed matter

Auxiliary particle theory of threshold singularities in photoemission and x-ray absorption spectra: Test of a conservingT-matrix approximation

T. Schauerte, Johann Kroha, P. Wölfle

Published Aug 15, 2000• 6 citations

Abstract

We calculate the exponents of the threshold singularities in the photoemission spectrum of a deep core hole and its x-ray absorption spectrum in the framework of a systematic many-body theory of slave bosons and pseudofermions (for the empty and occupied core level). In this representation, photoemission and x-ray absorption can be understood on the same footing; no distinction between orthogonality catastrophe and excitonic effects is necessary. We apply the conserving slave particle T-matrix approximation (CTMA), recently developed to describe both Fermi and non-Fermi liquid behavior in systems with strong local correlations, to the x-ray problem as a test case. The numerical results for both photoemission and x-ray absorption are found to be in agreement with the exact infrared power law behavior in the weak as well as in the strong coupling regions. We point out a close relation of the CTMA with the parquet equation approach of Nozi\`eres et al.

DOI: 10.1103/physrevb.62.4394

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We calculate the exponents of the threshold singularities in the photoemission spectrum of a deep core hole and its x-ray absorption spectrum in the framework of a systematic many-body theory of slave bosons and pseudofermions (for the empty and occupied core level). In this representation, photoemission and x-ray absorption can be understood on the same footing; no distinction between orthogonality catastrophe and excitonic effects is necessary. We apply the conserving slave particle T-matrix approximation (CTMA), recently developed to describe both Fermi and non-Fermi liquid behavior in systems with strong local correlations, to the x-ray problem as a test case. The numerical results for both photoemission and x-ray absorption are found to be in agreement with the exact infrared power law behavior in the weak as well as in the strong coupling regions. We point out a close relation of the CTMA with the parquet equation approach of Nozi`eres et al.

We use the numerical renormalization group to calculate the auxiliary spectral functions of the $U=\ensuremath{\infty}$ Anderson impurity model. The slave-boson and pseudo-fermion spectral functions diverge at the threshold with exponents ${\ensuremath{\alpha}}_{b}$ and ${\ensuremath{\alpha}}_{f}$ given by the x-ray photoemission and the x-ray absorption exponents, respectively. In contrast to the noncrossing approximation, the exact exponents obtained here depend on the impurity occupation number. In general, vertex corrections in the convolution formulas for physical Green's functions are singular at the threshold and may not be neglected in the Fermi liquid regime.

The zero-temperature core-level photoemission spectrum of a Hubbard system is studied across the metal to Mott insulator transition using dynamical mean-field theory and Wilson's numerical renormalization group [Rev. Mod. Phys. 47, 773 (1975)]. An asymmetric power-law divergence is obtained in the metallic phase with an exponent $\ensuremath{\alpha}(U,Q)\ensuremath{-}1$, which depends on the strength of both the Hubbard interaction

U

and the core-hole potential

Q

. For $Q\ensuremath{\lesssim}{U}_{c}∕2$, $\ensuremath{\alpha}$ decreases with increasing

U

and vanishes at the transition $(U\ensuremath{\rightarrow}{U}_{c})$ leading to a symmetric peak in the insulating phase. For $Q\ensuremath{\gtrsim}{U}_{c}∕2$, $\ensuremath{\alpha}$ remains finite, close to the transition, but the integrated intensity of the power law vanishes and there is no associated peak in the insulator. The weight and position of the remaining peaks in the spectra can be understood within a molecular orbital approach.

We study the singularities in x-ray absorption spectra of one-dimensional Hubbard and

t

-

J

models. We use boundary conformal field theory and the Bethe ansatz solutions of these models with both periodic and open boundary conditions to calculate the exponents describing the power-law decay near the edges of x-ray absorption spectra in the case where the core-hole potential has bound states.

The photon-absorption edge in a weakly interacting one-dimensional electron gas is studied, treating backscattering of conduction electrons from the core hole exactly. Close to threshold, there is a power-law singularity in the absorption, I(\ensuremath{\epsilon})\ensuremath{\propto}${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$, with \ensuremath{\alpha}=3/8+${\mathrm{\ensuremath{\delta}}}_{+}$/\ensuremath{\pi}-${\mathrm{\ensuremath{\delta}}}_{+}^{2}$/2${\mathrm{\ensuremath{\pi}}}^{2}$, where ${\mathrm{\ensuremath{\delta}}}_{+}$ is the forward-scattering phase shift of the core hole. In contrast to previous theories, \ensuremath{\alpha} is finite (and universal) in the limit of weak core-hole potential. In the case of weak backscattering U(2

{\mathit{k}}_{\mathit{F}}

), the exponent in the power-law dependence of absorption on energy crosses over to a value \ensuremath{\alpha}=${\mathrm{\ensuremath{\delta}}}_{+}$/\ensuremath{\pi}-${\mathrm{\ensuremath{\delta}}}_{+}^{2}$/2${\mathrm{\ensuremath{\pi}}}^{2}$ above an energy scale ${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{*}}$\ensuremath{\sim}[U(2

{\mathit{k}}_{\mathit{F}}

)${]}^{1/\ensuremath{\gamma}}$, where \ensuremath{\gamma} is a dimensionless measure of the electron-electron interactions.