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arXiv:cond-mat/9702234 · Physical review. B, Condensed matter

X-ray edge singularity in integrable lattice models of correlated electrons

Fabian H. L. Eßler, Holger Frahm

Published Sep 15, 1997• 25 citations
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Abstract

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.

DOI
10.1103/physrevb.56.6631

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We study the spectral function of interacting one-dimensional fermions for an integrable lattice model away from half-filling. The divergent power-law singularity of the spectral function near the single-particle or single-hole energy is described by an effective x-ray edge type model. At low densities and for momentum near the zone boundary, we find a second divergent singularity at higher energies which is associated with a two-particle bound state. We use the Bethe ansatz solution of the model to calculate the exact singularity exponents for any momentum and for arbitrary values of chemical potential and interaction strength in the critical regime. We relate the singularities of the spectral function to the long-time decay of the fermion Green's function and compare our predictions with numerical results from the time-dependent density-matrix renormalization group (tDMRG). Our results show that the tDMRG method is able to provide accurate time decay exponents in the cases of power-law decay of the Green's function. Some implications for the line shape of the dynamical structure factor away from half-filling are also discussed. In addition, we show that 85% of the total spectral weight in the zero field Heisenberg model near wave-vector $q=\ensuremath{\pi}$ is given by the two-spinon contribution.
We use the Bethe Ansatz solution for the one dimensional Hubbard model with open boundary conditions and applied boundary fields to study the spectrum of bound states at the boundary. Depending on the strength of the boundary potentials one finds that the true ground state contains a single charge or, for boundary potentials comparable to the Hubbard interaction, a pair of electrons in a bound state. If these are left unoccupied one finds holon and spinon bound states. We compute the finite size corrections to the low lying energies in this system and use the predictions of boundary conformal field theory to study the exponents related to the orthogonality catastrophe.
Bariev's model for correlated hopping in one dimension is integrable and soluble within the framework of two nested Bethe Ansätze. For repulsive interaction and finite magnetization the low energy excitation spectrum is a two-component Luttinger liquid with conformal charges c=1. In zero magnetic field the spectrum of spin excitations is gapped. For higher energy excitations the single-hole (single-particle) spectral functions display edge singularities that deviate from those expected from a Luttinger liquid. The system can be described as a mobile impurity coupled to the Luttinger liquid yielding an effective x-ray edge type model. Using the Bethe Ansatz solution for repulsive interaction we obtain expressions for the critical exponents for the single-particle and single-hole Green's functions in zero magnetic field.
We use the Bethe ansatz solution for the one-dimensional Hubbard model with open boundary conditions and applied boundary fields to study the spectrum of bound states at the boundary. Depending on the strength of the boundary potentials, one finds that the true ground state contains a single charge or, for boundary potentials comparable with the Hubbard interaction, a pair of electrons in a bound state. If these are left unoccupied one finds holon and spinon bound states. We compute the finite size corrections to the low-lying energies in this system and use the predictions of boundary conformal field theory to study the exponents related to the orthogonality catastrophe.
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 UUU and the core-hole potential QQQ. For $Q\ensuremath{\lesssim}{U}_{c}∕2$, $\ensuremath{\alpha}$ decreases with increasing UUU 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.