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arXiv:cond-mat/9601021 · Physical review. B, Condensed matter

Resonance in the Fermi-edge singularity of one-dimensional systems

Yuval Oreg, Alexander M. Finkel’stein

Published Apr 15, 1996• 29 citations
arXivHTMLPDF

Abstract

The problem of the Fermi-edge singularity in a one-dimensional Tomonaga-Luttinger liquid is reconsidered. The backward scattering of the conduction band electrons on the impuritylike hole in the valence band is analyzed by mapping the problem onto a Coulomb gas theory. For the case when the electron-electron interaction is repulsive, the obtained exponent of the one-dimensional Fermi-edge singularity appears to be different from the exponent found in previous studies. It is shown that the infrared physics of the Fermi-edge singularity in the presence of backward scattering and electron-electron repulsion resembles the physics of the Kondo problem. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.10928

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The problem of the Fermi-edge singularity in a one-dimensional Tomonaga-Luttinger liquid is reconsidered. The backward scattering of the conduction band electrons on the impuritylike hole in the valence band is analyzed by mapping the problem onto a Coulomb gas theory. For the case when the electron-electron interaction is repulsive, the obtained exponent of the one-dimensional Fermi-edge singularity appears to be different from the exponent found in previous studies. It is shown that the infrared physics of the Fermi-edge singularity in the presence of backward scattering and electron-electron repulsion resembles the physics of the Kondo problem. \textcopyright{} 1996 The American Physical Society.
Interacting electron systems in one dimension with spin-split Fermi points are studied theoretically with the aid of the Tomonaga-Luttinger bosonization. The spin splitting of the systems considered is classified into three categories: (i) the momentum-dependent spin splitting without the time-reversal symmetry, (ii) the momentum-dependent spin splitting with the time-reversal symmetry, and (iii) the momentum-independent spin splitting. Calculating analytically the current-current correlation functions in the three cases, the critical exponents of the optical Fermi-edge singularity (FES) are obtained by using the forward-scattering Tomonaga-Luttinger model. The phase diagrams for the sign of the FES exponent are drawn to clarify the relation between the shape of optical edge spectra and the electron-electron or the electron-hole interactions. Response to circularly polarized incident light is also revealed. In the case of the momentum-dependent spin splitting, the power-law divergence of the edge spectra is possible even for repulsive electron-hole interactions. This is in striking contrast to the FES in spin-degenerate (no spin splitting) electron systems. The FES in a spin-split chiral Luttinger liquid is also discussed. \textcopyright{} 1996 The American Physical Society.
Effects of long-range Coulomb interactions on the Fermi-edge singularity in optical spectra are investigated theoretically for one-dimensional spin-1/2 fermion systems with the use of the Tomonaga-Luttinger bosonization technique. Low-energy excitation spectrum near the Fermi level shows that dispersion of the charge-density fluctuation remains gapless but is nonlinear when the electron-electron (e-e) Coulomb interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., an infinite force range). Temporal behavior of the current-current correlation function is calculated analytically for arbitrary force ranges, ${\ensuremath{\lambda}}_{\mathit{e}}$ and ${\ensuremath{\lambda}}_{\mathit{h}}$, of the e-e and the electron-hole (e-h) Coulomb interactions. (i) When both the e-e and the e-h interactions have large but finite force ranges (${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\infty} and ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\infty}), the correlation function yields a power-law decay only for a long-time regime, which is determined by the force ranges as t>max[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]/vF{\mathit{v}}_{\mathit{F}}vF​. Corresponding optical spectrum near the Fermi edge (within an energy range of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/max[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]) exhibits the power-law divergence or the power-law convergence, which is an ordinary Fermi-edge singularity. (ii) When either the e-e or the e-h interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., ${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\rightarrow}\ensuremath{\infty} and/or ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\rightarrow}\ensuremath{\infty}), an exponent of the correlation function is dependent on time to lead the faster decay than that of any power laws. Then the optical spectra show no power-law dependence and always converge (become zero) at the Fermi edge, which is in striking contrast to the ordinary power-law singularity. Relations between the spectral shape and a measurement time (frequency resolution) are also clarified. \textcopyright{} 1996 The American Physical Society.
We study the critical properties of one-dimensional electron systems with the 1/rα1/r^α1/rα-type long-range interaction (α≥1α\ge 1α≥1). Using bosonization methods, we discuss how the 1D system exhibits a crossover from the Tomonaga-Luttinger liquid to the Wigner crystal when α→1α\to 1α→1. In particular, a detailed analysis is given to the Fermi-edge singularity phenomena when a {\it mobile} core hole is created suddenly in the valence band. It is found that the effect due to the core-hole motion on the Fermi-edge singularity is gradually suppressed as the system approaches the Wigner crystal, and its critical behavior is dominated by the ordinary impurity forward-scattering inherent in a static hole.
We review some of our recent results on the potential scattering in a weakly interacting one-dimensional(1D) electron gas. The technique we developed is a poor man's renormalization group procedure in the scattered wave basis. This technique can treat the renormalizations of the scattering on the barrier and the scattering between the electrons in a coherent way, and it allows us to find the scattering amplitudes on a localized potential of arbitrary strength for electrons at any energy. The obtained phase shifts are used to study the Fermi-edge singularity in an interacting 1D electron system, where anomalous exponent of the power-law singularity in the vicinity of the edge is found. The transmission coefficient is directly related to the conductance of a 1D channel by the Landauer formula. Simple formulas that describe the conductance at any temperature are derived. In spin-[Formula: see text] systems, the electron–electron backscattering induces renormalizations of the interaction constants, which causes the low-temperature conductance to deviate from the results of the Luttinger liquid theory. In particular, the temperature dependence of the conductance may become nonmonotonic. In the presence of a magnetic field, backscattering gives rise to a peak in the differential conductance at bias equal to the Zeeman splitting.
Using an elementary method, we demonstrate that (1) the duality of the correlation exponents near impurity site x=0x=0x=0 between ultraviolet and infrared fixed points is also valid for channel number larger than one; (2) the infrared fixed points ${\ensuremath{\delta}}_{c}^{i}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ do not correspond to two real half-infinity Tomonaga-Luttinger liquids even if the conductance is zero; (3) the electron's density of state at x=0x=0x=0 is enhanced compared with its neighbor ones; (4) as channel number is larger than (or equal to) two, the Fermi-edge singularity induced by backward scattering will be swept off by even small repulsive interaction of the electrons.