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arXiv:cond-mat/9311018 · Physical review. B, Condensed matter

Fermi-edge singularities and backscattering in a weakly interacting one-dimensional electron gas

C. L. Kane, К. А. Матвеев, L. I. Glazman

Published Jan 15, 1994• 70 citations
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Abstract

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.

DOI
10.1103/physrevb.49.2253

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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}\ensuremathϵ\ensuremath−\ensuremathα{\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}\ensuremathϵ\ensuremath−\ensuremathα, with \ensuremath{\alpha}=3/8+\ensuremathδ+{\mathrm{\ensuremath{\delta}}}_{+}\ensuremathδ+​/\ensuremath{\pi}-\ensuremathδ+2{\mathrm{\ensuremath{\delta}}}_{+}^{2}\ensuremathδ+2​/2\ensuremathπ2{\mathrm{\ensuremath{\pi}}}^{2}\ensuremathπ2, where \ensuremathδ+{\mathrm{\ensuremath{\delta}}}_{+}\ensuremathδ+​ 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(2kF{\mathit{k}}_{\mathit{F}}kF​), the exponent in the power-law dependence of absorption on energy crosses over to a value \ensuremath{\alpha}=\ensuremathδ+{\mathrm{\ensuremath{\delta}}}_{+}\ensuremathδ+​/\ensuremath{\pi}-\ensuremathδ+2{\mathrm{\ensuremath{\delta}}}_{+}^{2}\ensuremathδ+2​/2\ensuremathπ2{\mathrm{\ensuremath{\pi}}}^{2}\ensuremathπ2 above an energy scale \ensuremathϵ∗{\mathrm{\ensuremath{\epsilon}}}^{\mathrm{*}}\ensuremathϵ∗\ensuremath{\sim}[U(2kF{\mathit{k}}_{\mathit{F}}kF​)]1/\ensuremathγ{]}^{1/\ensuremath{\gamma}}]1/\ensuremathγ, where \ensuremath{\gamma} is a dimensionless measure of the electron-electron interactions.
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.
We theoretically investigate the Fermi-edge singularity in a spin-incoherent Luttinger liquid. Both cases of finite and infinite core hole mass are explored, as well as the effect of a static external magnetic field of arbitrary strength. For a finite mass core hole the absorption edge behaves as (omega-omega th)alpha/square root of absolute value (ln(omega-omega th)) for frequencies omega just above the threshold frequency omega th. The exponent alpha depends on the interaction parameter g of the interacting one dimensional system, the electron-hole coupling, and is independent of the magnetic field strength, the momentum, and the mass of the excited core hole (in contrast to the spin-coherent case). In the infinite mass limit, the spin-incoherent problem can be mapped onto an equivalent problem in a spinless Luttinger liquid for which the logarithmic factor is absent, and backscattering from the core hole leads to a universal contribution to the exponent alpha.
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.
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.
Scattering from a localized potential of arbitrary strength is found for weakly interacting electrons. Poor man's renormalization group procedure allows us to calculate the renormalized transmission amplitudes at any energies. Simple formulas that describe the conductance at any temperature are derived. In the presence of the electron-electron backscattering, the low-temperature conductance deviates from the results of 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.