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Physical review. B, Condensed matter

Fermi-edge singularity of the Tomonaga-Luttinger liquids with spin-split Fermi points

Hiroshi Otani, Tetsuo Ogawa

Published Feb 15, 1996• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.53.4684

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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.
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.
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.
In a magnetic field an interacting electron gas in one dimension may be described as a Tomonaga-Luttinger model comprising two components with different Fermi velocities due to the Zeeman splitting. This destroys the spin-charge separation, and even the quantities such as the density-density correlation involve the spins. Specifically, we have shown that the ratio of the up-spin and down-spin conductivities in a dirty system diverges at low temperatures like an inverse power of the temperature, resulting in a spin-polarized current. In finite, clean systems the conductance becomes different for up and down spins as another manifestation of the electron-electron interaction. \textcopyright{} 1996 The American Physical Society.
Using higher-dimensional bosonization we study correlation functions of fermions with singular forward scattering. Following Bares and Wen [Phys. Rev. B 48, 8636 (1993)] we consider density-density interactions in ddd dimensions that diverge in Fourier space for small momentum transfers |q| as $|\mathbf{q}{|}^{\ensuremath{-}\ensuremath{\eta}}$ with $\ensuremath{\eta}=2(d\ensuremath{-}1).$ In this case the single-particle Green's function shows Luttinger-liquid behavior. However, in contrast to d=1,d=1,d=1, in higher dimensions the singularity of the momentum distribution nk{n}_{\mathbf{k}}nk​ for wave vectors k close to the Fermi wave vector kF{k}_{F}kF​ is characterized by a different exponent than the singularity of the density of states \ensuremath{\rho}(\ensuremath{\omega}) for frequencies \ensuremath{\omega} close to zero. We also calculate the irreducible polarization ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\mathbf{q},\ensuremath{\omega})$ for $|\mathbf{q}|\ensuremath{\approx}{2k}_{F}.$ Whereas in the one-dimensional Tomonaga-Luttinger model ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\ifmmode\pm\else\textpm\fi{}{2k}_{F}+q,\ensuremath{\omega})$ exhibits anomalous scaling, we show that in d>1d>1d>1 the leading singular corrections cancel. Thus, even though singular density-density interactions in d>1d>1d>1 lead to Luttinger liquid behavior of the single-particle Green's function, gauge invariant correlation functions such as the polarization or the conductivity show conventional Fermi-liquid behavior. We discuss consequences for the effect of disorder on higher-dimensional Luttinger liquids.