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Physical review. B, Condensed matter

Multichannel impurity scattering in a Tomonaga-Luttinger liquid

Yu-Liang Liu

Published May 15, 1998• 0 citations
Publisher

Abstract

Using an elementary method, we demonstrate that (1) the duality of the correlation exponents near impurity site $x=0$ 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 $x=0$ 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.

DOI
10.1103/physrevb.57.12104

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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 \ensuremathδci=\ifmmode±\else\textpm\fi\ensuremathπ/2{\ensuremath{\delta}}_{c}^{i}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2\ensuremathδci​=\ifmmode±\else\textpm\fi\ensuremathπ/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.
Using bosonization and phase shift representation, we rigorously treat backward scattering of electrons on an impurity in a one-dimensional interacting electronic system, and demonstrate that correlation exponents of the system depend on a phase shift induced by the backward scattering, and usual exponent duality of the correlation functions between ultraviolet and infrared fixed points comes from the phase shift dependence of the correlation exponents. Finally, we study the tunnelling conductance of the system at zero temperature and obtain a modified Landauer-Bütiker formula.
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.
Using bosonization and path-integral methods, we study general low-temperature behavior of nonmagnetic and magnetic impurity scattering in Tomonaga-Luttinger liquid, and calculate the electron Green function for a general backward scattering potential. We demonstrate that the electron density of state near the impurity site is suppressed by backward scattering, but it mainly remains invariant far away from the impurity, and that at zero temperature the electrons are completely reflected on the impurity site; the system breaks into two subsystems, but right- and left-moving electron fields have a twisted boundary condition. We also show that a testing charge can only be partially screened by conduction electrons, and in a strong interaction region the impurity susceptibility has a 1/T1/T1/T-type low-temperature behavior.
The orthogonality catastrophe in a Tomonaga-Luttinger liquid with an impurity is reexamined for the case when the interaction parameter or the dimensionless conductance is g=1/2.g=1/2.g=1/2. By transforming bosons back to fermions, the Hamiltonian is reduced to a quadratic form, which allows for explicit calculation of the overlap integral and the local density of states at the defect site. The exponent of the orthogonality catastrophe due to a backward scattering center is found to be 1/8, in agreement with previous studies using different approaches. The time dependence of the core-hole Green's function is computed numerically, which shows a clear crossover from a nonuniversal short-time behavior to a universal long-time behavior. The local density of states vanishes linearly in the low-energy limit at g=1/2.g=1/2.g=1/2.
We report the exact calculation of the Fermi-edge singularity exponent for correlated electrons in one dimension (Luttinger liquid). Focusing on the special interaction parameter g=1/2g=1/2g=1/2, the asymptotic long-time behavior can be obtained using the Wiener-Hopf method. The result confirms the previous assumption of an open boundary fixed point. In addition, a dynamic kkk-channel Kondo impurity is studied via Abelian bosonization for k=2k=2k=2 and k=4k=4k=4. It is shown that the corresponding orthogonality exponents are related to the orthogonality exponent in a Luttinger liquid.