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arXiv:cond-mat/9702195 · Physical review. B, Condensed matter

Local perturbation in a Tomonaga-Luttinger liquid at g=1/2: Orthogonality catastrophe, Fermi-edge singularity, and local density of states

Akira Furusaki

Published Oct 15, 1997• 42 citations
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Abstract

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 $g=1/2.$ 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 $g=1/2.$

DOI
10.1103/physrevb.56.9352

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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.
The response of the Luttinger liquid to a local potential is studied at zero temperature. It exhibits a crossover behavior. On the largest time scale and in the case of repulsive electron-electron interaction, the backscattering from the potential contributes to the core-hole Green's function G(t) the factor (t0{\mathit{t}}_{0}t0​/t)1/8{)}^{1/8})1/8---a universal (potential independent) decay law. The orthogonality catastrophe remains of the same analytical type as for the simple metals.
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.
In this thesis, we have explored various aspects of the dynamics of an impurity moving in more than one 1D bath. A recurring theme has been the investigation of the orthogonality catastrophe (OC) that follows the injection of the impurity in the system. This phenomenon has been studied by calculating the Green’s function of the impurity, which also described the time evolution and spectral features of the latter. The distinctive signature of the OC is that the Green’s function shows a power-law tail at long times. This function has been calculated using suitable perturbative expansions in the impurity-bath coupling, namely the Linked Cluster Expansion (LCE) and a time-dependent perturbation theory around a nontrivial dynamics. All of our results are nonperturbative in the inter-bath hopping. In the two-bath scenario we have performed a detailed asymptotic expansion of the LCE Green's function at long times, which turned out to be very accurate in comparison with its numerical evaluation. The expansion has allowed us to obtain the renormalisation of the dispersion of the impurity bands, as well as the exponent of the power-law decay and the lifetime of the odd mode. One of our main results is that the OC, leading to the breakdown of the quasiparticle picture, survives the inclusion of a second 1D bath and dominates the long-time behaviour of all the components of the Green's function. In particular, the exponent characterising the long-time behaviour of the Green's function is given by half of the average of the exponents of the individual baths and, notably, is the same for the intra-bath Green's functions and for that connecting the two baths. In the case of two asymmetric baths, the Green's function is nonuniversal, acquiring a high-frequency component at short times and exhibiting persistent oscillations at longer times. In real experiments, the temperature of the baths is always finite, so we have extended the LCE treatment to this scenario. The effect of the temperature is to suppress the Green's function, limiting the possibility to observe its coherent oscillations and the power-law tail. At sufficiently long times, we found analytically that this suppression is exponential in time, with a different decay constant for the even and odd modes. For low temperatures, the two decay constants are approximately coinciding, and are proportional to the temperature. Using a perturbation theory in the inter-band part of the interaction we have been able to reproduce the LCE Green’s function and the OC with a method which also allows us to access the evolution of the whole impurity-bath system. The advantage of this approach is that it provides an analytic expression for the time evolution of the state of the whole impurity-baths system. This has allowed us to compute the time evolution of observables beyond the impurity Green's function, including the often-neglected properties of the baths. Moreover, the numerical effort required by the approach is sufficiently low so that we have been able to treat the case in which the impurity is initialised in a wave packet of an (almost) arbitrary shape. On the impurity side, we have analysed the time evolution of its population within each bath, observing how the persistent oscillations of the free impurity are damped and slowed down by the interaction with the baths. The impurity momentum is subjected to damping, as well. We have observed that momentum is transferred to the baths in two steps: a short transient, connected to the bath relaxation and independent of the inter-bath hopping, and a much slower decay caused by the emission of phonons during the deexcitation of the odd mode. Finally, we have also examined the time evolution of the probability density of finding the impurity in a given position and bath, for various Gaussian wave packets. We have studied the time evolution of observables describing the dynamics of the baths, which are rarely discussed in the literature on mobile impurities, but are nonetheless accessible to experiments. We have looked at the number of excited phonons, which shows a slow logarithmic divergence in time, related to the OC, superimposed with the faster growth caused by the emission of phonons from the odd mode decay. The particle density of the baths shows a semiclassical behaviour, intuitively similar to that of a pond in which a stone has been thrown. 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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.
When a local potential changes abruptly in time, an electron gas responds by shifting to a new state which at long times is orthogonal to the one in the absence of the local potential. This is known as Anderson's orthogonality catastrophe and it is relevant for the so-called x-ray edge or Fermi-edge singularity, and for tunneling into an interacting one-dimensional system of fermions. It often happens that the finite frequency response of the photon absorption or the tunneling density of states exhibits a singular behavior as a function of frequency: [Formula: see text], where ω(th) is a threshold frequency and α is an exponent characterizing the singular response. In this review singular responses of spin-incoherent Luttinger liquids are reviewed. Such responses most often do not fall into the familiar form above, but instead typically exhibit logarithmic corrections and display a much higher universality in terms of the microscopic interactions in the theory. Specific predictions are made, the current experimental situation is summarized and key outstanding theoretical issues related to spin-incoherent Luttinger liquids are highlighted.
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.