Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0703293 · Physical Review B

Theory of inelastic scattering from quantum impurities

L. Borda, Lars Fritz, Natan Andrei, Gergely Zaránd

Published Jun 15, 2007• 51 citations
arXivHTMLPDF

Abstract

We use the framework setup recently to compute nonperturbatively inelastic scattering from quantum impurities [G. Zar\'and et al., Phys. Rev. Lett. 93, 107204 (2004)] to study the energy dependence of the single-particle $S$ matrix and the inelastic scattering cross section for a number of quantum impurity models. We study the case of the spin $S=1∕2$ two-channel Kondo model, the Anderson model, and the usual $S=1∕2$ single-channel Kondo model. We discuss the difference between non-Fermi-liquid and Fermi-liquid models and study how a crossover between the non-Fermi-liquid and Fermi-liquid regimes appears in the case of channel anisotropy for the $S=1∕2$ two-channel Kondo model. We show that for the most elementary non-Fermi-liquid system, the two-channel Kondo model, half of the scattering remains inelastic even at the Fermi energy. Details of the derivation of the reduction formulas and a simple path integral approach to connect the $T$ matrix to local correlation functions are also presented.

DOI
10.1103/physrevb.75.235112

Related papers

Fermi-liquid theory for the single-impurity Anderson model

Christophe Mora, Cătălin Paşcu Moca, Jan von Delft +1 more

We generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozi\`eres that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives with respect to the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe ansatz and numerical renormalization group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature, and small bias voltage, and compute the coefficients of the $\ensuremath{\sim}{B}^{2}, \ensuremath{\sim}{T}^{2}$, and $\ensuremath{\sim}{V}^{2}$ terms exactly in terms of the Fermi-liquid parameters. The coefficients of ${T}^{2}, {V}^{2}$, and ${B}^{2}$ are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.

arXiv:1409.3451•Physical Review B•Aug 10, 2015•54 citations

One-and-a-half-channel Kondo model and its family of non-Fermi liquids

Anna I. Tóth

I construct non-Fermi liquid (NFL) quantum impurity models that are similar to the overscreened multi-channel Kondo models with the difference that an odd number of electron species screen the impurity. The simplest of them, named sesqui-channel (i.e. one-and-a-half-channel) Kondo (1.5CK) model, has less degrees of freedom and is simpler than the two-channel Kondo model, and yet it exhibits NFL physics. Using representation theory I derive the 1.5CK model for a spin-half impurity surrounded with electrons in cubic crystal field and solve it with the numerical renormalization group.

arXiv:1708.08019•Aug 26, 2017•0 citations

Non-Fermi Liquid Behavior in Kondo Models

Ian K. Affleck

Despite the fact that the low energy behavior of the basic Kondo model cannot be studied perturbatively it was eventually shown by Wilson, Anderson, Nozieres and others to have a simple "local Fermi liquid theory" description. That is, electronic degrees of freedom become effectively non-interacting in the zero energy limit. However, generalized versions of the Kondo model involving more than one channel or impurity may exhibit low energy behavior of a less trivial sort which can, nonetheless, be solved exactly using either Bethe ansatz or conformal field theory and bosonization techniques. Now the low energy limit exhibits interacting many body behavior. For example, processes in which a single electron scatters off the impurity into a multi electron-hole state have a non-vanishing (and sometimes large) amplitude at zero energy. This corresponds to a rare solveable example of non-Fermi liquid behavior. Essential features of these phenomena are reviewed.

arXiv:cond-mat/0409173•Journal of the Physical Society of Japan•Jan 1, 2005•23 citations

Instability of Non-Fermi Liquid Behavior in the Two-Channel Kondo Model

Yuan Qing-Shan, Chen Hong, Yumei Zhang

The effects of interchannel scattering of conduction electrons by the impurity and repulsion of conduction electrons at the impurity site on the two-channel Kondo model are simultaneously considered in this paper. It is shown that these two perturbations will substantially modify the usual local non-Fermi liquid behavior of the two-channel Kondo model. With bosonization and unitary transformations we find that the system can be transformed into a single channel Kondo model with anisotropy between longitudinal and transverse exchange couplings. Whatever for originally antiferromagnetic or ferromagnetic isotropic coupling, the system always flows to strong-coupling limit, which exhibits local Fermi liquid behavior at low temperatures.

Communications in Theoretical Physics•May 15, 2001•0 citations

Hamiltonian theory of the strongly coupled limit of the Kondo problem in the overscreened case

Domenico Giuliano, Arturo Tagliacozzo

By properly generalizing Nozie`res' Fermi liquid theory, we construct an Hamiltonian approach to the scattering of conduction electrons off a spin-1/2 impurity in the ovescreneed Kondo regime, as T -> 0. We derive the S-matrix at the interacting fixed point, and the corresponding phase shifts, together with leading energy corrections to the unitary limit. We apply our results to obtain the low-temperature dependence of the 2-channel Kondo conductance, and we relate it to possible transport experiments in a Quantum Dot

arXiv:cond-mat/0407624•Journal of Physics Condensed Matter•Aug 14, 2004•4 citations

Renormalization-group approach to spectral properties of the two-channel Anderson impurity model

Frithjof B. Anders

The impurity Green function and dynamical susceptibilities for the two-channel Anderson impurity model are calculated. An exact expression for the self-energy of the impurity Green function as ratio of two correlation functions is given. The imaginary part of the self-energy scales as $\sqrt{\ensuremath{\mid}\ensuremath{\omega}∕{T}_{K}\ensuremath{\mid}}$ for $T\ensuremath{\rightarrow}0$, serving as a hallmark for non-Fermi behavior. The many-body resonance is pinned to a universal value of $1∕(2\ensuremath{\pi}\ensuremath{\Delta})$ at $\ensuremath{\omega}=0$ for arbitrary local occupation in contrary to the single-channel Anderson impurity model. Its shape becomes increasingly more symmetric for the Kondo regimes of the model. The dynamical spin and channel susceptibilities are governed by two energy scales ${T}_{K}$ and ${T}_{h}$, and their ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$ approach a constant value for $\ensuremath{\omega}\ensuremath{\rightarrow}0$.

arXiv:cond-mat/0407169•Physical Review B•Mar 15, 2005•24 citations
We use the framework setup recently to compute nonperturbatively inelastic scattering from quantum impurities [G. Zar'and et al., Phys. Rev. Lett. 93, 107204 (2004)] to study the energy dependence of the single-particle SSS matrix and the inelastic scattering cross section for a number of quantum impurity models. We study the case of the spin S=1∕2S=1∕2S=1∕2 two-channel Kondo model, the Anderson model, and the usual S=1∕2S=1∕2S=1∕2 single-channel Kondo model. We discuss the difference between non-Fermi-liquid and Fermi-liquid models and study how a crossover between the non-Fermi-liquid and Fermi-liquid regimes appears in the case of channel anisotropy for the S=1∕2S=1∕2S=1∕2 two-channel Kondo model. We show that for the most elementary non-Fermi-liquid system, the two-channel Kondo model, half of the scattering remains inelastic even at the Fermi energy. Details of the derivation of the reduction formulas and a simple path integral approach to connect the TTT matrix to local correlation functions are also presented.
We generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozi\`eres that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives with respect to the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe ansatz and numerical renormalization group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature, and small bias voltage, and compute the coefficients of the $\ensuremath{\sim}{B}^{2}, \ensuremath{\sim}{T}^{2}$, and $\ensuremath{\sim}{V}^{2}$ terms exactly in terms of the Fermi-liquid parameters. The coefficients of T2,V2{T}^{2}, {V}^{2}T2,V2, and B2{B}^{2}B2 are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.
I construct non-Fermi liquid (NFL) quantum impurity models that are similar to the overscreened multi-channel Kondo models with the difference that an odd number of electron species screen the impurity. The simplest of them, named sesqui-channel (i.e. one-and-a-half-channel) Kondo (1.5CK) model, has less degrees of freedom and is simpler than the two-channel Kondo model, and yet it exhibits NFL physics. Using representation theory I derive the 1.5CK model for a spin-half impurity surrounded with electrons in cubic crystal field and solve it with the numerical renormalization group.
Despite the fact that the low energy behavior of the basic Kondo model cannot be studied perturbatively it was eventually shown by Wilson, Anderson, Nozieres and others to have a simple "local Fermi liquid theory" description. That is, electronic degrees of freedom become effectively non-interacting in the zero energy limit. However, generalized versions of the Kondo model involving more than one channel or impurity may exhibit low energy behavior of a less trivial sort which can, nonetheless, be solved exactly using either Bethe ansatz or conformal field theory and bosonization techniques. Now the low energy limit exhibits interacting many body behavior. For example, processes in which a single electron scatters off the impurity into a multi electron-hole state have a non-vanishing (and sometimes large) amplitude at zero energy. This corresponds to a rare solveable example of non-Fermi liquid behavior. Essential features of these phenomena are reviewed.
The effects of interchannel scattering of conduction electrons by the impurity and repulsion of conduction electrons at the impurity site on the two-channel Kondo model are simultaneously considered in this paper. It is shown that these two perturbations will substantially modify the usual local non-Fermi liquid behavior of the two-channel Kondo model. With bosonization and unitary transformations we find that the system can be transformed into a single channel Kondo model with anisotropy between longitudinal and transverse exchange couplings. Whatever for originally antiferromagnetic or ferromagnetic isotropic coupling, the system always flows to strong-coupling limit, which exhibits local Fermi liquid behavior at low temperatures.
By properly generalizing Nozie`res' Fermi liquid theory, we construct an Hamiltonian approach to the scattering of conduction electrons off a spin-1/2 impurity in the ovescreneed Kondo regime, as T -> 0. We derive the S-matrix at the interacting fixed point, and the corresponding phase shifts, together with leading energy corrections to the unitary limit. We apply our results to obtain the low-temperature dependence of the 2-channel Kondo conductance, and we relate it to possible transport experiments in a Quantum Dot
The impurity Green function and dynamical susceptibilities for the two-channel Anderson impurity model are calculated. An exact expression for the self-energy of the impurity Green function as ratio of two correlation functions is given. The imaginary part of the self-energy scales as $\sqrt{\ensuremath{\mid}\ensuremath{\omega}∕{T}_{K}\ensuremath{\mid}}$ for $T\ensuremath{\rightarrow}0$, serving as a hallmark for non-Fermi behavior. The many-body resonance is pinned to a universal value of $1∕(2\ensuremath{\pi}\ensuremath{\Delta})$ at $\ensuremath{\omega}=0$ for arbitrary local occupation in contrary to the single-channel Anderson impurity model. Its shape becomes increasingly more symmetric for the Kondo regimes of the model. The dynamical spin and channel susceptibilities are governed by two energy scales TK{T}_{K}TK​ and Th{T}_{h}Th​, and their ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$ approach a constant value for $\ensuremath{\omega}\ensuremath{\rightarrow}0$.