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/0301106 · Physical review. B, Condensed matter

Spectral function of the Kondo model in high magnetic fields

Achim Rosch, Theo A. Costi, Jens Paaske, P. Wölfle

Published Jul 29, 2003• 57 citations
arXivHTMLPDF

Abstract

Using a recently developed perturbative renormalization group (RG) scheme, we calculate analytically the spectral function of a Kondo impurity for either large frequencies $\ensuremath{\omega}$ or large magnetic field B and arbitrary frequencies. For large $\ensuremath{\omega}\ensuremath{\gg}\mathrm{max}[{B,T}_{K}]$ the spectral function decays as $1/{\mathrm{ln}}^{2}[\ensuremath{\omega}{/T}_{K}]$ with prefactors which depend on the magnetization. The spin-resolved spectral function displays a pronounced peak at $\ensuremath{\omega}\ensuremath{\sim}B$ with a characteristic asymmetry. In a detailed comparison with results from numerical renormalization group (NRG) and bare perturbation theory in next-to-leading logarithmic order, we show that our perturbative RG scheme is controlled by the small parameter $1/\mathrm{ln}[\mathrm{max}(\ensuremath{\omega}{,B)/T}_{K}].$ Furthermore, we assess the ability of the NRG to resolve structures at finite frequencies.

DOI
10.1103/physrevb.68.014430

Related papers

Comment on “Scaling feature of magnetic field induced Kondo-peak splittings”

Sebastian Schmitt, Frithjof B. Anders

In a recent work, Zhang and coworkers [Phys. Rev. B 82, 075111 (2010)] studied the Zeeman splitting of the Kondo resonance for the single impurity Anderson model in a finite magnetic field $B$ with the numerical renormalization group (NRG) method. They report a discrepancy between the position of the Kondo resonance in the total spectral function and the position in the spin-resolved spectral function at large magnetic fields. Additionally, the position of the Kondo maximum exceeded the Zeeman energy for $B/{T}_{K}\ensuremath{\gtrsim}5\ensuremath{-}10$, where ${T}_{K}$ is the low-energy Kondo scale of the model. In this comment, we argue that both these findings are a result of the specific choice of NRG parameter values. However, we reproduce the crossover in the splitting from Kondo-like behavior to a nonuniversal splitting larger than the Zeeman energy, but this crossover occurs at much larger fields of the order of the charge scale.

arXiv:1102.2187•Physical Review B•May 23, 2011•4 citations

Spectral function of the Anderson impurity model at finite temperatures

Aldo Isidori, David Roosen, Lorenz Bartosch +2 more

Using the functional renormalization group (FRG) and the numerical renormalization group (NRG), we calculate the spectral function of the Anderson impurity model at zero and finite temperatures. In our FRG scheme spin fluctuations are treated nonperturbatively via a suitable Hubbard-Stratonovich field, but vertex corrections are neglected. A comparison with our highly accurate NRG results shows that this FRG scheme provides a good description of the spectral line shape at zero and finite temperatures both in the weak- and strong-coupling regimes, although at zero temperature the FRG is not able to reproduce the known exponential narrowing of the Kondo resonance at strong coupling.

arXiv:1003.3188•Physical Review B•Jun 17, 2010•20 citations

Padé approximant approach for obtaining finite-temperature spectral functions of quantum impurity models using the numerical renormalization group technique

Žiga Osolin, Rok Žitko

We introduce an alternative approach for obtaining smooth finite-temperature spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique. It is based on calculating first the Green's function on the imaginary-frequency axis, followed by an analytic continuation to the real-frequency axis using Pad\'e approximants. The arbitrariness in choosing a suitable kernel in the conventional broadening approach is thereby removed and, furthermore, we find that the Pad\'e method is able to resolve fine details in spectral functions with less artifacts on the scale of $\ensuremath{\omega}\ensuremath{\sim}T$. We discuss the convergence properties with respect to the NRG calculation parameters (discretization $\ensuremath{\Lambda}$, $z$-averaging, truncation cutoff) and the number of Matsubara points taken into account in the analytic continuation. We test the technique on the single-impurity Anderson model and the Hubbard model (within the dynamical mean-field theory). For the Anderson impurity model, we discuss the shape of the Kondo resonance and its temperature dependence. For the Hubbard model, we discuss the inner structure of the Hubbard bands in metallic and insulating solutions at half-filling, as well as in the doped Mott insulator. Based on these test cases, we conclude that the Pad\'e approximant approach provides improved results for spectral functions at low-frequency scales of $\ensuremath{\omega}\ensuremath{\lesssim}T$ and that it is capable of resolving sharp spectral features also at high frequencies.

arXiv:1302.3334•Physical Review B•Jun 28, 2013•35 citations

Scaling feature of magnetic field induced Kondo-peak splittings

Hui Zhang, X. C. Xie, Qing‐Feng Sun

By using the full density matrix approach to spectral functions within the numerical renormalization group method, we present a detailed study of the magnetic field induced splittings in the spin-resolved and the total spectral densities of a Kondo correlated quantum dot described by the single level Anderson impurity model. The universal scaling of the splittings with magnetic field is examined by varying the Kondo scale either by a change of local level position at a fixed tunnel coupling or by a change of the tunnel coupling at a fixed level position. We find that the Kondo-peak splitting $\ensuremath{\Delta}/{T}_{K}$ in the spin-resolved spectral function always scales perfectly for magnetic fields $B<8{T}_{K}$ in either of the two ${T}_{K}$-adjusted paths. Scaling is destroyed for fields $B>10{T}_{K}$. On the other hand, the Kondo peak splitting $\ensuremath{\delta}/{T}_{K}$ in the total spectral function does slightly deviate from the conventional scaling theory in whole magnetic field window along the coupling-varying path. Furthermore, we show the scaling analysis suitable for all field windows within the Kondo regime and two specific fitting scaling curves are given from which certain detailed features at low field are derived. In addition, the scaling dimensionless quantity $\ensuremath{\Delta}/2B$ and $\ensuremath{\delta}/2B$ are also studied and they can reach and exceed 1 in the large magnetic field region, in agreement with a recent experiment [T. M. Liu et al., Phys. Rev. Lett. 103, 026803 (2009)].

arXiv:1006.4987•Physical Review B•Aug 9, 2010•6 citations

Dynamical spin-spin correlation functions in the Kondo model out of equilibrium

Dirk Schuricht, Herbert Schoeller

We calculate the dynamical spin-spin correlation functions of a Kondo dot coupled to two noninteracting leads held at different chemical potentials. To this end we generalize a recently developed real-time renormalization group (RG) method in frequency space [H. Schoeller, Eur. Phys. J. Spec. Top. 168, 179 (2009)] to allow the calculation of dynamical correlation functions of arbitrary dot operators in systems describing spin and/or orbital fluctuations. The resulting two-loop RG equations are analytically solved in the weak-coupling regime. This implies that the method can be applied provided either the voltage $V$ through the dot or the external magnetic field ${h}_{0}$ are sufficiently large, $\mathrm{max}{V,{h}_{0}}\ensuremath{\gg}{T}_{K}$, where the Kondo temperature ${T}_{K}$ is the scale where the system enters the strong-coupling regime. Explicitly, we calculate the longitudinal and transverse spin-spin correlation and response functions as well as the resulting fluctuation-dissipation ratios. The correlation functions in real-frequency space can be calculated in Matsubara space without the need of any analytical continuation. We obtain analytic results for the line shape, the small- and large-frequency limits, and several other features like the height and width of the peak in the transverse susceptibility at $\ensuremath{\Omega}\ensuremath{\approx}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$ denotes the renormalized magnetic field. Furthermore, we discuss how the developed method can be generalized to calculate dynamical correlation functions of other operators involving reservoir degrees of freedom as well.

arXiv:0905.3095•Physical Review B•Aug 26, 2009•25 citations

Spin precession and real-time dynamics in the Kondo model: Time-dependent numerical renormalization-group study

Frithjof B. Anders, Avraham Schiller

A detailed derivation of the recently proposed time-dependent numerical renormalization-group (TD-NRG) approach to nonequilibrium dynamics in quantum-impurity systems is presented. We demonstrate that the method is suitable for fermionic as well as bosonic baths. Comparisons with exact analytical results for the charge relaxation in the resonant-level model and for dephasing in the spin-boson model establish the accuracy of the method. The real-time dynamics of a single spin coupled to each type of bath is investigated. We use the TD-NRG to calculate the spin relaxation and spin precession of a single Kondo impurity. The short- and long-time dynamics are studied as a function of temperature in the ferromagnetic and antiferromagnetic regimes. The short-time dynamics agrees very well with analytical results obtained at second order in the exchange coupling $J$. In the ferromagnetic regime, the transient spin decay is described by the scaling variable $x=2{\ensuremath{\rho}}_{F}\ensuremath{\mid}J(T)\ensuremath{\mid}Tt$. In the antiferromagnetic regime, the long-time decay is governed for $T<{T}_{K}$ by the Kondo time scale $1∕{T}_{K}$. Here ${\ensuremath{\rho}}_{F}$ is the conduction-electron density of states, ${T}_{K}$ is the Kondo temperature, and $J(T)$ is the effective exchange coupling at temperature $T$. Results for spin precession are obtained by rotating the external magnetic field from the $x$ axis to the $z$ axis.

arXiv:cond-mat/0604517•Physical Review B•Dec 18, 2006•250 citations
Using a recently developed perturbative renormalization group (RG) scheme, we calculate analytically the spectral function of a Kondo impurity for either large frequencies \ensuremathω\ensuremath{\omega}\ensuremathω or large magnetic field B and arbitrary frequencies. For large \ensuremathω\ensuremath≫max[B,TK]\ensuremath{\omega}\ensuremath{\gg}\mathrm{max}[{B,T}_{K}]\ensuremathω\ensuremath≫max[B,TK​] the spectral function decays as 1/ln2[\ensuremathω/TK]1/{\mathrm{ln}}^{2}[\ensuremath{\omega}{/T}_{K}]1/ln2[\ensuremathω/TK​] with prefactors which depend on the magnetization. The spin-resolved spectral function displays a pronounced peak at \ensuremathω\ensuremath∼B\ensuremath{\omega}\ensuremath{\sim}B\ensuremathω\ensuremath∼B with a characteristic asymmetry. In a detailed comparison with results from numerical renormalization group (NRG) and bare perturbation theory in next-to-leading logarithmic order, we show that our perturbative RG scheme is controlled by the small parameter 1/ln[max(\ensuremathω,B)/TK].1/\mathrm{ln}[\mathrm{max}(\ensuremath{\omega}{,B)/T}_{K}].1/ln[max(\ensuremathω,B)/TK​]. Furthermore, we assess the ability of the NRG to resolve structures at finite frequencies.
In a recent work, Zhang and coworkers [Phys. Rev. B 82, 075111 (2010)] studied the Zeeman splitting of the Kondo resonance for the single impurity Anderson model in a finite magnetic field BBB with the numerical renormalization group (NRG) method. They report a discrepancy between the position of the Kondo resonance in the total spectral function and the position in the spin-resolved spectral function at large magnetic fields. Additionally, the position of the Kondo maximum exceeded the Zeeman energy for $B/{T}_{K}\ensuremath{\gtrsim}5\ensuremath{-}10$, where TK{T}_{K}TK​ is the low-energy Kondo scale of the model. In this comment, we argue that both these findings are a result of the specific choice of NRG parameter values. However, we reproduce the crossover in the splitting from Kondo-like behavior to a nonuniversal splitting larger than the Zeeman energy, but this crossover occurs at much larger fields of the order of the charge scale.
Using the functional renormalization group (FRG) and the numerical renormalization group (NRG), we calculate the spectral function of the Anderson impurity model at zero and finite temperatures. In our FRG scheme spin fluctuations are treated nonperturbatively via a suitable Hubbard-Stratonovich field, but vertex corrections are neglected. A comparison with our highly accurate NRG results shows that this FRG scheme provides a good description of the spectral line shape at zero and finite temperatures both in the weak- and strong-coupling regimes, although at zero temperature the FRG is not able to reproduce the known exponential narrowing of the Kondo resonance at strong coupling.
We introduce an alternative approach for obtaining smooth finite-temperature spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique. It is based on calculating first the Green's function on the imaginary-frequency axis, followed by an analytic continuation to the real-frequency axis using Pad\'e approximants. The arbitrariness in choosing a suitable kernel in the conventional broadening approach is thereby removed and, furthermore, we find that the Pad\'e method is able to resolve fine details in spectral functions with less artifacts on the scale of $\ensuremath{\omega}\ensuremath{\sim}T$. We discuss the convergence properties with respect to the NRG calculation parameters (discretization $\ensuremath{\Lambda}$, zzz-averaging, truncation cutoff) and the number of Matsubara points taken into account in the analytic continuation. We test the technique on the single-impurity Anderson model and the Hubbard model (within the dynamical mean-field theory). For the Anderson impurity model, we discuss the shape of the Kondo resonance and its temperature dependence. For the Hubbard model, we discuss the inner structure of the Hubbard bands in metallic and insulating solutions at half-filling, as well as in the doped Mott insulator. Based on these test cases, we conclude that the Pad\'e approximant approach provides improved results for spectral functions at low-frequency scales of $\ensuremath{\omega}\ensuremath{\lesssim}T$ and that it is capable of resolving sharp spectral features also at high frequencies.
By using the full density matrix approach to spectral functions within the numerical renormalization group method, we present a detailed study of the magnetic field induced splittings in the spin-resolved and the total spectral densities of a Kondo correlated quantum dot described by the single level Anderson impurity model. The universal scaling of the splittings with magnetic field is examined by varying the Kondo scale either by a change of local level position at a fixed tunnel coupling or by a change of the tunnel coupling at a fixed level position. We find that the Kondo-peak splitting $\ensuremath{\Delta}/{T}_{K}$ in the spin-resolved spectral function always scales perfectly for magnetic fields B<8TKB<8{T}_{K}B<8TK​ in either of the two TK{T}_{K}TK​-adjusted paths. Scaling is destroyed for fields B>10TKB>10{T}_{K}B>10TK​. On the other hand, the Kondo peak splitting $\ensuremath{\delta}/{T}_{K}$ in the total spectral function does slightly deviate from the conventional scaling theory in whole magnetic field window along the coupling-varying path. Furthermore, we show the scaling analysis suitable for all field windows within the Kondo regime and two specific fitting scaling curves are given from which certain detailed features at low field are derived. In addition, the scaling dimensionless quantity $\ensuremath{\Delta}/2B$ and $\ensuremath{\delta}/2B$ are also studied and they can reach and exceed 1 in the large magnetic field region, in agreement with a recent experiment [T. M. Liu et al., Phys. Rev. Lett. 103, 026803 (2009)].
We calculate the dynamical spin-spin correlation functions of a Kondo dot coupled to two noninteracting leads held at different chemical potentials. To this end we generalize a recently developed real-time renormalization group (RG) method in frequency space [H. Schoeller, Eur. Phys. J. Spec. Top. 168, 179 (2009)] to allow the calculation of dynamical correlation functions of arbitrary dot operators in systems describing spin and/or orbital fluctuations. The resulting two-loop RG equations are analytically solved in the weak-coupling regime. This implies that the method can be applied provided either the voltage VVV through the dot or the external magnetic field h0{h}_{0}h0​ are sufficiently large, $\mathrm{max}{V,{h}_{0}}\ensuremath{\gg}{T}_{K}$, where the Kondo temperature TK{T}_{K}TK​ is the scale where the system enters the strong-coupling regime. Explicitly, we calculate the longitudinal and transverse spin-spin correlation and response functions as well as the resulting fluctuation-dissipation ratios. The correlation functions in real-frequency space can be calculated in Matsubara space without the need of any analytical continuation. We obtain analytic results for the line shape, the small- and large-frequency limits, and several other features like the height and width of the peak in the transverse susceptibility at $\ensuremath{\Omega}\ensuremath{\approx}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$ denotes the renormalized magnetic field. Furthermore, we discuss how the developed method can be generalized to calculate dynamical correlation functions of other operators involving reservoir degrees of freedom as well.
A detailed derivation of the recently proposed time-dependent numerical renormalization-group (TD-NRG) approach to nonequilibrium dynamics in quantum-impurity systems is presented. We demonstrate that the method is suitable for fermionic as well as bosonic baths. Comparisons with exact analytical results for the charge relaxation in the resonant-level model and for dephasing in the spin-boson model establish the accuracy of the method. The real-time dynamics of a single spin coupled to each type of bath is investigated. We use the TD-NRG to calculate the spin relaxation and spin precession of a single Kondo impurity. The short- and long-time dynamics are studied as a function of temperature in the ferromagnetic and antiferromagnetic regimes. The short-time dynamics agrees very well with analytical results obtained at second order in the exchange coupling JJJ. In the ferromagnetic regime, the transient spin decay is described by the scaling variable $x=2{\ensuremath{\rho}}_{F}\ensuremath{\mid}J(T)\ensuremath{\mid}Tt$. In the antiferromagnetic regime, the long-time decay is governed for T<TKT<{T}_{K}T<TK​ by the Kondo time scale 1∕TK1∕{T}_{K}1∕TK​. Here ${\ensuremath{\rho}}_{F}$ is the conduction-electron density of states, TK{T}_{K}TK​ is the Kondo temperature, and J(T)J(T)J(T) is the effective exchange coupling at temperature TTT. Results for spin precession are obtained by rotating the external magnetic field from the xxx axis to the zzz axis.