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:1309.3757 · Physical Review B

Local susceptibility and Kondo scaling in the presence of finite bandwidth

M. Hanl, Andreas Weichselbaum

Published Feb 21, 2014• 29 citations
arXivHTMLPDF

Abstract

The Kondo scale ${T}_{\mathrm{K}}$ for impurity systems is expected to guarantee universal scaling of physical quantities. However, in practice, not every definition of ${T}_{\mathrm{K}}$ necessarily supports this notion away from the strict scaling limit. Specifically, this paper addresses the role of finite bandwidth $D$ in the strongly correlated Kondo regime. For this, various theoretical definitions of ${T}_{\mathrm{K}}$ are analyzed based on the inverse magnetic impurity susceptibility at zero temperature. While conventional definitions in that respect quickly fail to ensure universal Kondo scaling for a large range of $D$, this paper proposes an altered definition of ${T}_{\mathrm{K}}^{\mathrm{sc}}$ that allows universal scaling of dynamical or thermal quantities for a given fixed Hamiltonian. If the scaling is performed with respect to an external parameter that directly enters the Hamiltonian, such as magnetic field, the corresponding ${T}_{\mathrm{K}}^{\mathrm{sc}},\mathrm{B}$ for universal scaling differs, yet becomes equivalent to ${T}_{\mathrm{K}}^{\mathrm{sc}}$ in the scaling limit. The only requirement for universal scaling in the full Kondo parameter regime with a residual error of less than 1% is a well-defined isolated Kondo feature with ${T}_{\mathrm{K}}\ensuremath{\lesssim}0.01\phantom{\rule{0.16em}{0ex}}D$ irrespective of specific other impurity parameter settings. By varying $D$ over a wide range relative to the bare energies of the impurity, for example, this allows a smooth transition from the Anderson to the Kondo model.

DOI
10.1103/physrevb.89.075130

Related papers

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

Numerical study of the Kondo cloud using finite-U slave bosons

L. Costa Ribeiro, George Balster Martins, G. Gómez-Silva +1 more

In this work, using the finite-U slave boson mean-field approximation to solve the single-impurity Anderson model, the authors apply two different strategies to study the Kondo cloud, by analyzing quantities that are dependent on the distance to the magnetic impurity and then finding a universal distance scale ${\ensuremath{\xi}}_{K}$ through the collapse of the results into an universal function. The first method is based on the analysis of the local density of states of the conduction electrons (denoted as ${\ensuremath{\xi}}_{K}^{L})$, while the second relies on the analysis of spin correlations $({\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}})$. Our calculations show that there is exact quantitative agreement, in the way ${\ensuremath{\xi}}_{K}$ depends on $U/\mathrm{\ensuremath{\Gamma}}$, between ${\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}}$ and the results obtained through the heuristic expression ${\ensuremath{\xi}}_{K}\ensuremath{\propto}{v}_{F}/{T}_{K}$, while there is very close quantitative agreement between ${\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}}$ and ${\ensuremath{\xi}}_{K}^{L}$. The use of the slave boson technique to calculate the spin correlations, which eliminates finite size effects, allowed us to study very large Kondo clouds, something that is very difficult using other techniques, like the density matrix renormalization Group method, for example. In addition, the very smooth curves obtained for the spin correlations allowed us to qualitatively identify a region in the Kondo cloud, adjacent to the impurity, that had been connected to the free orbital fixed point in previous numerical renormalization group calculations [Phys. Rev. B 84, 115120 (2011)].

Physical review. B./Physical review. B•Feb 26, 2019•12 citations

Exact scaling functions of the multichannel Kondo model

Serge Florens

We reinvestigate the large degeneracy solution of the multichannel Kondo problem, and show how in the universal regime the complicated integral equations simplifying the problem can be mapped onto a first-order differential equation. This leads to an explicit expression for the full zero-temperature scaling functions at---and away from---the intermediate non-Fermi-liquid fixed point, providing complete analytic information on the universal low---and intermediate---energy properties of the model. These results also apply to the widely used noncrossing approximation of the Anderson model, taken in the Kondo regime. An extension of this formalism for studying finite-temperature effects is also proposed and offers a simple approach to solve models of strongly correlated electrons with relevance to the physics of heavy fermion compounds.

arXiv:cond-mat/0310179•Physical Review B•Mar 16, 2004•4 citations

Ground state magnetic susceptibility of Kondo impurities

Bedoor Alkurtass

Impurity models describe localized impurities interacting with the spins of electrons. Strong correlations lead to non-trivial behaviours of the model. In this work we present 3 different impurity models: (1) the single-impurity Kondo model; (2) the two-impurity Kondo model; (3) the two-channel Kondo model. For each model we present a spin chain emulation and utilize the density matrix renormalization group (DMRG). We study the magnetization and zero-temperature susceptibility of the impurity when a local magnetic field is applied. We get the expected relations theoretically.

Kuwait Journal of Science•Jun 5, 2024•1 citation

Scaling properties of the Anderson model in the Kondo regime studied by σGσW formalism

Catalin D. Spataru

The symmetric Anderson model for a single impurity coupled to two leads is studied at strong interaction using the GW approximation within the $\ensuremath{\sigma}G\ensuremath{\sigma}W$ formalism. We find that the low-energy properties show universal scaling behavior in the asymptotic regime. While the GW scaling functions are similar in form to the scaling functions known from the numerically exact solution, they are characterized by a different parameter value indicating that GW fails to describe correctly spin correlations between the impurity and lead electrons. We also compare the GW and exact Kondo scales for a broad range of the interaction strength. In contrast to the exponential behavior shown by the exact solution, the GW Kondo scale depends algebraically on the interaction strength.

arXiv:1011.1275•Physical Review B•Nov 10, 2010•7 citations

Field-dependent dynamics of the Anderson impurity model

David E Logan, Nigel L Dickens

Single-particle dynamics of the Anderson impurity model in the presence of a magnetic field H are considered, using a recently developed local moment approach that encompasses all energy scales, field and interaction strengths. For strong coupling in particular, the Kondo scaling regime is recovered. Here the frequency (ω/ω K ) and field ( H /ω K ) dependence of the resultant universal scaling spectrum is obtained in large part analytically, and the field-induced destruction of the Kondo resonance investigated. The scaling spectrum is found to exhibit the slow logarithmic tails recently shown to dominate the zero-field scaling spectrum. At the opposite extreme of the Fermi level, it gives asymptotically exact agreement with results for statics known from the Bethe ansatz. Good agreement is also found with the frequency and field-dependence of recent numerical renormalization group calculations. Differential conductance experiments on quantum dots in the presence of a magnetic field are likewise considered, and appear to be well accounted for by the theory. Some new exact results for the problem are also established.

arXiv:cond-mat/0202379•Journal of Physics Condensed Matter•Oct 18, 2001•45 citations
The Kondo scale TK{T}_{\mathrm{K}}TK​ for impurity systems is expected to guarantee universal scaling of physical quantities. However, in practice, not every definition of TK{T}_{\mathrm{K}}TK​ necessarily supports this notion away from the strict scaling limit. Specifically, this paper addresses the role of finite bandwidth DDD in the strongly correlated Kondo regime. For this, various theoretical definitions of TK{T}_{\mathrm{K}}TK​ are analyzed based on the inverse magnetic impurity susceptibility at zero temperature. While conventional definitions in that respect quickly fail to ensure universal Kondo scaling for a large range of DDD, this paper proposes an altered definition of TKsc{T}_{\mathrm{K}}^{\mathrm{sc}}TKsc​ that allows universal scaling of dynamical or thermal quantities for a given fixed Hamiltonian. If the scaling is performed with respect to an external parameter that directly enters the Hamiltonian, such as magnetic field, the corresponding TKsc,B{T}_{\mathrm{K}}^{\mathrm{sc}},\mathrm{B}TKsc​,B for universal scaling differs, yet becomes equivalent to TKsc{T}_{\mathrm{K}}^{\mathrm{sc}}TKsc​ in the scaling limit. The only requirement for universal scaling in the full Kondo parameter regime with a residual error of less than 1% is a well-defined isolated Kondo feature with TK\ensuremath≲0.01D{T}_{\mathrm{K}}\ensuremath{\lesssim}0.01\phantom{\rule{0.16em}{0ex}}DTK​\ensuremath≲0.01D irrespective of specific other impurity parameter settings. By varying DDD over a wide range relative to the bare energies of the impurity, for example, this allows a smooth transition from the Anderson to the Kondo model.
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)].
In this work, using the finite-U slave boson mean-field approximation to solve the single-impurity Anderson model, the authors apply two different strategies to study the Kondo cloud, by analyzing quantities that are dependent on the distance to the magnetic impurity and then finding a universal distance scale ${\ensuremath{\xi}}_{K}$ through the collapse of the results into an universal function. The first method is based on the analysis of the local density of states of the conduction electrons (denoted as ${\ensuremath{\xi}}_{K}^{L})$, while the second relies on the analysis of spin correlations $({\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}})$. Our calculations show that there is exact quantitative agreement, in the way ${\ensuremath{\xi}}_{K}$ depends on $U/\mathrm{\ensuremath{\Gamma}}$, between ${\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}}$ and the results obtained through the heuristic expression ${\ensuremath{\xi}}_{K}\ensuremath{\propto}{v}_{F}/{T}_{K}$, while there is very close quantitative agreement between ${\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}}$ and ${\ensuremath{\xi}}_{K}^{L}$. The use of the slave boson technique to calculate the spin correlations, which eliminates finite size effects, allowed us to study very large Kondo clouds, something that is very difficult using other techniques, like the density matrix renormalization Group method, for example. In addition, the very smooth curves obtained for the spin correlations allowed us to qualitatively identify a region in the Kondo cloud, adjacent to the impurity, that had been connected to the free orbital fixed point in previous numerical renormalization group calculations [Phys. Rev. B 84, 115120 (2011)].
We reinvestigate the large degeneracy solution of the multichannel Kondo problem, and show how in the universal regime the complicated integral equations simplifying the problem can be mapped onto a first-order differential equation. This leads to an explicit expression for the full zero-temperature scaling functions at---and away from---the intermediate non-Fermi-liquid fixed point, providing complete analytic information on the universal low---and intermediate---energy properties of the model. These results also apply to the widely used noncrossing approximation of the Anderson model, taken in the Kondo regime. An extension of this formalism for studying finite-temperature effects is also proposed and offers a simple approach to solve models of strongly correlated electrons with relevance to the physics of heavy fermion compounds.
Impurity models describe localized impurities interacting with the spins of electrons. Strong correlations lead to non-trivial behaviours of the model. In this work we present 3 different impurity models: (1) the single-impurity Kondo model; (2) the two-impurity Kondo model; (3) the two-channel Kondo model. For each model we present a spin chain emulation and utilize the density matrix renormalization group (DMRG). We study the magnetization and zero-temperature susceptibility of the impurity when a local magnetic field is applied. We get the expected relations theoretically.
The symmetric Anderson model for a single impurity coupled to two leads is studied at strong interaction using the GW approximation within the $\ensuremath{\sigma}G\ensuremath{\sigma}W$ formalism. We find that the low-energy properties show universal scaling behavior in the asymptotic regime. While the GW scaling functions are similar in form to the scaling functions known from the numerically exact solution, they are characterized by a different parameter value indicating that GW fails to describe correctly spin correlations between the impurity and lead electrons. We also compare the GW and exact Kondo scales for a broad range of the interaction strength. In contrast to the exponential behavior shown by the exact solution, the GW Kondo scale depends algebraically on the interaction strength.
Single-particle dynamics of the Anderson impurity model in the presence of a magnetic field H are considered, using a recently developed local moment approach that encompasses all energy scales, field and interaction strengths. For strong coupling in particular, the Kondo scaling regime is recovered. Here the frequency (ω/ω K ) and field ( H /ω K ) dependence of the resultant universal scaling spectrum is obtained in large part analytically, and the field-induced destruction of the Kondo resonance investigated. The scaling spectrum is found to exhibit the slow logarithmic tails recently shown to dominate the zero-field scaling spectrum. At the opposite extreme of the Fermi level, it gives asymptotically exact agreement with results for statics known from the Bethe ansatz. Good agreement is also found with the frequency and field-dependence of recent numerical renormalization group calculations. Differential conductance experiments on quantum dots in the presence of a magnetic field are likewise considered, and appear to be well accounted for by the theory. Some new exact results for the problem are also established.