Singular dynamics of underscreened magnetic impurity models

We give a comprehensive analysis of the singular dynamics and of the low-energy fixed point of one-channel impurity $s\text{−}d$ models with ferromagnetic and underscreened antiferromagnetic couplings. We use the numerical renormalization group (NRG) to perform calculations at $T=0$. The spectral densities of the one-electron Green’s functions and t-matrices are found to have very sharp cusps at the Fermi level $(\omega =0)$, but do not diverge. The approach of the Fermi level is governed by terms proportional to $1∕{\ln }^2(\omega ∕T_0)$ as $\omega \to 0$. The scaled NRG energy levels show a slow convergence as $1∕(N+C)$ to their fixed point values, where $N$ is the iteration number and $C$ is a constant dependent on the coupling $J$ from which the low energy scale $T_0$ can be deduced, with a renormalized coupling $\tilde{J}\left(\omega \right)=1∕\ln (\omega ∕T_0)$. We calculate also the dynamical spin susceptibility, and the elastic and inelastic scattering cross sections as a function of $\omega$. The inelastic scattering goes to zero as $\omega \to 0$, as expected for a Fermi liquid, but anomalously slowly compared to the fully screened case. We obtain the asymptotic forms for the phase shifts for elastic scattering of the quasiparticles in the high-spin and low-spin channels. The asymptotic forms found for the one-electron Green’s functions and $t$-matrices do not simply correspond to a perturbation expansion in powers of $\tilde{J}\left(\omega \right)$.

Figure 1

(Color online) This figure illustrates the notation used in this paper for the $s\text{−}d$ model (top) and the Anderson model (bottom). The site coupling to the impurity is denoted by $c$, and in the case of the Anderson model the Coulomb repulsion $U$ acts at the site $d$ only.

Figure 2

The flow of the energy difference $N\times (E_{p,1}(S+1∕2)-E_{p,1}(S-1∕2))$ of high-spin and low-spin one-particle excitations for different values of $S$ and ferromagnetic coupling $J=-0.1$. The symbols ($\times$ for $S=1$, 엯 for $S=3∕2$, and ◻ for $S=2$) show the flows rescaled by $1∕(S+1∕2)$.

Figure 3

The flow of the energy difference $N\times [E_{p,1}\left(S\right)-E_{p,1}(S-1)]$ of high-spin and low-spin one-particle excitations for different values of $S$ and antiferromagnetic coupling $J=+0.2$. The symbols ($\times$ for $S=3∕2$, 엯 for $S=2$, and ◻ for $S=5∕2$) show the flows rescaled by $1∕S$.

Figure 4

The flow of the energy difference $N\times (E_{p,1}\left(S\right)-E_{p,1}(S-1))$ of high-spin and low-spin one-particle excitations for $S=3∕2$ different values of the antiferromagnetic coupling $J$.

Figure 5

The effective ferromagnetic coupling $J_{\mathrm{eff}}$ as a function of the bare antiferromagnetic $J$ for $S=1(\times )$, $S=3∕2$ (엯), and $S=2$ (◻).

Figure 6

(Color online) Spectral densities ${\rho }_d$ of the Anderson model and ${\rho }_t$ of the Kondo model (top) as well as ${\rho }_c$ (bottom) for both models. A resonance in ${\rho }_d$ and ${\rho }_t$ corresponds to an antiresonance in ${\rho }_c$.

Figure 7

(Color online) Spectra ${\rho }_t$ (top) and ${\rho }_c$ (bottom) for $S=1∕2$ and various values of $J$ for ferromagnetic coupling. The antiresonance in ${\rho }_c$ corresponds to the resonance in ${\rho }_t$.

Figure 8

Low-energy behavior of the rescaled spectral function ${\rho }_t$ as a function of the reduced energy $\omega ∕T_0$ for FM coupling to a spin $S=1∕2$. After rescaling, the spectra corresponding to all coupling strengths fall on one curve that is well fitted by ${\rho }_t\left(\omega \right)=b∕{\ln }^2(T_0∕\omega )$. The range of fitting is always the interval [${10}^{-8}$, 0.5] independent of $J$ which takes the values $-0.1(\times )$, $-0.15$ (엯) to $-0.45(<)$.

Figure 9

(Color online) Energy scale $T_0$ as a function of the ferromagnetic coupling strength $J$ for different $S$. Estimates of $T_0$ from the fit of ${\rho }_t\left(\omega \right)=b∕{\ln }^2(\omega ∕T_0)$ over the interval [${10}^{-8}$,${10}^{-2}$] are shown as (엯) for $S=1∕2$ and as $(\times )$ for $S=1$. Estimates of $T_0$ from the flows through Eq. (13) are denoted by ◇ for $S=1∕2$ and $\ast$ for $S=1$. The full line joining the points is given by $T_0\left(J\right)=D\sqrt{2{\rho }_{0}J}\exp (-1∕\left(2{\rho }_{0}J\right))$ with $D=2.55$ and ${\rho }_0=0.47$. In the inset the parameter $b$ is plotted as a function of $J$. Here the data for $S=1$ are roughly $3\times$ those for $S=1∕2$.

Figure 10

(Color online) Spectral densities ${\rho }_t$ (top) and ${\rho }_c$ (bottom) of the underscreened $s\text{−}d$ model with $S=1$ resonance and antiresonance; former normalized by $J^2{\pi }^2{\rho }_0$.

Figure 11

(Color online) Spectral densities ${\rho }_t$ of the antiferromagnetically coupled $s\text{−}d$ model with $S=1∕2$ (dotted), $S=1$ (dashed), $S=3∕2$, $S=2$ (dotted-dashed), and $S=5∕2$. The inset shows the resonance on a lower energy scale, where the spectrum of the fully screened model (dotted) is almost flat. The value at $\omega =0$ is approximately the same for all values of $S$.

Figure 12

Rescaled spectral densities ${\rho }_t$ of the AFM $s\text{−}d$ model with $S=1$ (top) for antiferromagnetic couplings $J=0.1(\times )$, $J=0.15(엯),\dots$, $J=0.45$. One sees that after rescaling the spectrum $\rho$ and the frequency $\omega$, all points fall on the same line. The fits to the function ${\rho }_t\left(\omega \right)=a-b∕{\ln }^2(\omega ∕T_0)$ have been done on the interval [${10}^{-8}$,${\omega }_0$], where ${\omega }_0$ is taken such that ${\rho }_t\left({\omega }_0\right)∕{\rho }_t\left(0\right)\approx 0.9$.

Figure 13

(Color online) Energy scale $T_0$ as a function of the antiferromagnetic coupling $J$ for $S=1$ (diamonds) and $S=3∕2$ (squares), and $S=2$ (pentagrams). The full lines show fits to $T_0\left(J\right)=\overline{D}{\mid bJ\mid }^n\exp (b∕J)$ with different values of $\overline{D}$, $b$, and $n$. The dashed lines with the somewhat smaller symbols indicate $T_0$ as extracted from the flow diagrams. The dotted-dashed line shows $T_K=D\sqrt{2{\rho }_{0}J}\exp (-1∕\left(2{\rho }_{0}J\right))$ with $D=1$ and ${\rho }_0=0.5$.

Figure 14

Elastic (dashed), inelastic (dotted-dashed), and total (full) cross section for the fully screened model (left) and underscreened model ($S=1$, right) for $J=0.2$.

Figure 15

Elastic (dashed), inelastic (dotted-dashed), and total (full) cross section for the ferromagnetic model $J=-0.2$. The left-hand (right-hand) plot is for $S=1∕2(S=1)$. The dotted-dashed and the full lines virtually lie on top of each other indicating that the scattering is predominantly inelastic.

Figure 16

(Color online) Low-energy spectra of the impurity spin susceptibility multiplied by $\omega$ for ferromagnetic coupling to $S=1∕2$ (dark symbols) and $S=1$ (light symbols). The energies are rescaled by $T_{\chi }$ which is fitted from $-\omega \mathrm{Im}\chi \left(\omega \right)∕\pi =a∕{\ln }^2(\omega ∕T_{\chi })$ with a logarithmic least squares fit [Eq. (43)]. The ranges of the fits are $[{10}^{-8}:2\times {10}^{-2}]$ for $S=1∕2$ and $[{10}^{-8}:2\times {10}^{-3}]$ for $S=1$. $J$ ranges from $J=-0.05(\ast ),-0.1(\times ),\dots$ to $J=-0.45(>)$.

Figure 17

(Color online) Low-energy spectra of the impurity spin susceptibility multiplied by $\omega$ for antiferromagnetic underscreened models $S=1$ (dark symbols) and $S=3∕2$ (light red symbols) and $S=2$ (light gray symbols). The energies are rescaled by $T_{\chi }$ which is fitted from $-\omega \mathrm{Im}\chi \left(\omega \right)∕\pi =a∕{\ln }^2(\omega ∕T_{\chi })$ with a logarithmic least squares fit Eq. (43). The ranges of the fits vary significantly as a function of $J$ to fit the low-energy behavior. $J$ ranges from $J=0.05(\ast ),0.1(\times ),\dots$ to $J=0.45(>)$.

Figure 18

(Color online) Energy scale $T_{\chi }$ as a function of $J$ for ferromagnetic coupling of a spin $S=1∕2$ (엯) and $S=1(\times )$. The broken line is a fit to Eq. (42) with $a=4.7$ and $b=0.46$. The full line is a better fit to Eq. (42) with ${\left(bJ\right)}^{-1∕5}$ instead of the square root and $D=2.55$ and $b=0.50$.

Figure 19

Energy scale $T_{\chi }$ for underscreened $s\text{−}d$ models with $S=1$ (◇), $S=3∕2$ (◻), and $S=2(\star )$. The values of $T_{\chi }$ are obtained from low-energy fits to Eq. (43). Full lines are fits to Eq. (42) with the square root replaced by the exponent $n=2.5$ for $S=1$, $n=5.0$ for $S=3∕2$, and $n=7.5$ for $S=2$.