2.3 Results

One quantity that can be calculated analytically is the low-energy limit of the spin structure factor $S(\omega)\stackrel{\rm def}{=}\chi''_{zz}(\omega)/\omega$,

$$S(0)=\lim_{\omega\rightarrow 0} \frac{\chi''_{zz}(\omega)}{\omega}\ .$$ (2.20)

For a vanishing local magnetic field $S(0)$ is just the spin relaxation rate accessible in e.g. spin resonance experiments. With the result for $\chi _{zz}''(\omega )$ from (2.19) we obtain

$$S(0)=(g\mu_B)^2\:\frac{1}{\pi}\, \frac{\Delta_{\rm eff}^2(0)}{\left((g\mu_B h)^2+\Delta_{\rm eff}^2(0)\right)^2}\;\; ,$$ (2.21)

which leads to the curve shown in Fig. 2.2. Eq. (2.21) is of particular importance because it explicitly demonstrates universality, $T_{\rm K}^2S(\omega)=f(g\mu_B h/T_{\rm K})$, and allows to directly fit e.g. experimental data from ESR or NMR experiments and extract the Kondo

Figure 2.2: Universal curve for $S(0)$ as function of the local magnetic field $h$.

temperature. Note furthermore that the result (2.21) is not only valid in the Kondo limit, but also holds at the Toulouse point of the anisotropic Kondo model and everywhere in between. Since it does not depend on the details of $\Delta_{\rm eff}(\epsilon)$ it will also be true for general band structures $\epsilon_{\vec k}$ in (2.1) and thus is eventually the result for $S(0)$ in DMFT calculations.

The full frequency dependent $\chi _{zz}''(\omega )$ has to be calculated numerically using the form of the effective hybridization function in Fig. 2.1. The results for three values of the external field, $h=0$, $g\mu _B h=T_{\rm K}$ and $g\mu _B h=5T_{\rm K}$

Figure 2.3: $\chi _{zz}''(\omega )$ for three characteristic local magnetic fields $h=0$, $g\mu _B h=T_{\rm K}$ and $g\mu _B h=5T_{\rm K}$.

are displayed in Fig. 2.3. These correlation functions provide a good example for the usefulness of our mapping to the effective RLM since one can directly interpret the structures and their frequency and field dependencies in terms of analytical formulas derived for the RLM. E.g. the high-frequency behavior of $\chi _{zz}''(\omega )$ follows directly from equ. (2.19) and the behavior of the effective hybridization function $\Delta_{\rm eff}(\epsilon)$ at large energies (which is linear with logarithmic corrections, see Fig. 2.1): $\chi _{zz}''(\omega )$ decays like $1/\omega$ with logarithmic corrections, in agreement with (expensive) numerical results (23).

For the dependence of the dynamical susceptibility on the local magnetic field one makes use of the fact that the local magnetic field corresponds to the on-site energy in the effective RML. Therefore it is obvious that the observed shift of the resonance peak in $\chi _{zz}''(\omega )$ is due to the shifted center of the resonant level. Furthermore, the depletion of the maximum value is related to the decreasing occupation of the resonant level, which corresponds directly to the increasing local magnetization in the SIKM. At the same time, one observes a decrease of the total spectral weight in $\chi _{zz}''(\omega )$, which can be accounted for by a transfer to a finite expectation value of $\langle S_z\rangle$ in the SIKM. There is, however, also a non-trivial effect, namely the increasing broadening of the resonance peak with increasing magnetic field. For a RLM with a constant $\Delta_{\rm eff}(\epsilon)$ such a behavior does not occur; it is entirely related to the fact that with increasing magnetic field the system starts to notice the energy dependence of the effective hybridization.

The quantity not yet fixed in our calculation is $T_{\rm K}$, or more precisely the proportionality constant in $T_{\rm K}\propto \Delta_{\rm eff}(0)$. This can most conveniently be done by using Wilson's definition of the Kondo temperature(3)

$$\chi_0(h=0)=(g\mu_B)^2\,\frac{w}{4T_{\rm K}}\;\;,$$ (2.22)

where $\chi_0$ is the static magnetic susceptibility and $w=0.413$ the Wilson number. $\chi_0$ can be obtained from the imaginary part of the dynamic susceptibility (2.19) via

$$\chi_0=\frac{2}{\pi}\int\limits_0^\infty d\omega\,\frac{\chi''_{zz}(\omega)}{\omega}$$ (2.23)

and must in general be evaluated numerically. At the Toulouse point one can, however, give an analytic answer since $\Delta_{\rm eff}(\omega)={\rm const.}$ and thus

$$\chi_0(h)=(g\mu_B)^2\: \frac{1}{\pi}\,\frac{\Delta_{\rm eff}(0)}{h^2+\Delta_{\rm eff}(0)^2}\;\;.$$ (2.24)

Therefore at the Toulouse point the Korringa-Shiba relation(24)

$$R_{\rm S}=\frac{(g\mu_{\rm B})^2}{2\pi\chi_0^2}\lim_{\omega\to0} \frac{\chi_{zz}''(\omega)}{\omega}$$ (2.25)

is independent of the local magnetic field

$$R_{\rm S}= \frac{(g\mu_B)^2\,S(0)}{2\pi\chi_0^2(h)}=\frac{1}{2} \ .$$ (2.26)

In the following we will discuss $\chi _0(h)$ and the Korringa-Shiba relation for the Kondo limit $\rho_0 J\rightarrow 0$. The quantity $\chi _0(h)$ is particularly convenient for a comparison with NRG results.

Figure 2.4: The magnetic susceptibility $\chi _0(h)$ from equ. (2.23) (circles) and the same quantity obtained from an NRG calculation.

In Fig. 3.3 the circles represent the values of $\chi _0(h)$ calculated via (2.23) with the effective hybridization function from Fig. 2.1, and the full line represents the result of an NRG calculation. We observe excellent agreement for all values of the local magnetic field: notice that the curves agree without fit parameters. This example clearly demonstrates that the nontrivial form of the effective hybridization in Fig. 2.1 encodes the many-particle physics of the SIKM in a trivial noninteracting effective model.

The result in Fig. 3.3 can readily be combined with relations (2.17) and (2.21) to obtain the Wilson ratio(5)

$$R_{\rm W}=\frac{4\pi^2k_{\rm B}^2}{3(g\mu_{\rm B})^2} \frac{\chi_{\rm imp}(h)}{\gamma_{\rm imp}(h)} \ .$$ (2.27)

Our results for the Wilson ratio and the Korringa-Shiba relation obtained within the effective RLM are collected in Fig. 2.5. For the Wilson ratio we would actually have to calculate the quantity $\chi_{\rm imp}$ and not $\chi_0$.(5) However, for the case of small $\rho_0 J$ considered here, both quantities are equivalent.(25) One observes that both $R_{\rm W}$ and $R_{\rm S}$ are independent of the local magnetic field up to approximately $g\mu_B h\approx T_{\rm K}$, and then start to

Figure 2.5: Results for the Shiba ratio $R_{\rm S}$ (full line) and the Wilson ratio $R_{\rm W}$ (dashed line) as a function of a local magnetic field. The correct limiting values at $h\to 0$ are missed by approximately 5%.

decrease (Shiba ratio), respectively increase (Wilson ratio). The exact Bethe ansatz solution (4) gives $R_{\rm W}(h)=2$ independent of the magnetic field strength (see also Ref. (12)), and local Fermi liquid theory yields $R_{\rm S}=1$ for $h\rightarrow 0$.(3) Our limiting values as $h\to 0$ miss these exact results by approximately 5%. Notice that the term $(1-\Lambda'(0))$ in (2.17) is very important to obtain this correct value for $R_{\rm W}(h=0)$. Remarkably, our simple noninteracting effective model therefore correctly describes the Wilson ratio in the Kondo limit (for not too large magnetic fields), which is a hallmark of strong-coupling Kondo physics.

Let us finally analyze the accuracy of our effective model. Since Fig. 3.3 demonstrates that integral quantities like $\chi _0(h)$ are obtained with very good accuracy for all magnetic fields, one can infer from Fig. 2.5 that quantities depending on low-energy details in frequency space like $\gamma_{\rm imp}$ and $S(0)$ are more susceptible to our approximations for increasing magnetic fields. This suggests that for such low-energy quantities our effective model can be employed with very good accuracy (5% error) for magnetic fields below $T_{\rm K}$, and with good accuracy (20% error) still up to approximately  $5 T_{\rm K}$.