Local susceptibility and Kondo scaling in the presence of finite bandwidth

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 $T_{\mathrm{K}}\lesssim 0.01D$ 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.

Figure 1

Scaling of the frequency of the dynamical spin susceptibility ${\chi }^{\mathrm{d}}\left(\omega \right)/{\chi }_0^{\mathrm{d}}$ by the conventional impurity susceptibility $T_{\mathrm{K}}^{\mathrm{d}}\equiv 1/\left(4{\chi }_0^{\mathrm{d}}\right)$ (left) vs. the scale-preserving definition of Kondo temperature $T_{\mathrm{K}}^{\mathrm{sc}}\equiv 1/\left(4{\chi }_0^{\mathrm{sc}}\right)$ (right): all the densely lying curves of the left panels collapse onto a single universal curve in the right panels, respectively. (a) and (b) analyze the SIAM. The inset to (a) demonstrates the dependence of $T_{\mathrm{K}}^{\mathrm{d}}/T_{\mathrm{K}}^{\mathrm{sc}}$ vs. the onsite interaction $U$, while keeping the ratios $U/\Gamma =15$ and ${\varepsilon }_d=-U/2$ fixed. The color bar at the bottom of the inset relates the color of the lines in the main panel to the specific values of $U$ ranging from $U\ll 1$ to $U\gg 1$ (with $D\equiv 1$ the bandwidth). The limit ${lim}_{U\to 0}[T_{\mathrm{K}}^{\mathrm{d}}/T_{\mathrm{K}}^{\mathrm{sc}}]$ has been fitted, resulting in the value of 1, with excellent accuracy (actual value indicated together with the horizontal dotted line). The inset to (b) shows the dependence of $T_{\mathrm{K}}^{\mathrm{sc}}$ vs. $U$, which stretches over several orders of magnitude. In complete analogy, (c) and (d) analyze the Kondo model. In particular, the fitted limit ${lim}_{J\to 0}{T}_{\mathrm{K}}^{\mathrm{d}}/T_{\mathrm{K}}^{\mathrm{sc}}\simeq 1$ in the inset of (c) is the same as for the SIAM [cf. (a)] within the numerical error of significantly less than 1% [for comparison, the same calculation yet with the cheaper and less accurate setting of $\Lambda =2$ and $E_{\mathrm{tr}}=12$ (not shown) already resulted in $T_{\mathrm{K}}^{\mathrm{d}}/T_{\mathrm{K}}^{\mathrm{sc}}\simeq 0.98$, while $\Lambda =4$ and $E_{\mathrm{tr}}=20$ (not shown) already agreed well with the above results. In this sense, the above results for $\Lambda =4$ and $E_{\mathrm{tr}}=40$ are considered fully converged].

Figure 2

Temperature dependent scaling of the static spin susceptibility $\chi \left(T\right)$ (top) and the linear conductance $g\left(T\right)$ (in units of $2e^2/h$; bottom) for the SIAM (left and middle), as well as for the Kondo model (right). The color of the lines in the main panels matches the colors of the symbols in the inset, hence this indicates the respective parameter setting. The upper panels compare various definitions of the static spin susceptibility (${\chi }^{\mathrm{d}}$, ${\chi }^{\mathrm{FS}}$, ${\chi }^{\mathrm{sc}}$ in faint, dashed and solid, respectively). In the upper main panels, for clarity, the actual value of the relevant parameters [$\{D,\Gamma ,U\}$ for (a) and (b) and $D$ for (c)] are indicated in units of $T_{\mathrm{K}}^{\mathrm{sc}}$ for the largest and smallest $T_{\mathrm{K}}$ only. Similar to Fig. 1, the insets to the upper panels analyze the relation between $T_{\mathrm{K}}^{\mathrm{d}}$ and $T_{\mathrm{K}}^{\mathrm{sc}}$ as function of the parameters. Their ratio is fitted towards $T_{\mathrm{K}}\to 0$, resulting in a comparable value of 1 to very good accuracy as indicated for all three cases (a)–(c). The actual exponential range of $T_{\mathrm{K}}^{\mathrm{sc}}$ is shown in the insets to the lower panels. The lower panels show the static linear conductance $g\left(T\right)$ vs. $T/T_{\mathrm{K}}^{\mathrm{d}}$ (nonuniversal; dashed faint lines, but color match with symbols of inset otherwise) and vs. $T/T_{\mathrm{K}}^{\mathrm{sc}}$ (solid lines), which show proper scaling behavior, in that all lines collapse onto a single universal curve. With $T_{1/2}$ the temperature where $g\left(T\right)$ passes through $1/2$, in units of $T_{\mathrm{K}}^{\mathrm{d}}$, this ranges from $T_{1/2}^{\mathrm{d}}\equiv T_{1/2}/T_{\mathrm{K}}^{\mathrm{d}}=1.25$ down to 1.03 [indicated by the vertical dotted lines with the range of $T_{1/2}^{\mathrm{d}}$ specified with each panel (gray text at center right in each panel)]. In units of $T_{\mathrm{K}}^{\mathrm{sc}}$, this range collapses to the fixed value of $T_{1/2}^{\mathrm{sc}}\equiv T_{1/2}/T_{\mathrm{K}}^{\mathrm{sc}}\simeq 1.03$ to within residual relative variations of clearly less than 1% for all three cases [panels d-f; indicated by vertical solid light lines with their range specified by $T_{1/2}^{\mathrm{sc}}$ (black text)]. Using $\Lambda =4$ and $E_{\mathrm{tr}}=40$ as indicated, the value of $T_{1/2}^{\mathrm{sc}}\simeq 1.03$ above is considered well converged [for comparison, for $\Lambda =2$ and $E_{\mathrm{tr}}=8$ a similar calculation (not shown) resulted in $T_{1/2}^{\mathrm{sc}}\simeq 0.99$, while $\Lambda =2$ and $E_{\mathrm{tr}}=12$ resulted in $T_{1/2}^{\mathrm{sc}}\simeq 1.01$; while good overall scaling can already be observed for $E_{\mathrm{tr}}\lesssim 10$, the minor variations for smaller $E_{\mathrm{tr}}$ can be mostly eliminated by normalizing $g\left(T\right)$ by the numerical value $g\left(0\right)\approx 1$, which was not included here].

Figure 3

Linear conductance vs. magnetic field at $T=0$ for the SIAM (left and center panel), as well as for the Kondo model (right panel). Again the insets indicate the respective parameter setting of the lines in the main panels. Analogous to the analysis in Figs. 2–2, here, the main panels show the static linear conductance $g\left(B\right)$ vs. $B/T_{\mathrm{K}}^{\mathrm{d}}$ (nonuniversal; dashed faint lines, but color match with symbols of inset otherwise) and vs. $B/T_{\mathrm{K}}^{\mathrm{sc}},\mathrm{B}$ (solid lines), which demonstrate universal scaling. With $B_{1/2}$ the magnetic field where $g\left(B\right)$ passes through $1/2$, in units of $T_{\mathrm{K}}^{\mathrm{d}}$, changes from $B_{1/2}^{\mathrm{d}}\equiv B_{1/2}/T_{\mathrm{K}}^{\mathrm{d}}=1.84$ down to 1.55 for given data [indicated by the vertical dotted lines with their individual range specified with each panel (gray text at center right in each panel)]. In units of $T_{\mathrm{K}}^{\mathrm{sc}},\mathrm{B}$, this range collapses to the value $B_{1/2}^{\mathrm{sc}}\equiv B_{1/2}/T_{\mathrm{K}}^{\mathrm{sc}}=1.55$ to within relative uncertainties of clearly less than 1% for all three cases [panels d–f; indicated by vertical solid light lines with the range $T_{1/2}^{\mathrm{sc}}$ specified by the black text]. Using $\Lambda =4$ and $E_{\mathrm{tr}}=40$ as indicated, the data is considered fully converged (regarding minor variations for significantly lower $E_{\mathrm{tr}}\lesssim 10$ and thus much faster calculations, see caption to Fig. 2).

Figure 4

Contributions to the impurity susceptibility ${\chi }^{\mathrm{d}}$ as in Eq. (B1) for the data in Fig. 2 in the main text [(a)–(c) have exactly the same parameter setting as Figs. 2–2]. The lower panels replicate the same data as in the upper panel, yet switch to a logarithmic scale also on the vertical axis. The thick light solid line corresponds to a plain power-law fit, suggesting that the correction $T{\chi }^{\delta }$ decays like $1/T^2$, hence becomes irrelevant in the limit $T\to 0$. The insets in the lower panels have been replicated from Fig. 2 to indicate the parameter setting.

Figure 5

Dependence of single-particle energy level spectra ${\tilde{\varepsilon }}_{k\sigma }\left({\varepsilon }_d\right)$ on local occupation $\langle {\widehat{n}}_{\mathrm{loc},\sigma }\left({\varepsilon }_d\right)\rangle$ and level index $k$ for the SIAM [NRG (green dot-dashed)] as well as the RLM [quadratic solution (blue) and NRG (red dashed)] using a long even Wilson chain of length $N$ as specified. The local occupation $\langle {\widehat{n}}_{\mathrm{loc},\sigma }\left({\varepsilon }_d\right)\rangle$ and thus the phase shift is changed by varying the position of the impurity energy level ${\varepsilon }_{d(,\sigma )}$. While this level is swept from $+\infty$ to $-\infty$, $\langle {\widehat{n}}_{\mathrm{loc},\sigma }\left({\varepsilon }_d\right)\rangle$ changes smoothly from 0 to 1. Combining all energies in units of the energy scale ${\omega }_n$ vs. $x\equiv k-\langle {\widehat{n}}_{\mathrm{loc},\sigma }\left({\varepsilon }_d\right)\rangle$, this results in a single continuous antisymmetric curve $\varepsilon \left(x\right)$ that is linear for small $\left|x\right|$, yet is quickly dominated by exponential behavior for larger $\left|x\right|\gtrsim 2$ (see inset and text). The discrete levels ${\tilde{\varepsilon }}_{k\sigma }\left({\varepsilon }_d\right)<0$ (i.e., within the range $x<0$) correspond to single-particle levels below the Fermi energy and are thus occupied in the ground state. The data for the blue curve were obtained by numerical diagonalization of the quadratic Hamiltonian (RLM), hence all single-particle energies are easily obtained. In particular, their energies are not restricted to the energy range below the truncation energy, as is the case for the NRG-method (dashed and dot-dashed lines).