Sub-Ohmic two-level system representation of the Kondo effect

It has been recently shown that the particle-hole symmetric Anderson impurity model can be mapped onto a $Z_2$ slave-spin theory without any need of additional constraints. Here, we prove by means of numerical renormalization group that the slave-spin behaves in this model like a two-level system coupled to a sub-Ohmic dissipative environment. It follows that the $Z_2$ symmetry gets spontaneously broken at zero temperature, which we find can be identified with the onset of Kondo coherence, being the Kondo temperature proportional to the square of the order parameter. Since the model is numerically solvable, the results are very enlightening on the role of quantum fluctuations beyond mean field in the context of slave-boson approaches to correlated electron models, an issue that has been attracting interest since the 80${}^{\prime }$s. Finally, our results suggest as a byproduct that the paramagnetic metal phase of the Hubbard model at half-filling, in infinite coordination lattices and at zero temperature, as described for instance by dynamical mean-field theory, corresponds to a slave-spin theory with a spontaneous breakdown of a local $Z_2$ gauge symmetry.

Figure 1

The average value $\langle {\sigma }^x\rangle$ as a function of the external symmetry breaking field $h$ for several values of $\Gamma$ and $U=0.1$.

Figure 2

Square of the symmetry breaking order parameter and of $T_K/\Gamma$ plot in logarithmic scale versus $U/\Gamma$. Note the linear behavior at large $U/\Gamma$.

Figure 3

Spectral function of the physical impurity electron operator ${\sigma }^x{d}_{\sigma }$ at $U=0.1$ and $\Gamma =0.01$. The dots are the values of the impurity spectral function as calculated directly by the Anderson impurity model (1). We just plot few of these points since the two spectral functions are just coincident.

Figure 4

Left panel: spectral function of the impurity operator $d_{\sigma }$. Right panel: spectral function of the Ising operator ${\sigma }^x$, on a large scale, top, and zoomed close to zero energy, bottom. The parameter that are used are $U=0.1$ and $\Gamma =0.01$ in units of the conduction electron half-bandwidth.

Figure 5

Convolution of the spectral functions $A_{\sigma }\left(\epsilon \right)$ and $A_d\left(\epsilon \right)$ for $U=0.1$ and $\Gamma =0.01$. Note the absence of Hubbard bands, which should be appear at approximately $\pm U/2=0.05$ but are hidden below the broad background peaked around the particle-hole band edge at energy approximately $\pm 2D=\pm 2$.