**Ferroelectrics**, *11.2.2019*.

It is with much pleasure that I...

We, however, succeeded in finding a way to guarantee at least a qualitative consistency between the behavior of the spectral function and thermodynamic response functions. The critical behavior in the response function then correctly induces the respective symmetry breaking due the emergence of the order parameter in the spectral function [1].

We applied the general scheme on a single-impurity Andesrson model of the formation of magnetic moment in the limit of strong electron interaction. A quasiparticle peak develops at the Fermi energy with increasing electron coupling the half-width of which exponetially decreases and is inversely proportional to the lifetime of the magnetic moment, the Kondo effect.

This behavior can be detected also from the magnetic susceptibility that grows exponentially with interaction strength. We showed that the approximation we developed the exponential decrease in the central peak of the spectral function and the exponential increase of the magnetic susceptibility is qualitatively correctly described in accord with the exact Bethe-ansatz solution in both the spectral function and magnetic susceptibility [1,2].

The proposed scheme of the construction of nonperturbative approximations for strongly correlated electrons allows a direct application on extended systems with quantum phase transitions to magnetic or superconducting states. Such extensively consistent description of quantum phase transitions in strongly correlated electron systems has been missing.

Figure 1 Dynamical formation of magnetic moment due to strong electron correlation U on an impurities in metals expressed as an exponentially narrow quasiparticle peak at the Fermi energy. Half-width of the quasiparticle peak isn inversely proportional to the lifetime of the formed magnetic moment. Inset shows a magnified central peak as a function of the interaction strength. |

Figure 2 Local magnetic susceptibilities from the spectral self-energy chi and from the thermodynamic one chi ^{T} are in the consistent description inversely proportional to the half-width of the central peak in the spectral function and in accord with the exact solution of the Bethe ansatz, chi^{ex} grows exponentially with the interaction strength. The standard (inconsistent) approaches produce either an unphysical divergence, chi^{HF}, or miss the correct strong-coupling asymptotics. |

[2] V. Janiš, V. Pokorný and A. Kauch, Phys. Rev. B 95, 165113 (2017) 1-12.