Critical quasi-particle theory and scaling near a Quantum Critical Point of Heavy Fermion metals

We recently developed a theory of the critical properties of a heavy fermion metal near an antiferromagnetic (AFM) quantum phase transition governed by three-dimensional spin fluctuations. The critical spin fluctuations induce critical behavior of the electron quasi-particles (qp) as seen in a diverging effective mass, leading, e.g., to a diverging specific heat coefficient. This in turn gives rise to a modification of the spin excitation spectrum [1]. We use that the concept of electron quasi-particles is well-defined as long as the qp width is less than their excitation energy, which is still the case in the so-called non-Fermi liquid regime. Impurity scattering [1,2] and/or higher order loop processes in the clean system [3] cause a redistribution of the critical scattering at the hot lines all over the Fermi surface, leading to a weakly momentum dependent critical self-energy. We derive a self-consistent equation for the qp effective mass which allows for two physical solutions: the usual weak coupling spin density wave solution and a strong coupling solution featuring a power law divergence of the effective mass as a function of energy scale. The resulting spin excitation spectrum obeys E/T scaling with dynamical exponent z$=$4 and correlation length exponent $\nu =$1/3, in excellent agreement with data for YbRh$_2$Si$_2$ [1,2]. Results of our theory applied to three-dimensional metals featuring quasi-two-dimensional spin fluctuations will be presented with the aim of explaining the observed properties of the AFM quantum critical point of CeCu$_{\mathrm{6-x}}$Au$_{\mathrm{x}}$, in particular the E/T scaling exhibited by inelastic neutron scattering data. In that case we find z$=$8/3 and $\nu =$3/7 [3]. Finally, the microscopic underpinning of our theory will be addressed, including the issues of qp renormalization, vertex corrections, interaction of bosonic fluctuations in the renormalization group sense, and higher loop corrections [3].\\[4pt] [1] P. W\"{o}lfle, and E. Abrahams, Phys. Rev. B \textbf{84}, 041101 (2011); Ann. Phys. (Berlin) \textbf{523}, 591 (2011); Phys. Rev. B \textbf{80}, 235112 (2009).\\[0pt] [2] E. Abrahams and P. W\"{o}lfle, PNAS \textbf{109}, 3228 (2012).\\[0pt] [3] E. Abrahams, J. Schmalian, and P. W\"{o}lfle, to be published.