Origin for the disorder-induced quantum criticality in NbFe$_2$*

Using KKR-CPA ab-initio electronic-structure method founded on an optimal site-centered basis-set, we investigate the key features giving rise to the quantum critical transitions observed in NbFe$_2$ upon doping with 1.75{\%} Nb [1]. These phase transitions involve lowest-energy excitations at/near the Fermi surface. In particular, it is suggested to arise due to an accidental unconventional band critical point (uBCP) with vanishing quasi-particle velocity [2]. Moving off-stoichiometry by increasing Nb, or reducing electrons (e/a), we find the Fermi level E$_f$ increases (rather than decreases based only on band-filling) and meet the uBCP to produce excitations driving the anomalies. We detail the concentration-dependence electronic dispersion, density of state, E$_f$ shift, and energies for NbFe$_2$, and why disorder increase the E$_f$ with electron loss. At stoichiometry all our results agree with those from full potential calculations, including itinerant magnetism.\\[4pt] [1] D. Moroni-Klementowicz et al., Phys. Rev B 79, 224410 (2009).\\[0pt] [2] Brian Neal, Erik R. Ylviskar, and Warren E. Pickett, private communication (2009).