We study the effects of short-range random potential disorder on three-dimensional Dirac and Weyl semi-metals . We focus on the proposed quantum critical point (QCP) separating a semi-metal and diffusive metal phase driven by disorder. We will briefly review the existing evidence of such a QCP  and the effects of large disorder . We will then explore the non-perturbative effects of rare regions using Lanczos and the kernel polynomial method, from which we establish the existence of two distinct types of excitations in the weak disorder regime. The first are perturbatively renormalized dispersive Dirac states and the second are weakly dispersive quasi-localized “rare” eigenstates. We establish that these rare eigenstates contribute an exponentially small but non-zero density of states at zero energy, thus converting the semi-metal to diffusive metal transition into an avoided quantum critical point. Nonetheless, we show a quantum critical scaling regime still exists albeit only at finite temperature or energy. Lastly, we establish how to find the location of the avoided QCP and tune how strongly it is avoided.
 J. H. Pixley, David A. Huse, S. Das Sarma, arXiv:1602.02742 (2016).
 J. H. Pixley, Pallab Goswami, S. Das Sarma, Phys. Rev. B 93, 085103 (2016).
 J. H. Pixley, Pallab Goswami, S. Das Sarma, Phys. Rev. Lett. 115, 076601 (2015).