Emergent Anisotropic Non-Fermi Liquid at a Topological Phase Transition in Three Dimensions

Understanding correlation effects in topological phases and their transitions is a cutting-edge area of research in recent condensed matter physics. We study topological quantum phase transitions (TQPTs) between double-Weyl semimetals (DWSMs) and insulators, and argue that a novel class of quantum criticality appears at the TQPT characterized by emergent anisotropic non-Fermi-liquid behaviors, in which the interplay between the Coulomb interaction and electronic critical modes induces not only anisotropic renormalization of the Coulomb interaction but also strongly correlated electronic excitation in three spatial dimensions. Using the standard renormalization group methods, large $N_f$ theory, and the $\epsilon =4-d$ method with a fermion flavor number $N_f$ and spatial dimension $d$, we obtain the anomalous dimensions of electrons (${\eta }_f=0.366/N_f$) in large $N_f$ theory and the associated anisotropic scaling relations of various physical observables. Our results may be observed in candidate materials for DWSMs such as ${\mathrm{HgCr}}_2{\mathrm{Se}}_4$ or ${\mathrm{SrSi}}_2$ when the system undergoes a TQPT.

Figure 1

Phase diagram for the TQPT between the DWSM and insulator phases with the tuning parameter $m$. The insets show the energy dispersions for the (a) DWSM, (b) insulator, and (c) TQPT.

Figure 2

Feynman diagrams at one-loop order for the (a) fermion self-energy, (b) boson self-energy, and (c) vertex correction. A straight line with an arrowhead and a wavy line represent the fermion and boson propagators, respectively.

Figure 3

RG flows of $\alpha$ and $\gamma$ for (a) $N_f=1$ and (b) $N_f=10$ at $\epsilon =1$. The red dots represent the fixed points $({\alpha }^{*},{\gamma }^{*})$. For $N_f=1$, $({\alpha }^{*},{\gamma }^{*})=(0.671,0.748)$ and for $N_f=10$, $({\alpha }^{*},{\gamma }^{*})=(0.096,0.966)$ obtained from Eq. (18).