Disorder in tilted Weyl semimetals from a renormalization group perspective

Isolated Weyl cones in a disordered environment do not show the phenomenon of Anderson localization due to the absence of backscattering processes. However, besides the conventional three dimensional diffusive metal, an additional semimetallic phase can form. In this paper we study the effect of tilt on the physics of disorder in isolated Weyl cones. Our main finding is that, while the semimetallic phase remains a stable phase, tilt decreases the size of the semimetallic region. Conversely, disorder increases the effective tilt of the quasiparticle excitations.

Figure 1

Modified Weyl cone dispersion $E^{t,\lambda }$ for (a) the untitled case, (b) ordinary tilt, and (c) modified tilting.

Figure 2

Diagrams contributing to the self-energy correction ${\mathrm{\Sigma }}_{\chi }^{t,\lambda }$ to the modified naked Green function $G_{0;\chi }^{t,\lambda }$.

Figure 3

Vertex corrections to disorder couplings $\mathrm{\Gamma },\overline{\mathrm{\Gamma }}$.

Figure 4

Flow for disorder couplings $\gamma ,\overline{\gamma }$ and SCBA elliptic critical disorder line (see the Appendix) in (a) the untilted case, $t=0$, and (b) the tilted case, $t=0.25$. Red dots: disorder fixed points. Blue dots: phase transition points. Purple dot: untilted phase transition fixed point ${(\gamma ,\overline{\gamma })}_{\left[{\varphi }^{t=0}\right]}=(1/\sqrt{2},0)$.

Figure 5

Dependence of the phase transition disorder strength ${\mathrm{\Gamma }}_{\left[\varphi \right]}$ on the ordinary tilt $t$. The SCBA (see the Appendix) consistently overestimates the disorder required for the WSM-DM transition, whereas the boundary of the approximate weak-disorder parallelogram leads to a lower estimate.

Figure 6

Stream plot of anomalous tilting versus disorder for the appropriate terminal relations of $\gamma$ and $\overline{\gamma }$. The flow predicts that $\lambda$ will find a terminal value $0\le {\lambda }_{*}\le 1$. (a) $t=\overline{\gamma }=0$ and (b) $t=0.25,\gamma =\overline{\gamma }$.

Figure 7

Terminal observable tilt at the weak disorder fixed point ${\left(t_{\text{obs}}\right)}_{[*,\text{SM}]}$ reached as a function of disorder strength $\mathrm{\Gamma }$ for selected value of original tilt $t$.

Figure 8

(a)–(c) Comparison of $\tilde{\alpha }$ and $\tilde{\alpha }{g}^{2}f\left(\tilde{\alpha }\right)$ for the indicated composite couplings $g$.