Disorder-Driven Multifractality Transition in Weyl Nodal Loops

The effect of short-range disorder in nodal line semimetals is studied by numerically exact means. For arbitrary small disorder, a novel semimetallic phase is unveiled for which the momentum-space amplitude of the ground-state wave function is concentrated around the nodal line and follows a multifractal distribution. At a critical disorder strength, a semimetal to compressible metal transition occurs, coinciding with a multi- to single-fractality transition. The universality class of this critical point is characterized by the correlation length and dynamical exponents. At considerably higher disorder, an Anderson metal-insulator transition takes place. Our results show that the nature of the semimetallic phase in nonclean samples is fundamentally different from a clean nodal semimetal.

Figure 1

(a) The Fermi surface of the WNL is a continuous line in the plane $k_z=0$. The ground-state wave function has a width $\mathrm{\Gamma }(W,L)$ around the loop, for fixed linear system size ($L$) and disorder strength ($W$). (b) Schematic phase diagram as a function of $W$. For small $W$, the DOS at $E=0$ vanishes, ${\rho }_0=0$, and $\mathrm{\Gamma }$ vanishes with $L^{-1}$—the system is in a multifractal semimetallic phase. For $W$ larger than a critical disorder strength, ${\rho }_0\ne 0$ and $\mathrm{\Gamma }$ is $L$ independent—the system enters a single-fractal metallic phase. For larger $W$ the system becomes an Anderson insulator.

Figure 2

(a) DOS for different $W$ and varying $N_m$. For energies above the dashed vertical line, differences between $N_m=1000$ and $N_m=2000$ are below 1%. (b) ${\rho }^{\prime }(E)$, converged with $N_m$, as a function of $W$, for different $E$. Inset: Extrapolation to $E\to 0$ of ${\rho }^{\prime }(E)$ crossing points (Filled circle) and ${\rho }^{\prime }(E)$ maximum (Filled square).

Figure 3

(a) Exponent ${\tau }_k(q)$ vs $q$ [see Eq. (2)] for different $W$ and $L$. (b) Plot of the ground-state PD in momentum space $|{\mathrm{\Psi }}_{\mathit{k}}{|}^2$ for a configuration with $W=3.5$. (c) $|{\mathrm{\Psi }}_{\mathit{k}}{|}^2$ width around the loop $\mathrm{\Gamma }(W,L)$ vs $W$ for varying $L$. (d) $|{\mathrm{\Psi }}_{\mathit{k}}{|}^2$ as a function of $k^{\prime }$ for $W=1.75$ and $k_z=0$, where $k^{\prime }$ is measured relative to the loop. The system is in the MF regime and the PD curves collapse by rescaling $|{\mathrm{\Psi }}_{\mathit{k}}{|}^2\to |{\mathrm{\Psi }}_{\mathit{k}}{|}^{2}L$ and $k^{\prime }\to k^{\prime }L$. $L_0=16$ is the smallest used linear system size.

Figure 4

(a) The quantity ${\varphi }_{\beta }(W,L)$ vs $W$ (see text), for $\beta =5$ and varying $L$. The red dashed line intersects the maxima of each $L$ curve. (b) $W_{\max }(L)$ vs $L$ for different $\beta$, with $L\in [20,26]$. The horizontal dashed, black line separates regimes where $W_{\max }(L)$ either increases or decreases with $L$.

Figure 5

(a) Collapse of scaling variable $\mathrm{\Gamma }$ by shifts of $\log {\xi }_s$, for $L\in [20,26]$. (b) Collapse of scaling variable ${\mathrm{\Gamma }}^{-1}/L$ by shifts of $\log {\xi }_m$, for $L\in [12,26]$. (c) Collapse of the $\rho (E)$ curves according to Eq. (6), obtained for different $W$ and $E\in [0.025,0.175]$, with parameters $\nu =1$, $z=1.9$ and $W_c=2.6$. The curves collapse in two different branches that connect at $W=W_c$, corresponding to the SM and M phases.