Single-particle excitations in disordered Weyl fluids

We theoretically study the single-particle Green function of a three-dimensional disordered Weyl semimetal using a combination of techniques. These include analytic $T$-matrix and renormalization group methods with complementary regimes of validity and an exact numerical approach based on the kernel polynomial technique. We show that at any nonzero disorder, Weyl excitations are not ballistic: They instead have a nonzero linewidth that for weak short-range disorder arises from nonperturbative resonant impurity scattering. Perturbative approaches find a quantum critical point between a semimetal and a metal at a finite disorder strength, but this transition is avoided due to nonperturbative effects. At moderate disorder strength and intermediate energies the avoided quantum critical point renormalizes the scaling of single-particle properties. In this regime we compute numerically the anomalous dimension of the fermion field and find $\eta =0.13\pm 0.04$, which agrees well with a renormalization group analysis ($\eta =0.125$). Our predictions can be directly tested by ARPES and STM measurements in samples dominated by neutral impurities.

Figure 1

(a) Schematic crossover diagram for disordered Weyl semimetals as a function of momentum ($k$) and disorder ($W)$, with the diffusive metal (DM) regime at low momentum, the semimetal (SM) regime at weak disorder and intermediate momentum, and the quantum critical (QC) fan at intermediate disorder and momentum. The cutoff momentum $\mathrm{\Lambda }/v$ bounds the low energy regime. The shape of the crossover boundaries follows from $E\left(\mathbf{k}\right)$ and Refs. [41] and [48]. (b)–(d) Electronic dispersion curves of the spectral function $A(\mathbf{k},\omega )=-G^{\prime \prime }(\mathbf{k},\omega )/\pi$ for momentum $\mathbf{k}=(k,0,0)$ with $\left|\mathbf{k}\right|<1$, for $L=40$ and KPM expansion order $N_C=1024$ at $W/t=0.2$ (b), 0.75 (c), and 0.9 (d). Each curve is shifted vertically by $(10kL/2\pi )$. The quasiparticle excitations are sharp in the SM regime (b) and marginally broadened in the QC regime (c).

Figure 2

(a) The convergence of $\gamma \left(0\right)$ with $L$ and $N_C$ for a gaussian disorder distribution. Data for different $N_C$ is with $L=80$ and data with $L\ge 120$ is for $N_C=2^{14}$. $\gamma \left(0\right)$ is well converged for $W\ge 0.625t, L=176$, and $N_C=16384$, whereas for smaller values of $W$ we are not able to clearly discern between the artificial KPM broadening or the intrinsic broadening due to disorder. (Inset) Zoomed in on the low $W$ region displaying the strong finite $L$ and $N_C$ effects. (b) The damping $\gamma \left(0\right)$ of the disordered Weyl point on a log scale computed from the $T$ matrix in the dilute limit for $H_W$ and the KPM for a gaussian disorder distribution for $H_L$ (converged in $L$ and $N_C$) versus $1/W^2$. Dashed lines are fits to the rare region form in Eq. (18). For the $T$ matrix, we consider $n_{\text{imp}}{b}^3=0.039$ to get the best comparison with the KPM results, and $t$ is replaced by the unit of energy $E_0=\hslash v/b$. The offset arises because although the leading $W$ dependence of $\gamma \left(0\right)$ is universal, the pre-exponential factor is sensitive to the difference between Eqs. (1) and (2). (Inset) $\gamma \left(0\right)$ versus $W$ on a linear scale.

Figure 3

(a) The quasiparticle residue at the Weyl node $Z(\mathbf{k}=0)$ as a function of the disorder strength computed within the three methods we use in this paper. The dashed line shows the qualitative expectation of the RG, $Z\left(0\right)\sim {(1-W/W_c)}^{\nu (z-1-\eta )}$, with $\nu (z-1-\eta )=0.375$ [58]. We find $Z\left(0\right)$ has a minimum near the AQCP ($W_c/t\approx 0.75$ for Gaussian) but does not go to zero. (Inset) $Z\left(0\right)$ up to large disorder strengths passing through the Anderson localization transition ($W_l\approx 3.75t$, Ref. [47]) at $E=0$. (b) The quasiparticle residue at $\mathbf{k}=0$ for Gaussian (G) and binary (B) disorder as a function of disorder for various expansion orders. We find $Z\left(0\right)$ is well converged and has a minimum near the avoided quantum critical point, with a value that is dictated by the strength of avoidance.

Figure 4

Momentum dependence of the quasiparticle residue (a) and the damping (b) from the $T$-matrix formalism. We set $n_{\text{imp}}{b}^3=0.34$ among all the data. The legend is shared across the two plots, the unit of energy is $E_0=\hslash v/b$, and open symbols denote the contributions from only the $j=1/2$ momentum sector and filled symbols denote the contributions from both $j=1/2$ and $j=3/2$ sectors. Dashed lines are a fit to $\gamma \left(\mathbf{k}\right)=\gamma \left(0\right)+{a\left|\mathbf{k}\right|}^2$. The quasiparticle residue becomes larger than one for some finite $k$. The $j=3/2$ contribution plays an important role for $k>0.3$.

Figure 5

Momentum dependence of the single-particle excitations for various Gaussian disorder strengths for $L=80, N_C=2^9$, and momentum $\mathbf{k}=(k,0,0)$. (a) The residue, (b) spectral line width, and (c) dispersion for various disorder strengths. The solid gray lines are the power law fits at the AQCP ($W_c/t\approx 0.75$), and the solid blue (red) line is a fit of $\gamma \left(\mathbf{k}\right) \left[E\right(\mathbf{k}\left)\right]$ to $\gamma \left(0\right)+{a\left|\mathbf{k}\right|}^2 \left[v\right|\mathbf{k}\left|\right]$ at low momentum.

Figure 6

Off-shell rare region contribution to the spectral function. (a) The average spectral function for $L=50, N_C=512, k=6k_0=6(2\pi /L)$, and various disorder strengths. At weak disorder the spectral function is peaked near $k$ but there is a clear peak near $\omega =0$. (b) Spectral function for a single sample that is either rare (R) or typical (T) with $L=25, W/t=0.5, N_C=1024$, and various momentum ($k_0=2\pi /L$). We find the rare sample has a clear peak near zero energy at all momentum we consider here that is absent in the typical sample.

Figure 7

Computing the anomalous dimension $\eta$ at the AQCP ($W_c/t=0.86\pm 0.01$) for binary disorder and a system size of $L=80$: (Inset) Fit of $1/G^{\prime }(k=0,\omega )$ to a power law in the quantum critical regime yields $\eta =0.13\pm 0.04$. Data collapse in the KPM expansion order $N_C$, showing excellent single parameter scaling for over two decades of $N_C\omega$.