Ballistic transport in disordered Dirac and Weyl semimetals

We study the dynamics of Dirac and Weyl electrons in disordered point-node semimetals. The ballistic feature of the transport is demonstrated by simulating the wavepacket dynamics on lattice models. We show that the ballistic transport survives under a considerable strength of disorder up to the semimetal-metal transition point, which indicates the robustness of point-node semimetals against disorder. We also visualize the robustness of the nodal points and linear dispersion under broken translational symmetry. The speed of wavepackets slows down with increasing disorder strength, and vanishes toward the critical strength of disorder, hence becomes the order parameter. The obtained critical behavior of the speed of wavepackets is consistent with that predicted by the scaling conjecture.

We study the dynamics of Dirac and Weyl electrons in disordered point-node semimetals. The ballistic feature of the transport is demonstrated by simulating the wavepacket dynamics on lattice models. We show that the ballistic transport survives under a considerable strength of disorder up to the semimetal-metal transition point, which indicates the robustness of point-node semimetals against disorder. We also visualize the robustness of the nodal points and linear dispersion under broken translational symmetry. The speed of wavepackets slows down with increasing disorder strength, and vanishes toward the critical strength of disorder, hence becomes the order parameter. The obtained critical behavior of the speed of wavepackets is consistent with that predicted by the scaling conjecture.
In this Letter, we visualize a characteristic feature of disordered PNSMs: ballistic transport under disorder. The simulation of wavepacket dynamics clearly shows the ballistic feature, and the calculated spectral function shows that the "linear dispersion" in PNSM is robust against disorder. It is also shown that the speed of the ballistic transport obeys the scaling theory [31][32][33][34][35] near the PNSM-metal transition point.
Model for Dirac semimetals. For the numerical simulations, we consider the following two models of simple cubic lattice describing disordered PNSMs. The DSM phase, where the doubly degenerated 3D Dirac cones arise at high-symmetry points in the Brillouin zone, can be realized on the topological-trivial phase boundary of topological insulators. As a simple model for topological insulators, we employ the Wilson-Dirac type tightbinding Hamiltonian [33,[36][37][38][39][40][41][42], where r is the position of lattice sites and e µ (µ = x, y, z) is the lattice vector in µ direction. u is the nearest neigh-bor transfer with spin-orbit interactions, m 0 is referred to as "mass", and m 2 is the Wilson parameter. The length unit is set to the lattice constant. Dirac matrices α µ and β are an anticommuting set of 4 × 4 matrices satisfying α 2 µ = β 2 = 1 4 , with 1 4 the identity matrix. We introduce an on-site random potential V (r), which is uniformly distributed in [− W 2 , W 2 ]. We take m 2 as the energy unit, and set u/m 2 = 2. We tune the parameters (m 0 /m 2 , W/m 2 ) along the line of DSM phase starting from (−2, 0), where the Dirac points locate at (k x , k y , k z ) = (π, 0, 0), (0, π, 0), and (0, 0, π). The DSM phase, which is the phase boundary between the weak and strong topological insulators, is identified by the transfer matrix method [39]. Note that the slope of Dirac cones (i.e. group velocity at the nodal point) is u in the clean limit.
Model for Weyl semimetals. The time-reversal-broken type WSM phase, where pairs of 3D Dirac cones arise, can be realized in the tight-binding Hamiltonian [19,23,[42][43][44], where σ µ are the Pauli matrices. The Weyl nodes are split in k x direction with separation 2k 0 , where k y , k z = 0 or π. The parameters u and m 2 are the same as in the DSM. We tune the parameters (m 0 /m 2 , W/m 2 ) on the line starting from (−1, 0) where the Weyl points locate at (k 0 = ±π/2, 0, 0) (detail of the tuning is described later). The clean limit group velocity at the Weyl points is m 2 in x direction, and is u in y and z directions. Time evolution simulation. We study the dynamics of wavepackets in PNSMs in the presence of randomness by a direct time-evolution simulation on the lattice models.

arXiv:2003.01929v1 [cond-mat.dis-nn] 4 Mar 2020
For simplicity, we focus on the transport in a specific (x) direction without loss of generality. Periodic boundary conditions are imposed in the directions transverse to the transport (y and z), and the system length is sufficiently large so that the boundaries in x direction can be neglected. That is, we consider a quasi-one-dimensional system with the cross-section of L × L sites. The 3D dynamics is obtained from the limit L → ∞.
Chebyshev expansion. The time-evolution operator U (∆t) for the wavefunction ψ(t), can be written for a time-independent Hamiltonian H, where = 1. The exponential function is expanded by Chebyshev polynomial T n [45], where M is an order of truncation [46], J n is the Bessel function of the first kind, and C 0 = 1, C n≥1 = 2. We have rescaled the Hamiltonian and time step so that the eigenvalues ofH are in the range (−1, 1), where λ max is set to be slightly larger than the possible maximum absolute value in eigenvalues of H. Initial wavepacket. We focus on the time-evolution of the Dirac/Weyl states near the nodal points. Thus we prepare the state corresponding to the eigenstate at a nodal point k 0 in a clean system, exp(ik 0 · r)χ. Then we make an initial wavepacket by multiplying a gaussian factor centered at x 0 , We adopt the eigenstate of α x as the spinor χ of the initial state, so that the wavepacket moves in x direction according to the spin-momentum locking. We set the width of the wavepacket ξ = 40 for weak disorders and ξ = 10 near the critical point, in order to see the wavepacket dynamics efficiently.
Wavepacket dynamics. Figure 1 shows the time-evolution of a wavepacket in disordered DSM. The plotted probability density F (x, t) = s |ψ s (x, y, z, t)| 2 y,z,dis is summed over all internal degrees of freedom, and averaged over the cross-section at x and over disorder realizations (e.g. 4000 simulations for W = 6). The wavepacket travels linearly with time, that is, ballistic, even in a strong disorder (note that the DSM-metal transition occurs at W = W DSM c 6.4 [33]), while the speed slows down with increasing disorder Scaling behavior of speed. In a system with dispersion E = v α |k| α , the number of states below k behaves as and the DOS is given by On the other hand, the scaling theory [33] predicts that near the critical disorder strength W c , that is, δ = |Wc−W | Wc 0, DOS obeys a scaling formula, where z is the dynamical exponent and ν is the critical exponent. In the systems where α is well-defined, the scaling formula Eq. (11) should have the same energy dependence as Eq. (10), Assuming the linear dispersion α = 1 in disordered PNSM phases, the scaling behavior of the speed of Dirac/Weyl electrons, v = v 1 , is obtained by comparing Eq. (10) with Eq. (12), We evaluated the travelled distance of wavepacket peak r after smoothing the curves by taking averages, and estimated the speed of the ballistic modes v(L) = dr dt for a specific size of cross-section. In order to estimate the speed v for 3D limit, L → ∞, we estimated the speed for different sizes (L = 35,45,55,65,75,85,95) and fitted the data with the formula where a 1 and a 2 > 0 are fitting parameters. Then the obtained speeds v (see Fig. 2) coincide with the scaling ansatz Eq. (13) with W DSM c = 6.4, ν DSM = 0.8, and z = 1.5 [33] near the critical point, while they reproduce the slopes of the linear bands, 2 and 1 for DSM and WSM, respectively, in the clean limit.
Ballistic transport in disordered WSM. The ballistic transport is expected to be a common feature for PNSMs. We demonstrate the ballistic transport occurs also in the disordered WSMs where the point-nodes arise in pairs and the time-reversal symmetry is broken (class A [47][48][49]), while the DSM preserves the time-reversal symmetry (class AII). The behavior of the wavepackets and of their speed is qualitatively the same as in the case of DSM. We estimated the speed of wavepackets dynamics in WSMs in the same way (see Fig. 3). Assuming z = 1.5 [11,14,15,24,33], the speed can be fitted by Eq. (13) with W WSM c = 6.3, ν WSM = 0.9, which is consistent with Refs. 20 and 21.
Robustness of nodal points. The presence of ballistic mode under disorder implies that the states near the nodal points are robust against disorder. Here we visualize this numerically. In disordered systems, we cannot define the dispersion and therefore the k-space position of nodal points, in principle. However, a spectral feature of "linear dispersion" and "nodal point" in real (say, disordered) materials is observed by angle-resolved photoemission spectroscopy (ARPES) measurements. The spectral function ρ k (E), which is equivalent to the imaginary part of the retarded Green function, − 1 π ImTr s G R (E, k, s), can be understood as the local density of states in mo- mentum space, where N is the number of lattice sites and S the internal degrees of freedom (four in DSM and two in WSM).
We can obtain the spectral function by using the kernel polynomial method [50,51], which enables large-scale calculations using the Chebyshev polynomial expansion, whereẼ = E/λ max . Here we focus on the bulk spectral function by taking r|k, s = ψ k,s (x, y, z) as ψ k,s (x, y, z) = e ikxx e ikyy e ikzz χ s , where χ s is one of the orthonormal bases. Figure 4 shows the density plot of the spectral function Eq. (15) at k y = k z = 0. In the clean system W = 0, the dispersion is reproduced [Figs. 4(a) and 4(b)]. With increasing disorder, the "band" gets blurred. However, we can still find a clear "nodal point" and "linear dispersion" below the critical disorder strength of the DSM-metal transition W DSM c 6.4 or the WSM-metal transition W WSM c 6.3. The slope of the "linear dispersion" decreases as increasing disorder; the evolution of the slope is consistent with the self-consistent Born approximation [20,[52][53][54] and coincides with that of the speed of wavepacket dynamics (shown in Fig. 3). In addition, the "gap" of cosine bands, i.e., the energy range where only the "linear dispersion" arises, is narrowed as increasing disorder; the cosine bands are broadened by the disorder and the "gap" closes in diffusive metallic phase (W > W c ). We note that the surface spectral function, which is accessible by the ARPES measurements, can be obtained by the same way and gives the same dependence on the disorder. We have also utilized the spectral function to fix the effective mass for the disordered WSMs. The spectrum in WSMs (as in DSMs) shows the linear band structures and nodal points up to the critical point, and enables us to estimate the position of Weyl nodes. We have tuned the mass m 0 so that the Weyl nodes locate at (±π/2, 0, 0) with a sufficient precision.
Critical superdiffusion. Before concluding, we mention some interesting properties that signal the PNSM-metal phase transition. The first point is the vanishing velocity of the ballistic modes (see Fig. 3). The second point is the breakdown of the linear dispersion [see Figs. 4(d) and 4(h)]. The third point is superdiffusion. The scaling formula for the mean displacement at E = 0 is [33] ||r|| ∼ δ −ν f r (tδ zν ).
At the critical point δ = 0, δ dependence should be cancelled to avoid singularity, Since 1/z 0.67 > 1/2, a superdiffusion is expected at the critical point. Although it was technically difficult to see the critical superdiffusion directly (since the wavepacket speed vanishes at the critical point), a subballistic transport has been obtained in the vicinity of the critical point. It can be also seen in Fig. 4 as a superlinear "dispersion" α > 1 around the critical point.
Conclusion. We have studied the wavepacket dynamics of disordered DSM/WSMs and found that the PNSMs show a ballistic transport, reflecting the robustness of PNSMs against disorder. Similar ballistic transport is also observed in topological insulators on percolative lattice [55], and is a universal behavior. It is to be noted that since a wavepacket is not an energy eigenstate, the ballistic transport feature is not restricted to E = 0. This fact can be also seen in the "linear dispersion" (spectral function) of disordered PNSMs. We have confirmed that the speed of the ballistic mode slows down and vanishes toward the PNSM-metal transition point. The behavior of the speed is consistent with the prediction of the scaling theory. This means that the speed of the ballistic mode can be an order parameter characterizing the PNSM-metal transition. Indeed, we have estimated the critical point and the critical exponent for the WSMmetal transition.
We thank Kentaro Nomura and Yuya Ominato for useful discussions. This work was supported by KAKENHI (Grants No. JP16J01981 No. JP19K14607, and No. JP19H00658) from the Japan Society for the Promotion of Science.