Resonance states and beating pattern induced by quantum impurity scattering in Weyl/Dirac semimetals

Currently, Weyl semimetals (WSMs) are drawing great interest as a new topological nontrivial phase. When most of the studies concentrated on the clean host WSMs, it is expected that the dirty WSM system would present rich physics due to the interplay between the WSM states and the impurities embedded inside these materials. We investigate theoretically the change of local density of states in three-dimensional Dirac and Weyl bulk states scattered off a quantum impurity. It is found that the quantum impurity scattering can create nodal resonance and Kondo peak/dip in the host bulk states, remarkably modifying the pristine spectrum structure. Moreover, the joint effect of the separation of Weyl nodes and the Friedel interference oscillation causes the unique battering feature. We in detail an- alyze the different contribution from the intra- and inter-node scattering processes and present various scenarios as a consequence of competition between them. Importantly, these behaviors are sensitive significantly to the displacement of Weyl nodes in energy or momentum, from which the distinctive fingerprints can be extracted to identify various semimetal materials experimentally by employing the scanning tunneling microscope.

Progress in material preparation and experimental techniques has led to a surge of interest in two-dimensional (2D) Dirac materials such as graphene and surface states of topological insulators. Very recently, this concept is extended to 3D systems, known as topological Dirac semimetals (DSMs), which are newly-discovered bulk analog of graphene as a new topological states of matter. Recent experiments have identified a class of materials 1-3 (Bi 1−x In x ) 2 Se 3 , Na 3 Bi, and Cd 3 As 2 to be the DSMs. In these new Dirac materials, 3D massless Dirac fermions are excited around the doubly degenerate Dirac cones, which are protected by time-reversal symmetry (TRS) or inversion symmetry (IS).
Breaking either the TRS or the IS will drives the DSMs into a Weyl semimetal (WSM) phase, which is manifested as the splitting of a pair of degenerate Weyl nodes with opposite chirality in momentum or energy space. As a new topological nontrivial phase, these massless WSM fermions are drawing great interest for their scientific and technological importance. The WSM Fermion states have been predicted theoretically and observed experimentally in a family of the noncentrosymmetric transition-metal monosphides 4-10 with preserving the TRS, e.g., TaAs, NbAs, NbP, and TaP. The nontrivial topology along with the node separation leads to many exotic phenomena and unique physical properties, such as the chiral anomaly [11][12][13] , the unique Fermi arc surface states [6][7][8][9][10]14 , the chiral Hall effect 13,15 , the chiral magnetic effects 11,16 , and the negative [17][18][19][20] and extremely large magnetoresistance 21 .
When most of the previous studies concentrated on the clean host WSM bulk states, it is expected that the dirty WSM system would present rich physics due to the interplay between the WSM Fermion states and the impurities embedded inside these materials. On one hand, the unique 3D spin-momentum locking can mediate the interaction between magnetic impurities in both Dirac and Weyl semimetals, leading to anisotropic Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling and rich spin textures [22][23][24] . On the other hand, the feedback effect of impurities on host bulk states can change the property of Weyl nodes, at which the differences between a Weyl phase and a normal metal are most pronounced. The stability of the nodal density of states (DOS) was investigated in the presence of various types of local impurities 25,26 , and nonzero DOS at the degeneracy point was predicted for the disorder strength beyond a certain critical value 27,28 . We would like to mention that these discussions, however, are limited to the classic impurity model and only the single node scattering is taken into account. The nodal resonance induced by impurities have also been extensively studied in graphene and topological insulators [29][30][31][32][33] . As a representational feature of quantum impurities, the Kondo effect has been intensely discussed in three-dimensional Dirac and Weyl systems [34][35][36][37] of dilute magnetic impurities. The results showed that the nature of the Kondo effect of impurity is only affected strongly by the linear dispersion of Dirac/Weyl host bulk states but it is in general blind to the momentum splitting of TRS-broken Weyl nodes. In ref. 35, they found that the spatial spin-spin correlation between the magnetic impurity and the conduction electron is sensitive to the displacement in the momentum splitting of Weyl nodes, where rich features are shown due to an extra phase factor. Even so, it is still challenging how to identify the TRS-broken WSM materials from the transport fingerprints.
In this paper, we study how the local density of states (LDOSs) in host WSM/DSM bulk states are modulated by the embedded quantum impurity in the resonance regime and in the Kondo regime. We specially pay attention to the response of nodal behavior to impurity scattering processes. It is found that the quantum impurity scattering can create a LDOS resonance or Kondo peak/dip in the host bulk states exactly at the Dirac point and thus remarkably destroy the pristine spectrum structure, which are sensitive to the degree of the splitting of two WSMs nodes. Compared with the single node scattering, the internode scattering possesses more information about the unique properties. Interestingly, by taking the intranode scattering into account, we find the unique battering feature for the TRS-broken WSMs, which is long-range measurable in real space with current scanning tunneling microscope technologies.
The rest of the paper is organized as follows. In Sec. II we present a general interaction model of Weyl fermions with Anderson quantum impurity and treat it by employing the standard equations of motion for Green's functions. The low-energy resonance, Kondo signature, and Friedel oscillation in host materials are discussed in Sec. III, and a short summary is given in the last section.

Model and Theory
Consider a 3D WSM with a pair of chirality-opposite Weyl nodes, whose low-energy Hamiltonian can be described as 22,34 = ∑ ⊕ with complex conjugation operation K is the time-reversal operator, P = τ x ⊗ σ 0 is the inversion operator, and τ  is the Pauli matrix on the chirality space. Breaking either TRS ≠ Q 0 or IS Q 0 ≠ 0 transforms a DSM into a Weyl system, the former splitting the two degenerate weyl points separately at different momentum = ± k Qv / node f but with the same energy while the latter shifting two Weyl nodes at different energy ω node = ± Q 0 but with the same momentum. This can be seen from the dispersion spectrum of Ĥ w , We utilize the typical Anderson impurity model to study the quantum impurity effect and spin-1/2 Kondo screening in 3D Dirac and Weyl semimetals. The full Hamiltonian can be written as = is characterized by a single-orbital energy ε 0 and the on- represents the hybridization between the impurity and the host material with the hybridization matrix T where the spinor = ↑ ↓ d d d ( , ) T and the coupling strength V x is assumed to be dependent on the Weyl node χ but regardless of  k and σ under the assumed wide-band approximation and spin-conserved hoping. Here, we also assume that the magnetic impurities are embedded inside the WSM such that the effect of Fermi arc surface states can be neglected safely.
Using the method of standard equations of motion, the retarded Green's function of Weyl electrons with respect to the full Hamiltonian Ĥ can be derived as , where ω + = ω + i0 + and all qualities are 4 × 4 matrix in the spin ⊗ chirality space. When performing the Fourier transform to the real space, its block matrix in chirality space is Scientific RepoRts | 6:36106 | DOI: 10.1038/srep36106 ) is still 2 × 2 matrix in subspace of the electron spin and  r is measured from the impurity as a scattering center, whose position is chosen to be the origin of coordinates. The expression in Eq. (5) recalls the extensively applied T-matrix approach 30,31 is expressed in terms of the Green's function of magnetic impurity, defined as } . Similar relation can be found in Anderson impurities interacting with topological insulator 29,33,38,39 or graphene 40 . In Eq. (5), At the impurity position =  r 0, the bare Green's function is given by with the cutoff energy D, unitstep function Θ (x), and ω χ = ω + − χQ 0 . Note that even for STR-broken case due to finite Q, ω χ + G (0, ) 0 is diagonal in spin space and independent of Q, remarkably different from the case of 2D topological insulator or graphene 33,41 by expanding the ± ⋅   e ik r in terms of spherical harmonics according to the Rayleigh equation 24,42 , and finally arrive at a simple analytical expression for i The next task is to calculate the impurity Green's function ω σσ′ .
Carrying out the equations of motion, we find To form a set of close iterative equations, we truncate them following the standard method 43 , where the operator pair with the same spin indices can be pulled out of the Green's function as an average and is calculated with the Fluctuation dissipation theory where f(ω) is the fermi distribution function. After carrying out lengthy but straightforward calculations, we finally derive the expression for the impurity Green's function in the deep Coulomb blockade regime, i.e., U → ∞ , as k f By comparison with the normal metals 44 or 2D Dirac materials 33 , the most distinction is the specific expressions of self-energies ∑ 0 (ω + ) and ∑ 1 (ω + ).

Results and Discussion
Resonance states in LDOS. Our purpose is to explore the unique local properties of the WSM when the conducting elections are scattered off a quantum impurity. As the Weyl nodes are separated in energy or momentum, a very interesting question is whether Q or Q 0 leads to some especial spectrum structures locally around the quantum impurity. Next, we focus on the LDOS in WSMs, which is defined as is the unperturbed LDOS, and δρ ω  r ( , ) contributed by the second term in Eq. (5), reflects the substantial modification of the LDOS by the doping impurity. Beyond the usual single-node treatments, we here emphasize the impurity scatter processes between two Weyl nodes. We find that the introduce of quantum impurity not only scatters the electrons within the same Weyl node but also between two nodes. Specifically, we can split LDOS as δρ ω δρ collects the contribution from scattering process between two nodes. After proceeding the calculations, we obtain readily the following analytical expressions intra f Equations (16) and (17) are our central results. In order to understand them deep, we in the following limit our discussions to the symmetrical coupling Γ + = Γ − , and first discuss the impurity effect in the DSMs, i.e., setting = = Q Q 0, 0 whose LDOS ρ(ω) for a fixed  r is illustrated in Fig. 1. Without the internode scattering (i.e., single-node case), seeing Fig. 1(a), there a pronounced resonance structure, whose position depends on the impurity level ε 0 . This resonance is a consequence of the backaction of the resonance in the impurity DOS, which is defined as ρ ω ω and depicted in the corresponding inset, indicating the single-level resonance tunneling between the impurity and the reservoirs. In Fig. 1(a), with the increase of ε 0 from − 0.2 to 0.2 in step of 0.1 the low-energy resonance is first shifted close to the Dirac point, accompanied with increasing magnitude, and then passes over the Dirac point into its other side, on whole exhibiting a symmetry with respect to the Dirac point. Intriguingly, a sharp pronounced resonance for ε 0 = 0 can be located exactly at the Dirac point, completely destroying the 3D typical ω 2 Dirac spectrum. Similar Dirac-point resonance appears in doping surface of topological insulators with quantum impurities 33 or quantum magnets 44,45 . Our further calculations confirm that the scenario of Dirac-point resonance cannot emerge for classic impurity model, i.e., replacing ω χχ′ pot 0 0 stands for a classic impurity potential. If the internode scattering is taken into account, the scenario is very different from the single-node case. We plot the LDOS ρ(ω) including both intra-and inter-node scattering in Fig. 1(b). By comparison with single-node case, most interesting in double-node case is that the resonance peak becomes weaker and weaker when close to the Dirac point and is completely smoothed away at the Dirac point, in which ρ(ω) ∝ ω 2 recovers the typical square dependence on energy. To understand it, we plot the change of DOS δρ inter (ω) and δρ intra (ω) for ε 0 = 0 in the inset of Fig. 1(b), from which we know that the negative δρ inter (ω) tends to suppress the resonance in δρ intra (ω) and, at Dirac point ω = 0 they have the same amplitude but opposite sign and thus cancel each other exactly. This point also can be seen from Eqs (16) and (17).
From the above discussions for DSMs, we are known that the competition between intra-and inter-node scatterings is crucial for the development of the Dirac-point resonance. In Fig. 2(a) we depict the change of LDOS δρ(ω) for the TRS-broken WSMs, i.e., Q 0 = 0 but ≠ Q 0. Here, we just choose Q along z-axis and so the degenerate Weyl nodes are shifted by ± Q z in the direction of  k z but Q x/y = 0. Obviously, the Dirac-point resonance for finite Q z recovers since δρ inter only partly offsets δρ intra , as shown in Fig. 2(a). From Eqs (16) and (17), one can notice that inter i ntra is less than − 1 for the chosen parameter. The variation of LDOS δρ ω  r ( , ) for different Q z is plotted in Fig. 2(b), in which the Dirac-point resonance peak increases first for small Q z ∈ (0, π/2r) and then exhibits a periodic function of Q z , seeing the inset. For Q z = (2n + 1)π/4r (n = 0, 1 … ) or ⊥ r Q, the internode scattering is prohibited due to destructive interference and thus δρ ω  r ( , ) intra dominates. Therefore, to probe the feature of TRS-broken WSMs, it is necessary to consider the impurity-induced scattering between Weyl nodes since Q only enters δρ inter but not δρ intra .
For noncentrosymmetric WSMs, i.e., Q 0 ≠ 0 and = Q 0, we from Eqs (16) and (17) see that Q 0 contributes to both δρ ω  r ( , ) intra and δρ ω  r ( , ) inter but with different ways, thus their zero-energy resonances cannot be completely compensated. Another most interesting effect for noncentrosymmetric WSMs is the emergence of Kondo resonance, which is expected to occur because of the nonzero LDOS at ω = 0 when two Weyl nodes are split to ω node = ± Q 0 . If we choose the proper parameters in Kondo regime, the impurity DOS presents a remarkable sharp Kondo resonance at ω = 0 as shown in the inset. The Kondo resonance is mainly attributed to the self-energy ω ∑ + ( ) Scientific RepoRts | 6:36106 | DOI: 10.1038/srep36106

Conclusions
On conclusions, we have investigated the influence of quantum impurity on the DSM and WSM materials by looking at the modification of LDOS around the impurity. It is found that the quantum impurity scattering can create the LDOS low-energy resonance, the Kondo signature, and the Friedel oscillation, all of which are sensitive to the displacement of Weyl nodes in energy or momentum. We in detail analyze the different contribution from the intra-and inter-node scattering processes and present different scenarios as a consequence of competition between them. We further study the spatial dependence of LDOS and find that the separation of Weyl nodes along with the Friedel interference oscillation leads to the unique battering feature, which arises in the intranode scattering for the IS-broken WSMs but in internode scattering for the TRS-broken WSMs. Especially, the beating feature for the TRS-broken WSMs is remarkably dependent on the spatial direction of the probing position, which is long-range measurable in real space by employing current scanning tunneling microscope technologies. (a-c) panels are plotted for θ r = 0, π/4, and π/2, respectively. Insets are δρ ω  r ( , ) inter scaled by r 2 . All structures are independent of the azimuthal angle ϕ r . Here, Q z = 0.05, Q 0 = 0, ω = 0.8, ε 0 = − 0.01, and the others parameters are the same as in Fig. 1.