Indirect magnetic interaction mediated by Fermi arc and boundary reflection near Weyl semimetal surface

We theoretically investigate the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction between magnetic impurities distributed in the vicinity of the surface of a Weyl semimetal. Contrary to previous studies, we further take into account the influence of interplay of the Fermi arc and bulk states, and interface refection. It is shown that the RKKY pattern is significantly mediated by the Fermi-arc surface state along with the interface reflection. The Fermi-arc surface state mediates the RKKY interaction by interfering with the bulk states. The resulting interference contribution in the short-range impurity distance R is comparable in magnitude to the bulk-band contribution and even dominates the latter near the surface. It either enhances or weakens the bulk contribution, depending on the relative orientation of impurities and Fermi energy. More importantly, for the long-range impurity distance the interference term dominates in that it can prolong the decay rate from the original bulk R−5-law to R−2 (R−3) for finite (zero) Fermi energy. The interface reflection not only enhances the magnitude of the RKKY interaction and changes its anisotropy from the original XXZ to XYZ or Ising spin model, but also generates extra twisted RKKY terms parallel to the line connecting Weyl nodes, lacked in the scenario without the interface effect. They originate from the interaction between the impurity and the mirror image of the other impurity. We further analyze in detail the spatial anisotropy of the decay rate and beating pattern. These findings provide a deeper insight into surface magnetic interaction mediated by Weyl fermions.

Besides the linear dispersion of bulk band, the most significant feature of WSMs is the splitting nodes, which are topologically robust as long as translational symmetry is preserved and can be interpreted as a composition of a magnetic monopole and anti-monopole. They can only be created or annihilated at pair. The stable topological feature of Weyl node pairs could be characterized by the non-zero Berry curvature in the momentum space. As a consequence, the WSMs exhibit many anomaly phenomena, such as anomalous Hall effects [1,[25][26][27][28][29], chiral

Model and method
To consider the influence of the Fermi arc on the RKKY interaction, we study a semi-infinite WSM as shown in figure 1, where a WSM is placed in the right half-plane (y>0) and the vacuum in the left half-plane (y<0). The surface is at y=0 and infinite along x-and z-directions. Along y direction, the momentum k y is not a good quantum number and replaced by an operator k y =−i∂ y . For the convenience of analysis, we employ a simplest continuum model for WSMs proposed by Okugawa and Murakamii [58], which is written as , , In order to obtain the surface state in this semi-infinite WSM, we set the magnetic term [59] = -¥ m for y<0 and finite in the WSM side. We assume an incident wave -Ce k y i y is injected along −y direction. Near the interface, if the wave is bound to the x-z surface, the solution of wave vector k y becomes imaginary. By considering the continuity conditions of the boundary between left and right regions, one can easily obtain the surface state at y=0 as acts as the decay factor and θ (y) denotes the Heaviside step function. The corresponding surface band reads as E sur =v f k x in the range of -< < m k m z and vanishes otherwise. The surface state forms a Fermi arc in k-space connecting two Weyl nodes if projected to the surface Brillouin zone.
The structure of the bulk states in the presence of the boundary differs from the case of an infinite sample in which k y is a good quantum number. Following the same processes but considering a real wave vector k , y the wave function of bulk bands can be calculated as, 2 , and χ=±1 denote the chirality of the two Weyl nodes. The wave function consists of two parts: an incident wave and a reflected wave. When we drop the conjugate terms in equation (3), the wave function reduces to the case without the boundary. In obtaining the analytical expression of equation ( where λ=v f /v z and the energy dispersion is E s =sk χ . We assume that two magnetic impurities positioned at points r A and r B interact with WSM electrons weakly enough not to change the host dispersion. The interaction is expressed as the standard s-d exchange Hamiltonian where S i is the local classic impurity spin. Using standard approximating method [60][61][62][63] to the second order term of the interaction J 0 , one at zero temperature can obtain an effective interaction between magnetic impurities,  Im  d Tr  , , , , , 5 i j is the retarded Green's function in frequency-real space and u F is Fermi energy. Obviously, before calculating the exchange interaction, we must first derive the retarded Green's functions. They can be constructed from the wave functions within the Lehmann's representation [64,65]: where Ψ n (r) represents the nth eigenstate in equations (2) and (3) with the corresponding eigenenergy E n . Due to the coexistence of surface and bulk wave functions, the full Green's function can be decomposed into two parts: , , , , , , , respectively, constructed by the wave functions of the surface band only or bulk states only. In this way, we can work out the analytical expressions for Green's functions. With the help of equation (2) and integrating over the momentum k, we can obtain the Green's function contributed by the surface band, . For convenience, above and hereinafter we denote , and ¢ ( ) R 1 2 denotes R 1 (2) in which Δz is replaced by λΔz. Also, one can use the wave functions in equation (3) to construct the Green's function of bulk states. After integrating over the momentum, the bulk Green's function can be written analytically as i t a n i 2 . Here, α 1 =σ 0 , α 2 =σ x , and ¢ = ¢ | | R R n n .

Results and discussion
Now let us turn our attention to the indirect exchange interaction between two magnetic impurities caused by the WSM electrons. Due to the presence of surface, the RKKY interaction will be mediated by the Fermi-arc surface state and the reflected bulk states. Although the Green's function can be split into the bulk and surface parts, the RKKY interactions in equation (5) cannot be simply separated to the bulk and surface components due to the interference between them. In the following, we discuss them separately. In numerical calculations, we adopt the model parameters extracted from [66]: By substituting the Green's function equation (7) or equation (2) into equation (5) and then performing careful calculations, the RKKY interaction can be written in form of Here, the range function J i is the in-plane components (the Heisenberg and Ising terms), and the others are twisted component: J i DM for DM term, J zx fr and J zy fr for the spin-frustrated terms They are determined by the states of host electrons and the configuration of two magnetic impurities.

Influence of reflected bulk states on the RKKY interaction
We first consider the exchange interaction mediated by the bulk states by substituting equation (2) into (5). Due to the strong anisotropy of WSM dispersion, the RKKY interaction is also heavily dependent on the spatial orientation of distance vector between two impurities. In the following, we give both numerical and analytic results for the impurities deposited along x-axis and only the numerical results for other impurity configurations.
For impurities arranged parallel to the x-axis and at u F =0, the non-zero range functions read = - . In equation (10), we divide each bulk RKKY component J i b into three parts. The first term is exactly equal to the interaction between two impurities without considering the boundary reflection. It remains unchanged no matter the distance  y of impurities away from the boundary. The second term in J i b originates from the mirror effect, characterized by power-law decaying with the relative distance R 2 between the impurity A (x y z , , A A A ) and the mirror image (x y z , , B B B ) of the impurity B with respect to the boundary. The last term in J i b is an extra mixed term between them. Naturally, the components in the second and the third terms are significantly dependent on the impurity distance  y with respect to the boundary. To explicitly exhibit the modification of RKKY interaction by the boundary, we in figure 2(a) display the evolutions of RKKY components for u F =0 as a function of  y . In numerical calculations, we fix the relative distance between impurities (i.e., Δx=const and Δz=Δy=0) and so  y denotes two times the distance of impurity pair from the boundary. Here and in the following, we take W = J 1 b , noticing two impurities deposited along x-axis). This also can be seen from equation (10), where for D   y x 1 or  ¥ R 2 , only the first term is remained in J i b . In this limit, the RKKY interaction reduces to be , 80   f  f  1  3  0   2  2  1   5  2  3  0   2  2  1 5 , and isotropic dispersion (λ=1) is taken for comparison. These results recover those of bulk RKKY interaction in the absence of the boundary [53,54]. This means that the boundary effect could be ignored if the impurity pair is located far enough away from the boundary. When the impurities move towards the boundary, all RKKY terms deviate from the above behaviors, accompanied by two significant changes. One is the quick increase of J x b with reducing  y while overlapping J y b and J z b decrease in somewhat different velocity, making the spin configuration develop from original XYY to XYZ model. The anisotropy of the exchange interaction reaches the strongest at the boundary ( =  y 0), where only the Ising term J x b survives while the other components are exactly compensated by the interface reflection. Another significant change is the appearance of a peculiar DM term in the vicinity of the boundary. It is noted that if the boundary is absent [54], the DM term vanishes for u F =0 and is finite only for electron-hole symmetry breaking (or ¹ u 0 F ) with its direction perpendicular to the line connecting the Weyl nodes. In contrast, in present Weyl system with boundary, the DM term emerges even for u F =0 when impurities are located in the vicinity of the boundary but not exactly at the boundary. Moreover, different from the DM term in bulk WSMs [54], here the DM term J z DM is parallel to the line connecting the Weyl nodes, which is contributed purely by the interaction between the impurity and the mirror image of the other impurity (indicating by R 2 ).
In figure 2(b), we depict the  y -dependence of RKKY components for impurities arranged along the z-axis. Compared with figure 2(a), the similar change happens for this case. The main differences are: (1) the obtained RKKY interaction is almost two orders of magnitude smaller; (2) J x b is overlapped with J y b for large  y and in intermediate  y the spin model XYZ becomes prominent. Besides, in this impurity configuration, the DM term vanishes completely. On the x-z surface, the dependence of DM term on spatial direction can be described with sin cos , sin sin , cos figure 1, where the position of one impurity is chosen at the origin point.
We display the variation of non-zero RKKY interaction with  y with finite Fermi energy u F for impurities parallel to x-axis in figures 3(a), (b) and to z-axis in figures 3(c), (d). Also, the relative distance of two impurities is fixed as in figure 2. Remarkably, the finite Fermi energy gives rise to oscillation of all the RKKY interactions and their oscillating amplitudes decay with the distance  y measured from the boundary. The underlying physics stems from the interference between the impurity-impurity interaction and the impurity-mirror-impurity interaction, which manifests themselves as a cosine function . At the same time, the RKKY interaction is several orders of magnitude larger than the case of u F =0 in figure 2. For impurities parallel to x-axis, one in figure 3(b) can notice that a new DM term J x DM develops for finite u F , perpendicular to the line connecting Weyl nodes due to the electron-hole asymmetry. It dominates quickly over the component J z DM . In addition, the spin-frustrated term J zx fr has non-zero components near the boundary in this case and vanishes for impurities far away from the boundary, resembling to J z DM . J zx fr originates from the interplay of mirror effect and the electron-hole asymmetry. Figures 3(c), (d) show the case of two impurities arranged along z-axis, where the same RKKY oscillation emerges but without any DM term.

Influence of the Fermi arc on the RKKY interaction
Two impurities deposited on WSM surface interact with each other not only via the bulk states but also the surface state near the Fermi energy. In the following, we will explore the contribution of Fermi-arc surface state on RKKY components. When substituting w ( ) G r r , , A B sur in equation (7) into equation (5), one can calculate the RKKY interaction contributed by surface state. It is easy to find that there is no overlapping between w ( ) G r r , , for any energy ω and so purely surface state does not contribute to the exchange interaction for present two-nodal model. Physically, such zero contribution can be attributed to the open Fermi arc in single conducting channel. Similar mechanism is also addressed in [67], where the open Fermi arc at a single surface of a WSM cannot host the quantum Hall effect. Even so, the Fermi arc also mediates the RKKY interaction significantly by the interference with the bulk states, characterized by a product of bulk-state and surface-state Green's functions.
For impurities parallel to the x-axis, the non-zero range functions at u F =0 read   figure 4(a) show the RKKY terms stemming from the interference effect as a function of spatial distance  y , where two impurities are arranged parallel to the x-axis and u F =0. Naturally, these interference interactions vanish if impurities are far away from the boundary (  ¥  y ). With the impurity pair close to the boundary, their magnitudes become visible and the spin structure behaves XYY-like model due to always = J J y i z i . More importantly, by comparing figure 4(a) with 2(a), it shows that the magnitude of the interference contribution is comparable with bulk contribution. In figures 4(b) and (c), we plot the bulk components (J i b and J z b,DM ) and the interference components (J i i and J z i,DM ) for comparison. Obviously, with the decrease of  y , the interference components play more and more important role. For example, in figure 4(b) when the distance  y is lower than a certain position (indicated with a circle), J x i dominates over the bulk value J x b . This specific position can be further mediated by u F . Above discussions are for two impurities resided parallel to the x-axis. For arbitrary impurity configurations, the competition between interference and bulk parts leads to rich behaviors. In the following, we keep at least one impurity situated at the boundary and discuss the orientation dependence of J x . In figures 5(a) and (b), we plot the dependence of J x b and J x i on polar angle q R 1 with two impurities placed at the x-z plane (i.e. =  y 0). For u F =0 in figure 5(a), the interference contribution J x i dominates in most regimes. For impurities deposited along the x-axis (seeing q p = 2 R 1 ), both J x b and J x i reach the maxima. Then they oscillates with q R 1 deviating from π/2. When reaching the z-axis (seeing q = 0 R 1 ), both components become quite small. J x b and J x i present either enhancing (the same sign) or compensating (opposite sign) behavior, depending on the impurity spatial direction q R 1 . Their relative magnitudes and signs and so the anisotropy of total RKKY interactions can be changed significantly by u F , as shown in figure 5(b). For example, for q p = J 2, and J x i can be tuned to be opposite sign and compensate each other. It is noticed that there is no dependence on the azimuthal angle f R 1 in previous works [53,54] as a consequence of the azimuthal symmetry around a pair of Weyl dispersions. Here, the appearance of boundary breaks this symmetry and leads to interesting f R 1 -dependent RKKY interaction. In figures 5(c) and (d), we display the dependence of J x b and J x i on f R 1 , where we keep one impurity at origin point and rotate the line connecting two impurities in the x-y plane. Remarkably, both J x b and J x i are sensitive to f R 1 in ), the interference term J x i dominates over J x b for u F =0 and compensate J x b for finite u F due to opposite sign, as shown in figure 5(d). Finally, we want to address the variation of bulk RKKY J b and interference RKKY interaction J i with impurity distance R 1 . Without the boundary or for  ¥  y , the bulk RKKY interaction, namely the first term of J x y z b in equation (10) or I 1 and I 2 in equation (11), show -R 1 5 decaying law, regardless the impurity configurations. The same results were also reported in [54]. When the boundary effect is taken into account, the second and third terms in equation (10) add two fast decaying rates, which are dependent on the distance  y with respect to the boundary. They decay as -R 1 6 or -R 1 7 for large  y and vanish at the surface. Thus, the decay rate of RKKY interactions still dominates by -R 1 5 -law. Remarkably different from this, we find that the interference terms J x y z i between Fermi arc and bulk states significantly change this scenario. To shed light on this, we derive an analytical expression for impurities situated at the interface along x-axis with R 1 ?1, is as the unit of J.
Intriguingly, the interference component causes quite slowly decaying rate compared with bulk component J x b . The decay rate with impurity relative distance R 1 depends on the Fermi energy u F , whichis -R 1 3 -law for u F =0 and -R 1 2 -law for finite u F . In this case, the interference component plays a crucial role and therefore dominates over the bulk contribution for large R 1 . This long range behave resembles to the case in two-dimensional metal material [68]. For impurities situated at the interface but along z-axis (parallel to the line connecting the two Weyl points), we find l l l l » - which instead shows -R 5 -decaying rate for u F =0 and R −4 -decaying rate for finite u F . Obviously, the RKKY along z-direction decays faster than along x-direction, exhibiting the significant anisotropy of the RKKY interaction.
Interestingly, for two impurities placed along z-axis, from equation (14) one can see that there are two periodic functions: one is 2T 1  . For arbitrary direction of impurity arrangement, their periods present the dependence on impurity relative direction q R 1 , as shown in figure 6(b). When impurities are along x-axis (q p = 2 R 1 ), the oscillation of T 2 vanishes and so do the beating frequency. This reveals that the beating pattern of indirect exchange interaction is anisotropic and depends tightly on the direction of impurities. The underlying mechanism stems from the spin-momentum entanglement. is as the unit of J.

Summary
We have explored the RKKY interaction between magnetic impurities mediated by Weyl fermions. Contrary to all previous studies [53][54][55], we further take into account the influence of interplay of Fermi arc and bulk states and interface refection. In the presence of the boundary, the bulk states and its interference with Fermi arc acquire non-trivial structure, that complicates the form of the indirect exchange interaction. It is found that there emerge rich magnetic interactions, including Heisenberg, Ising, DM, and spin-frustrated terms We first consider the contribution from boundary reflection. With impurities close to the boundary, the reflection effect becomes more and more important, on one hand, significantly increasing the size of RKKY interaction and on the other hand changing the anisotropy from the original XXZ to XYZ spin model and finally to Ising one. The boundary reflection also generate interesting DM term near the boundary. This is the consequence of the absence of the translational invariance perpendicular to the boundary. Then, we analyze the contribution from Fermi-arc surface state, which mainly interferes with bulk states and modifies the RKKY interaction with magnitude comparable with or exceeding the bulk contribution. It either enhances or weakens the bulk contribution, depending on the relative orientation of two impurities and Fermi energy. More importantly, for some specific configuration of magnetic impurities, the interference term can fall off as R −3 for zero Fermi energy or R −2 for finite Fermi energy, much more slowly than -R 5 in the bulk contribution, displaying a typical behavior in a two-dimensional metal. The direction dependence of the RKKY beating pattern are discussed. It is expected that these RKKY interactions can be probed experimentally by broadband electron spin resonance technique coupled with an optical detection scheme [69].