Electromagnetic fields induced by an electric charge near a Weyl semimetal
Weyl semimetals (WSM) are a new class of topological materials that exhibit a bulk Hall effect due to time-reversal symmetry breaking, as well as a chiral magnetic effect due to inversion symmetry breaking. These unusual electromagnetic responses can be characterized by an axion term with space and time dependent axion angle . In this paper we compute the electromagnetic fields produced by an electric charge near to a topological Weyl semimetal with two Weyl nodes in the bulk Brillouin zone. We find that, as in ordinary metals and dielectrics, outside the WSM the electric field is mainly determined by the optical properties of the material. The magnetic field is, on the contrary, of topological origin in nature due to the magnetoelectric effect of topological phases. We show that the magnetic field exhibits a particularly interesting behavior above the WSM: the field lines begin at the surface and then end at the surface (but not at the same point). This behavior is quite different from that produced by an electric charge near the surface of a topological insulator, where the magnetic field above the surface is generated by an image magnetic monopole beneath the surface, in which case, the magnetic field lines are straight rays. The unconventional behavior of the magnetic field is an experimentally observable signature of the anomalous Hall effect in the bulk of the WSM. We discuss a simple candidate material for testing our predictions, as well as two experimental setups which must be sensitive to the effects of the induced magnetic field.
Materials characterized by topological order, or simply topological materials have attracted great attention recently both from the theoretical and experimental fronts. The best studied of these are the topological insulators (TIs), which are characterized by a gapped bulk and protected boundary modes that are roboust in the presence of disorder Qi-Review (); Hassan-Review (). Up to recent times, one usually associated topologically nontrival properties with gapped systems; however, we have learned that gapless (semi)metallic states may be topologically nontrivial in the same sense as gapped insulators. A particularly interesting state of matter is the topological Weyl semimetal (WSM), which may be thought of as a 3D analog of graphene. These are states characterized by phases with broken time-reversal (TR) or inversion (I) symmetry, whose electronic structure contains a pair of Weyl nodes (band crossing points) in the Brillouin zone (BZ) provided the Fermi level is close to the Weyl nodes. WSMs possess protected gapless surface states on disconected Fermi arcs with end points at the projection of the bulk nodes onto the surface BZ Armitage-Review ().
In addition to their spectroscopic distinguishing features, topological phases also exhibit unusual electromagnetic (EM) responses, which are a direct macroscopic manifestation of the nontrivial topology of their band structure. It has been shown that the EM response of both topological insulators (Qi-TFT, ) and Weyl semimetals Burkov-TFT () is described by the so-called term in the action of the EM field, . For TIs, the only nonzero value compatible with TR symmetry is , and thus has no effect on Maxwell equations in the bulk. Its only real effect, a half-quantized Hall effect on the sample surfaces, becomes manifest only in the presence of surface magnetization. When TR and I symmetries are broken in the bulk, such as in a topological Weyl semimetal, the axion field may also acquire linear space and time dependence , where is the separation between the Weyl nodes in momentum space and is their energy offset (resulting from the chiral anomaly). Unlike the term for TIs, the analogous term for WSMs does modify Maxwell equations in the bulk and thus has additional observable physical consequences, namely the anomalous Hall effect (AHE) and the chiral magnetic effect (CME) Burkov (). A number of physical effects, mainly optical, have been predicted on the basis of this theory. For example, the magneto-optical Faraday and Kerr rotation Kerr-Faraday/WSM () and the Casimir effect Casimir/WSM (), as well as the appearance of plasmon polaritons Plasmons/WSM () and helicons Helicons/WSM () at the sample’s surface. In this paper we are concerned with a particular physical effect associated with the anomalous Hall effect.
One of the most striking consequences of the term in topological insulators is the image magnetic monopole effect, namely, the appearance of a magnetic field which resembles that produced by a magnetic monopole when an electric charge is put near the material surface Qi-Monopole (); Karch (); MCU-GreenTI (). Physically, the monopole magnetic field is induced by a circulating Hall current on the TI surface, centered at the position of the charge projected onto the TI. In this paper we tackle the analogous effect in topological Weyl semimetals. To be precise, we investigate the electromagnetic fields induced by an electric charge above a WSM with broken TR symmetry. We assume the charge to be located along the axis defined by the separation between the Weyl nodes in the BZ, i.e. near the surface without Fermi arcs. What is relevant in our configuration is that due to the magnetoelectric effect in WSMs, a magnetic field is induced. Outside the material, the magnetic field is particularly interesting as it is not perfectly radial (as that produced by a magnetic monopole). Indeed, the physical origin of the magnetic field is the anomalous Hall effect in the bulk, which as we will see, can be interpreted in terms of a family of -dimensional subsystems parametrized by the coordinate along the nodal separation. Each subsystem exhibits a quantum-like Hall effect, as such a WSM can be understood as a chain of 2D Dirac surface states. This result is in accordance with the heterostructure model of Weyl semimetals introduced by A. Burkov in Refs. Burkov-TFT (); Burkov (), consisting of alternating thin layers of trivial and topological insulators, which indeed is the best candidate material to test our predictions. We also discuss two experimental setups which could be sensitive to the induced magnetic field.
The rest of this paper is organized as follows. In Sec. II we briefly review the electromagnetic response of topological Weyl semimetals. The central part of this paper is presented in Sec. III, where we compute the EM fields produced by an electric charge above a WSM. In Sec. IV we compute the interaction energy and the force that the material exerts upon the static charge. We close with a brief summary of our results and conclusions in Sec. V, where we also propose two possible experimental setups to determine the magnetolectric effect via the measurement of the resultant magnetic field. Appendix A contains the details of the calculation of the required scalar and vector potentials determining the electromagnetic fields. In appendix B we successfully compare some limits of our results with others found in the literature. Throughout the paper we use Gaussian units.
Ii Electromagnetic response of Weyl semimetals
The low energy physics of a Weyl semimetal with two nodes is described by the linearized Hamiltonian Armitage-Review ()
where is the Fermi velocity and . The operator describes the node degree of freedom, while describes the conduction-valence band degree of freedom. The two Weyl nodes are located at in the BZ and shifted by in energy. A naive analysis reveals that the terms proportional to and b in the Hamiltonian (1) can be gauged away and it reduces to . The chiral transformation , with (and corresponding for ) indeed gauges away the terms and but it also changes the integration measure in the path integral and thus the seeming chiral symmetry of the fermionic field is broken, which is nothing more than the chiral anomaly. This gives rise to an unusual EM response described by an additional term in the action of the electromagnetic field Burkov-TFT ()
where is the fine-structure constant and is the so-called axion field Wilczek (). Topological response of WSMs is thus described by an action similar to that of axion-electrodynamics. It is useful to compare this with the term in the effective action of 3D topological insulators. In that case is the only nonzero value consistent with TR symmetry Qi-TFT (). The EM response of 3D TIs is rather simple, since the only nontrivial physical effect is to generate a half-quantized quantum Hall effect on the sampl’s surfaces. Indeed, a general method to describe the topological magnetoelectric effect in 3D TIs has been elaborated in Refs. MCU-GreenTI (); Martin (); MCU1 (); MCU2 (); MCU4 () by means of Green’s function techniques.
Unlike the term in 3D TIs, Eq. (2) does modify Maxwell equations in the bulk of a Weyl semimetal and thus provides additional observable consequences. The physical manifestation of the Chern-Simons-like term (2) can be best understood from the associated equations of motion. Varying the full action , where is the usual nontopological Maxwell action for electromagnetic fields in matter, we find that the axionic term (2) changes two of the four Maxwell equations, i.e.
with the constitutive relations and . Faraday’s law, , and the equation stating the absence of magnetic monopoles, , remain untouched. Here, and , where is the static permittivity, is the longitudinal conductivity and is the magnetic susceptibility. As in ordinary metals, in the linear response regime, the electric current and the electric field are related by , where the frequency-dependent conductivity tensor can be derived by using, for example, the semiclassical Boltzmann transport theory. Here, we assume that the susceptibility of the WSM is negligible.
The most salient features of Weyl physics are fully contained in the inhomogeneous Maxwell equations (3) and (4). For example, the b-dependent terms encode the anomalous Hall effect, associated with the chiral Fermi arc surface states, and which is expected to occur in a Weyl semimetal with broken TR symmetry. The -dependent term describes the chiral magnetic effect, which is the generation of an electric current driven by an applied magnetic field in the bulk of a Weyl semimetal with broken I. The existence of this CME has been a subject of debate in the condensed matter literature, since a current proportional to the magnetic field with no voltage applied cannot represent an equilibrium current. Therefore, as pointed out in Ref. Burkov2 (), in general, Eq. (4) needs to be supplemented by an equation describing the relaxation of the chiral charge. Moreover, CME vanishes as a static effect in the thermal equilibrium Vazifeh (); YChen () whereas it might exist in the nonequilibrium limit Tewari (); Chang (). For a detailed discussion of the CME see Ref. Burkov-TFT ().
Iii Calculation of the EM fields
iii.1 Statement of the problem
It is well known that an electric charge near the surface of a 3D TI induces a vortex Hall current (because of the in-plane component of the electric field produced by the charge), which generates a magnetic field which resembles that produced by a magnetic monopole Qi-Monopole (). Similar image monopole is predicted when a charge is near the surface of linear magnetoelectric material Dzyaloshinskii (). The problem we shall consider in this paper is that of an electric charge near the surface of a topological Weyl semimetal. Due to the broken symmetries in the bulk, additional nontrivial topological effects may result in a further complication of the electromagnetics as compared to the case of the TIs.
To begin with, let us consider the geometry as shown in Fig. 1. The lower half-space () is occupied by a topological Weyl semimetal with a pair of nodes separated along the -direction in the bulk BZ, while the upper half-space () is occupied by a dielectric fluid. An electric charge is brought near to the surface that does not support Fermi-arc electronic states, which is the -plane for ; see Fig. 1. Being this a static problem, it is appropriate to neglect all frequency dependence to the conductivities, such that the EM response of the WSM is fully captured by Eqs. (3) and (4) with . Since , there is no surface currents, and the result is just a material that is solely a bulk Hall material with current responses given by the transverse Hall conductivity .
For the sake of generality, in section III.2 we solve Maxwell equations (3) and (4) by considering two semi infinite bulk Hall materials, characterized by the parameters for and for , separated by the surface . The inhomogeneity in and is therefore limited to a finite discontinuity across the interface. In Appendix B we verify that our results correctly reproduce the ones reported in Ref. ChiralMatter () for an infinite chiral media, and the well-known electrostatic field produced by a charge near a dielectric medium Schwinger () as well. In the last section III.3 we take the limit , which yields the electromagnetic fields produced by an electric charge in a dielectric fluid above the surface of a WSM.
iii.2 General solution and consistency checks
Since the homogeneous Maxwell equations that relate the potentials to the fields are not modified in the presence of the term, the electrostatic and magnetostatic fields can be written in terms of the scalar and vector A potentials according to and . In the Coulomb gauge , for a pointlike electric charge of strength at with (that is, the charge lies in the medium 2), the electromagnetic potentials satisfy the equations of motion
where is the charge density. To obtain the general solution for the EM potentials, we must solve equations (5) and (6) in the bulk Hall systems and then impose the appropriate boundary conditions. For simplicitly, we work in cylindrical coordinates . Exploiting the axial symmetry of the problem, we introduce the reduced scalar potential through the representation
where and are the momentum and position parallel to the -plane. This integral can be simplified if we express the area element in polar coordinates, , and choose the axis in the direction of the vector . Noting that , the angular integration can be easily performed to obtain
where is the th order Bessel function of the first kind. Inserting this ansatz into the equations of motion and assuming the axial symmetry for the vector potential, we find that it is appropriate to introduce the analogous dimensional representation
which defines the vector potential through . Note that this choice naturally satisfies the Coulomb gauge . The problem now consists in determining the reduced functions and . To this end, we insert the above representations into the equations of motion to obtain the differential equations
where we have expressed the charge density as . Here, we have omitted the dependence of the dielectric function and Hall conductivity on for brevity. The differential equations (10) and (11), along with the appropriate boundary conditions at the interface and at the singular point , constitute a complete boundary value problem. To solve this problem, we employ stardard techniques of electromagnetism Schwinger (). Here, we present only the final results and relegate the details of the technical calculations to Appendix A. We obtain the following expressions for the reduced functions beneath the surface ()
and, above the surface (), we obtain
where we have defined for brevity
and . Here, is the complex wave number in the medium , with
and is an effective bulk Hall conductivity (with dimensions of inverse length). The imaginary part of , , implies that the electromagnetic fields are attenuated in the bulk, as in ordinary metals. The maximum penetration depth in the medium is found to be , which is fully determined by the bulk Hall conductivity and the dielectric constant .
The final expressions for the scalar and vector potentials in coordinate representation are obtained by inserting the reduced functions (12)-(15) into the representations (8)-(9) and computing the -integrals. Now we consider two consistency checks of our results (for technical details see Appendix B). First, we verify that when the two materials are topologically trivial (i.e. with vanishing bulk Hall conductivities), we recover the result obtained for an electric charge in front of a dielectric interface Schwinger (). In this limit we also find out that , as expected, since there is no magnetoelectric effect in the absence of the term. Second, we consider the case in which the electric charge is embedded in an infinite chiral medium (i.e. with the same optical and topological properties in the whole space). In Appendix B we succesfully reproduce the results reported in Ref. ChiralMatter () for the EM fields generated by an electric charge in chiral matter.
iii.3 EM fields induced by a charge near a WSM
Next we analyze the realistic case in which the electric charge is located in a dielectric fluid, above the surface of a topological Weyl semimental, as shown in Fig. 1. This situation is described by the reduced functions (12)-(15) in the limit . Let us first discuss the resulting electric field. Taking in Eqs. (12) and (14) and inserting the result into the representation (8) we find that, in coordinate representation, the electrostatic potential beneath the surface becomes
and, above the surface, we find
We observe that in the dielectric fluid (), the electric potential can be interpreted as due to the original electric charge of strength at , an image electric charge of strength at , and an additional term arising from the nontrivial topology of the WSM. Inside the Weyl semimetal (), the electric potential has not a simple interpretation. In Fig. 2a we plot the electrostatic potential (in units of ) as a function of the dimensionless distance , for and . We take , which is appropriate for the Weyl semimetal EuIrO Sushkov (), and (continuous line) and (dashed line). As anticipated, we observe that the electrostatic potential is attenuated inside the WSM due to the metallic character of the material. In order to compare this behavior with that of ordinary metals, we compute the penetration depth inside the material. To this end, we evaluate numerically the surface potential and estimate the penetration depth . From Fig. 2a we read and , thus implying that
where . If we insert the numerical value into the expression for the penetration depth, we find . This means that if m Casimir/WSM (), then m, which is larger than the typical values of penetration depths in ordinary metals [Stainless steel m, Titanium m, Aluminium m, Silver m, for instance]. From Eq. (18), one can further see that, in the limit , we obtain that and , as in a perfect conductor.
The electric field can be directly computed from the electrostatic potential (18)-(19) as . In Fig. 2b we illustrate the electric field E (in units of ) generated by an electric charge at (red sphere) close to a Weyl semimetal as a function of the dimensionless coordinates and , for and . We observe that the electric field outside the WSM is similar to that generated by the original electric charge, with deviations close to the interface due to the screening of the field inside the material. In fact, the electric field is (in practical terms) indistinguishable from that produced by an electric charge close to a nontopological material (e.g. an ordinary metal or a dielectric) because it is dominated by the dielectric response. Nevertheless, the electric field beneath the surface is more complicated than in the nontopological cases. For example, the electric field within a uniform and isotropic dielectric is a radially directed field (with the charge outside the material as its source); while the field inside an ordinary metal is zero. In the present case, as shown in Fig. 2b, the electric field is remarkably different because of the curved field lines inside.
Now we discuss the induced magnetic field. The vector potential is given by , with the function defined by Eq. (9) and the reduced function given by Eqs. (13) and (15) in the limit . In coordinate representation, the function beneath the surface is
and, above the surface, we obtain
The magnetic field can be obtained from these expressions as . In Fig. 2c we show the magnetic field B (in units of ) induced by an electric charge at close to a Weyl semimetal as a function of the dimensionless coordinates and , for and . Clearly, the field lines do not have a simple form. The magnetic field generated by an electric charge close to a TI should serve as the benchmark for understanding the subtlety of our result. In that case, the monopole magnetic field beneath (above) the surface is radially directed with the magnetic monopole above (beneath) the surface as its origin Qi-Monopole (); Karch (); MCU-GreenTI (). In the present case, however, the behavior of the field lines is radically different. Above the surface, the magnetic field lines begin at the surface and end at the surface (but not at the same point). This behavior is of course different from the radially directed field predicted in topological insulators. The situation beneath the surface is even more complicated. Indeed, it strongly differs from that of the topological insulator. In the last Sec. V we discuss two experimental setups which could be used to test this nontrivial magnetic field.
To understand the physical origin of the induced magnetic field above the surface we rewrite the Maxwell equation (4) as , where the bulk Hall current, given by , is induced by the in-plane component of the electric field produced by the charge. Since we have taken , we find that the current is indeed circulating around the symmetry axis, i.e. . In Fig. 3 we show a stream density plot of the bulk Hall current (in units of ) for different values of and . We observe that each cross section of the bulk Hall current resembles the surface Hall current induced by an electric charge near to a topological insulator. Naively, this suggests that a 3D Weyl semimetallic phase can be understood as an infinite number of 2+1 Dirac subsystems (one for each value of in the bulk) supporting a surface Hall current. Indeed, as we shall discuss in the last section, the simplest material exhibiting Weyl physics consists of a topological insulator-trivial insulator heterostructure, thus supporting our interpretation of the bulk Hall current in terms of 2+1 subsystems.
Iv Interaction energy and force
Now let us compute the force between the electric charge and the Weyl semimetal. The interaction energy between a charge distribution and a WSM is given by Schwinger ()
where is the electrostatic potential in the absence of the term. The first contribution represents the total energy of a charge distribution in the presence of the WSM, including mutual interactions. We evaluate this energy for the problem of an electric charge above a WSM. Making use of Eq. (19), the interaction energy becomes
which we interpret as follows. The first term has a clear physical meaning: it corresponds to the interaction energy between the original charge at and the image charge at Schwinger (). The second term is less clear. An interpretation in a similar fashion is not immediate; however, we are certain that it is a consequence of the nontrivial bulk topology of the material since it vanishes as the bulk Hall conductivity goes to zero. We observe that as the charge approaches the interface (), the nontopological contribution will dominate the behavior of the interaction energy (24) provided ; and therefore , as usual. However, this trivial contribution vanishes for , which is achieved by embedding the charge in a dielectric fluid with the same permittivity to that of the Weyl semimetal. This idea was recently employed in Refs. MU (); MC () to cancel out the trivial electrostatic effects when studying the interaction between an hydrogen-like ion and a planar topological insulator. In the following we concentrate on this case, which is appropriate to isolate the topological effects. A distinguishing feature of this interaction energy is that it does not diverge as the charge approaches the interface. Indeed, we can compute the surface interaction energy analytically, with the result (setting )
This finite value of the interaction energy at the interface is a signature that the electric field cannot be interpreted in terms of a symmetrically located image charge, as in metals, dielectrics and topological insulators. In Fig. 4 we show a plot of the ratio between the interaction energy and the surface energy as a function of the dimensionless distance . We observe that the maximum value is precisely at the surface, and it decreases asymptotically to zero as the charge move away the surface. The force that the Weyl semimetal exerts upon the charge can be computed by taking the derivative of the interaction energy with respect to , i.e. . To get an insight of the magnitude of this force, in the inset of Fig. 4 we plot the force (in units of , which is the force that a perfect metallic surface exerts upon the charge) as a funtion of the dimensionless coordinate . As we can see, the force between the Weyl semimetal and the charge tends asymptotically to the force between the charge and a perfect metallic surface.
V Summary and Discussion
In summary, in this paper we have computed the electromagnetic fields produced by an electric charge near to a topological Weyl semimetal with two nodes in the bulk Brillouin zone (i.e. with broken time-reversal symmetry). We found that, outside the WSM, the electric field behaves as that generated by the original electric charge, with deviations close to the interface due to the screening of the field inside the material (see Fig. 2b). This behavior is of course dominated by the dielectric properties of the semimetal, in such a way that the topological contribution is always hidden. The magnetic field is, on the contrary, of topological origin in nature due to the magnetoelectric effect of topological phases. In particular, we showed that the magnetic field exhibits a particularly interesting behavior above the WSM: the field lines begin at the surface and then end at the surface (but not at the same point), as depicted in Fig. 2c. Clearly, this field has a quite different behavior from that produced by an electric charge near the surface of a topological insulator, where the magnetic field above the surface is generated by an image magnetic monopole beneath the surface. The unconventional behavior of the magnetic field above the surface of the WSM is an experimentally observable signature of the anomalous Hall effect in the bulk, and thus its detection is in order. To this end, in what follows we discuss the simplest candidate materials for testing our predictions, as well as some distinguishing features which hopefully provide the precise route for its detection.
Generic Weyl semimetal phase have more than only two Weyl nodes (but they should have an even number of them because of the Nielsen-Ninomiya theorem NN1 (); NN2 (); NN3 ()). In order to take into account multiple pairs of Weyl nodes, we must use the superposition principle to obtain the corresponding electromagnetic fields. However, this can be a difficult task because the Weyl nodes will not be all aligned with each other and then our results can not be directly applied. This limitation can be partially cured by using materials which are not Weyl semimetals, but where the Weyl nodes appear once time-reversal is broken by an external magnetic field. In this regard, in the Dirac materials CdAs DiracCdAs1 (); DiracCdAs2 (); DiracCdAs3 () and NaBi DiracNaBi1 (); DiracNaBi2 (); DiracNaBi3 (), each Dirac point is expected to split into two Weyl nodes with a separation proportional (in magnitude and direction) to the magnetic field. This gives rise to a WSM phase with multiple pair nodes aligned with each other, but regrettably at different energies.
Perhaps the simplest candidate for testing our predictions is a topological insulator-trivial insulator multilayer heterostructure Burkov-TFT (); Burkov (). The advantages of this model are, on the one hand, its simplicity for experimetal realization and on the other hand, that the Weyl semimetal it realizes contains only two Weyl nodes in the direction of the material growth. This system may be described as a chain of 2D Dirac surface states of the TI layers, which are coupled by tunneling matrix elements between surface states on the same TI layer and between different layers. In the low-energy regime, this model leads to the well-known Weyl Hamiltonian (1) whose electromagnetic response is the one described in this paper. For details of the microscopic model see Refs. Burkov-TFT (); Burkov (). Within this multilayer framework, and knowing that an electric charge near to a TI surface produces a monopolar magnetic field due to the surface Hall effect, the magnetic field induced by a charge near to a Weyl semimetal phase can be qualitatively interpreted in terms of an infinite number of image magnetic monopoles located beneath the surface. The effects produced by the images induced by the images themselves, destroy the characteristic divergent field lines associated with a chain of monopoles, and then the magnetic field will behave in a different fashion, presumably as that depicted in Fig. 2c. Now, let us discuss two specific fingerprints of the induced magnetic field above the Weyl semimetal which, at least in principle, could be measured experimentally.
Angle-resolved measurement. The force that the Weyl semimetal exerts upon the charge is calculated as , where the interaction energy is given by Eq. (24). This force, of course, corresponds to the static part of the Lorentz force, i.e. , where is the electric field above the WSM evaluated at the position of the original charge. Not surprisingly, this force attracts the charge toward the surface in the perpendicular direction to it. However, interesting phenomena appear when we consider the dynamics of the external charge. For example, consider a steady electron beam drifting at a distance above the surface of the WSM. If the motion of the electrons is slow enough with respect to the Fermi velocity in the solid, the induced polarization and magnetization of the material rearranges infinitely fast, in such a way that the solution for the electromagnetic fields we have computed in this paper are still valid. In this case, where the charge is moving with a uniform velocity v above the surface of the WSM, the force acting upon the charge will acquire an additional term of the form due to the induced magnetic field. For an electron beam moving along the -direction (with velocity ) we find
Remarkably, this anomalous force is orthogonal to the electrons motion as well as to the electric contribution . As a result, these effects can be distinguished each other. Experimentally, the required probe can be provided by the steady electron beam emitting from low-energy electron gun (low-energy electron diffraction). While drifting above the WSM, the anomalous force (26) would deflect the trajectory of the electron beam, and this in-plane deflection can be traced by angle-resolved measurement. A similar effect has been recently predicted to occur for an electron beam moving above the surface of a cylindrical topological insulator Martin ().
Scanning SQUID magnetometry. Another possible technique for measuring the induced magnetic field could be scanning SQUID (Superconducting Quantum Interference Device) magnetometry. Roughly speaking, a SQUID is a very sensitive magnetometer used to measure extremely subtle magnetic fields, based on superconducting loops containing Josephson junctions. Technically, we have to compute the magnetic flux through a loop (of radius and parallel to the surface) placed at a distance above the Weyl semimetal, i.e. , where is the surface of the loop. A simple calculation shows us that , where the function is given by Eq. (22). The magnetic flux from a topological insulator through a Josephson junction, , serves as the benchmark for comparing our result. In that case, the magnetic flux grows from 0 (at ) to the constant value (as ), where , with is the magnetic monopole strength MCU-GreenTI (). This is so because the magnetic field is radially directed away from the image magnetic monopole beneath the surface and therefore the loop will always enclose field lines. This interesting tendency of the magnetic flux to a constant value can be thought as a distinctive feature of the induced magnetic field in topological insulators. The case of a Weyl semimetal is quite different, as we shall discuss in the following.
In Fig. 5 we show a plot of the magnetic flux (in units of ) as a function of the dimensionless radius for , and different values of . Of course, at . Furthermore, in the limit , the function (22) is a highly oscillatory integral and therefore , as shown in Fig. 5. This behavior implies the existence of a maximum flux at a critical radius . The fact that the magnetic flux tends to zero as the radius goes to infinity can be easily understood from the fact that the magnetic field lines, which start at the WSM surface, go back again to the surface, as discussed before. The existence of the maximum, as well as the asymptotic vanishing of the flux, are distinguishing features of this problem which could serve as a possible experimental test. One of the key challenges for the experimental detection of this flux profile would be to find a way to fix and localize the charge above the surface. We finally point out that the strength of the magnetic flux generated by an electric charge above a Weyl semimetal, , is larger than the flux from a topological insulator by a factor of . Therefore, the detection of the magnetoelectric effect of topological phases is in principle easier in a Weyl semimetal than with TIs.
Acknowledgements.We thank Alberto Cortijo for useful comments and suggestions. A. M. was supported by the CONACyT postdoctoral Grant No. 234774. L.F.U. has been supported in part by the project CONACyT (México) # 237503. M.C. has been partially supported by UNAB DGID under grant # DI-33-17/RG.
Appendix A Detailed solution
In this section we present the detailed solution of the equations of motion Eqs. (10) and (11) in the general case where two bulk Hall materials are in contact, as described in the main text. To this end, we first derive the corresponding boundary conditions at the interface and at the singular point ; and next we use standard electromagnetic techniques to obtain the general solution in the whole space.
Boundary Conditions. The boundary conditions for and can be determined as usual. Assuming that the reduced functions are bounded when is in the infinitesimal neighborhood of , integration of Eqs. (10) and (11) over the interval between and with , yields the continuity of and there. Then, the continuity of and at follows. In a similar fashion, one can show that the singularity in Eq. (10) requires, at , that be continuous, while be discontinuous, i.e. . Analogously, integrating twice Eq. (11) yields the continuity of and at the singular point.
General solution. The solutions to equations (10) and (11) subject to the above discussed boundary conditions can be expressed in terms of the solutions, , , and , of the corresponding homogeneous equations. Here, labels the two media and is the complex wave number, with and given by Eq. (17).
Now we proceed to compute the reduced functions and . To this end, we first partition the -axis in the three regions: (I) , (II) and (III) . Next, we write an appropriate linear combination of the solutions to the homogeneous equation for each region, and finally we apply the corresponding boundary conditions. On the one hand, for the reduced scalar potential , the forms of the solutions in the three regions are as follows:
where the signs in the exponentials (27) and (29) are required by the boundary condition that goes to zero for . On the other hand, we observe that Eq. (11) dictates the relation between and ; namely, for and for . Using this result we find that eq. (27) implies that, for the region I, . In a similar fashion we obtain the corresponding expressions for and .
Imposing the boundary conditions and solving the resulting system of equations we find for the coefficients
Using these results, we can write the reduced functions beneath the surface as and , whose explicit forms are