Macroscopic degeneracy of zero-mode rotating surface states in 3D Dirac and Weyl semimetals under radiation
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Macroscopic degeneracy of zero-mode rotating surface states in 3D Dirac and Weylsemimetals under radiation
Jos´e Gonz´alez and Rafael A. Molina
Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, Madrid 28006, Spain
We investigate the development of novel surface states when 3D Dirac or Weyl semimetals areplaced under circularly polarized electromagnetic radiation. We find that the hybridization betweeninverted Floquet bands opens in general a gap, which closes at so-called exceptional points foundfor complex values of the momentum. This corresponds to the appearance of midgap surface statesin the form of evanescent waves decaying from the surface exposed to the radiation. We observea phenomenon reminiscent of Landau quantization by which the midgap surface states get a largedegeneracy proportional to the radiation flux traversing the surface of the semimetal. We showthat all these surface states carry angular current, leading to an angular modulation of their chargethat rotates with the same frequency of the radiation, which should manifest in the observation ofa macroscopic chiral current in the irradiated surface.
Introduction.—
In recent years we have witnessed thediscovery of several types of materials characterized byhaving electron quasiparticles with linear momentumdispersion. Graphene was certainly the first of thosematerials[1], but afterwards we learned about the topo-logical insulators[2, 3], to end up more recently with theinvestigation of 3D semimetals whose low-energy excita-tions behave as Dirac[4–6] or Weyl fermions[7, 8].These materials have attracted a lot of attention fortheir potential to develop a new type of electronic trans-port without dissipation. The key idea is that of topo-logical protection, which has its precedent in the edgestates of the quantum Hall effect. The surface states inthe novel materials may also have a well-defined chirality,protecting them against backscattering. Both propertiesare fundamental for revolutionary applications in spin-tronics and fault-tolerant quantum computation [2, 3].In this search, the interaction between light and mat-ter can play an important role, since the electromagneticradiation may be a versatile resource to change and con-trol topological states of matter. In 2D semimetals, itcan open a gap in the bulk, leading to chiral currents atthe boundary of the electron system[9–18], as observedon the surface of 3D topological insulators[19]. The ef-fect of the radiation has been also investigated in thecase of 3D Dirac and Weyl semimetals, focusing on bulkproperties[20–22].In this paper, we investigate the development of novelsurface states when a 3D Dirac or Weyl semimetal isplaced under circularly polarized electromagnetic radia-tion. We will show that such states are intimately relatedto avoided crossings at the gap that opens up from thehybridization of inverted Floquet bands, as representedin Fig. 1. The gap closes at so-called exceptional points(EPs) [23, 24], which are celebrated in the context ofnon-Hermitian Hamiltonians and appear here for com-plex values of the momentum describing the evanescenceinto the 3D semimetal. The stability of the novel sur-face states is then guaranteed by a new mechanism oftopological protection, relying on the fact that each EP comes as a branch point in the spectrum which cannotbe removed by small perturbations.The genuine feature of the novel surface states is thatthey all carry a significant angular current, with the samechirality of the photon polarization and the frequency ofthe radiation. Such states may prove especially useful inthe current drive towards increasing the frequency limitsof electronic devices [25].
FIG. 1. Left: Schematic representation of the first Floquetbands with quasi-energy ǫ and ǫ ± ~ Ω (Ω being the radia-tion frequency) for a model of 3D semimetal with two nodesalong the momentum axis k z (red color denotes positive en-ergy while light blue denotes negative energy, the directiontransverse to k z represents additional components of the mo-mentum). Right: Geometry of 3D semimetal with the surfaceexposed to the radiation where the evanescent states appear. Model and Floquet theory.—
A low-energy Hamilto-nian for a 3D Dirac semimetal around the Brillouin zonecenter, considering terms up to quadratic order in thequasimomentum k , can be written as [26, 27]. H = ǫ ( k ) I + M ( k ) σ z + ~ v ( ζk x σ x − k y σ y ) (1) ǫ ( k ) = c + c k z + c ( k x + k y ) (2) M ( k ) = m − m k z − m ( k x + k y ) , (3)where I is the 2 × σ i , i = x, y, z , arethe Pauli matrices, and ζ = ± E ± = ǫ ( k ) ± q M ( k ) + ~ v ( k x + k y ) . (4)With the parameters m , m , m < k c =(0 , , ± p m /m ). Ignoring the part proportional to theunit matrix ǫ ( k ), we can expand the Hamiltonian lin-early around each Dirac point to obtain a model for 3Dmassless Dirac fermions with anisotropic linear disper-sion E ( k ) = ± q ~ v ( k x + k y ) + 4 m m ( k z − k c,z ) .Consider illumination by circularly polarized off-resonant light of frequency Ω and field amplitude F . Inthe case of polarization in the x − y plane, light pro-duces the vector potential in the dipolar approximation A ( t ) = A ( η sin Ω t, cos Ω t, η = ± A = F / Ω. We make the Peierls substitution k → k + A ( t )and use Floquet theory in order to compute the bandstructure in the presence of the radiation field[28]. So-lutions of the time-dependent Schr¨odinger equation inthe case of time-periodic Hamiltonians have the form | Ψ( t ) i = e − iǫt/ ~ | Φ( t ) i , with a conserved quantity, thequasi-energy ǫ , playing a similar role to the energy inthe time-independent Sch¨odinger equation. The Floquetstates | Φ( t ) i are periodic in time with the same periodas the Hamiltonian and they can be developed in Fourierseries, | Φ( t ) i = P m e − im Ω t | u mα i . This transforms thetime-dependent Schr¨odinger equation into X n H mn | u nα i = ( ǫ α + m ~ Ω) | u mα i , (5)where the Floquet Hamiltonian matrix elements are givenby H mn = T R T dtH ( t ) e i ( m − n )Ω t , T = 2 π/ Ω being theperiod of the time-dependent Hamiltonian[28].As a relevant example, we perform calculations in aninfinite wire in the z direction with a finite square sec-tion in the x − y plane. Using the Floquet formalism,we calculate the band structure as a function of k z dis-cretizing the model Hamiltonian with a standard tight-binding regularization including the Floquet subbands.Results with parameters of the model in the topologi-cal Dirac semimetal regime are shown in Fig. 2, wherewe compare the energy of the bulk bands (left) with thequasi-energy of the bands in the wire (right) as a functionof k z . The structure of the bands wrapping up in theFloquet-Brillouin zone is very apparent. In the middleof the Floquet-Brillouin zone at ǫ = 0, a gap opens be-tween states with Floquet modes n = − n = 1(blue), the hybridization being a second order effect lead-ing to a quite small gap, proportional to A . The muchbigger gap (proportional to A ) between modes n = 0(green) and n = 1 is in one of the edges of the Floquet-Brillouin zone at ǫ = ~ Ω /
2, with a mirror structure (notshown) at ǫ = − ~ Ω / n = 0 and n = − E ( e V ) k z (Å −1 ) 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 E ( e V ) k z (Å −1 ) FIG. 2. Band structure for model parameters c = c = c =0 . m = − . m = − .
605 eV ˚A , m = − . , ~ v = 1 . ~ Ω = 0 . A = 0 .
05 ˚A − , in the bulk (leftpanel) and for a wire of section S = 150 ×
150 ˚A in the x − y plane (right panel). The position of the Dirac cone at zerofield is k c,z = p m /m = 0 .
909 ˚A − . The color representsthe predominant Floquet mode of the particular state in aRGB coding where the intensity of each mode is mapped tothe intensity of each fundamental color, n = − , , k z of Im( k z ) = 0 .
01 ˚A − is shown with black circles with a dot.A few high-energy bands with n = − In order to explore the possibility of evanescent statesin the z direction, we solve the Schr¨odinger equation inthe same geometry but including a finite imaginary partin k z . The Hamiltonian becomes then non-Hermitianand the eigenenergies acquire a finite imaginary part.These are obviously non-physical solutions of the prob-lem. However, non-Hermitian Hamiltonians may haveeigenenergies that are strictly real, in the form of EPs[23, 24]. These correspond to physical evanescent solu-tions with the imaginary part of the momentum givingthe decay length of a wave in a semi-infinite geometry.The black circles in Fig. 2 mark the presence of theevanescent states in the gaps induced by the radiation.Not all the avoided crossings present evanescent states,that depends on the properties of the actual bands in-volved. However, the number of avoided crossings is pro-portional to A and to the area of the section of the wire.This may imply in general a large degeneracy of evanes-cent states, whose origin is clarified in the continuumlimit discussed in what follows. Quantization and localization of zero-mode surfacestates.—
In order to unveil the properties of the novelsurface states, we focus now on the low-energy physicsaround any of the nodes of the Hamiltonian (1), tak-ing moreover the projection for a given chirality ζ . Thisamounts to linearize M ( k ) about the node at k = k c ,which leads in real space to the Hamiltonian for a singleWeyl quasiparticle H W = − i ~ ( vσ x ∂ x + vσ y ∂ y + v z σ z ∂ z )+ ~ v A ( σ x cos(Ω t ) + σ y sin(Ω t )) (6)with a velocity v z = v along the z -axis[29]. H W can be translated into a time-independent Hamil-tonian after applying the unitary transformation U = e − iJ z Ω t/ ~ , with the projection of the total angular mo-mentum J z = − i ~ x∂ y + i ~ y∂ x + ~ σ z / e H W = U † H W U − i ~ U † ∂ t U = − i ~ ( vσ x ∂ x + vσ y ∂ y + v z σ z ∂ z ) − Ω J z + ~ v A σ x (7)Each eigenvector χ of (7) corresponds then to a solutionΨ( t ) of the original time-dependent Schr¨odinger equa-tion, given by Ψ( t ) = e − iJ z Ω t/ ~ e − iεt/ ~ χ ( ε being theeigenvalue of (7)). This shows that the eigenvalues j z of the projection J z can be used in this approach to labelthe different side bands arising from the irradiation.After applying the gauge transformation P =exp( − i A x ) to (7), the Hamiltonian becomes (in polarcoordinates r, θ and with σ ± = ( σ x ± iσ y ) / e H ′ W = − i ~ v X s = ± e − siθ (cid:18) ∂ r − si r ∂ θ (cid:19) σ s − i ~ v z σ z ∂ z − ~ Ω (cid:16) − i∂ θ + σ z (cid:17) − ~ Ω A r sin( θ ) (8)The last term in (8) is responsible for the coupling be-tween states with different angular momenta j z . Then,the avoided crossing of the bands with j z = ± ~ / ǫ = ~ Ω / U has the effect of shifting the energy of the side bands inthe present approach by half-integer multiples of ~ Ω.The development of the gap can be captured (for nottoo large amplitude A ) by solving in the space spannedby a linear combination of spinors with projection of thetotal angular momentum j z = ± ~ / χ = (cid:18) φ ( r ) e iθ φ ( r ) (cid:19) e ik z z + (cid:18) e − iθ φ ( r ) φ ( r ) (cid:19) e ik z z (9)We have considered specifically a cylindrical geometrywith the section at z = 0 exposed to the radiation, asrepresented in Fig. 1 [31]. A typical shape of the gapis shown in Fig. 3(a), where we see that it has severaloscillations before recovering the linear dispersion of theoriginal cones beyond a certain k z . This is indeed thegeneric behavior, with an increasing number of oscilla-tions as the radius R of the cylinder grows.The origin of the oscillations in the gap is clarifiedby noting that they arise from the existence of EPs inthe spectrum of the Hamiltonian (8), corresponding tovalues of k z inside the complex plane where the gapcloses. Around each oscillation of the gap, the low-est eigenvalue ε behaves as a function of complex k z as ε ∼ √ k i − k z p k i − k z , with EPs at complex conjugatemomenta k i , k i [28]. Evanescent states with Im( ε ) = 0exist then in the segment | Im( k z ) | ≤ | Im( k i ) | , for each - k z ¶ - - H k z L I m H ¶ L (a) (b)FIG. 3. (a) Plot of the gap of the Hamiltonian (8) (in eV)as a function of real k z (measured in units of the inverse ofthe typical microscopic length scale a in the material) for ~ v/a = 1 . ~ Ω = 0 . A = 0 . a − , and radius R = 200 a of the cylindrical geometry considered in the text.(b) Imaginary part of the lowest eigenvalue of the Hamiltonian(8) (in eV) as a function of Re( k z ), for complex momenta withIm( k z ) = 0 . a − and the same parameters as in (a). pair k i , k i . We have represented for instance in Fig. 3(b)the behavior of the imaginary part of the lowest eigen-value, when Im( k z ) = 0, as a function of the real partof k z . We observe the recurrent development of com-plex momenta at which the eigenvalue becomes purelyreal, leading to a set of evanescent eigenstates in perfectcorrespondence with the minima of the gap in Fig. 3(a).We find then that the evanescent states are preservedby a mechanism of topological protection, as the branchcuts cannot be undone in the complex plane unless thebranch points coalesce in pairs. It can be seen that theimaginary part of the EPs has a very smooth dependenceon the frequency Ω, while it grows linearly with the am-plitude A [32]. The number of zero-mode evanescentstates may be actually characterized from the numberof branch points that the lowest band ε ( k z ) has in thecomplex plane.It can be checked that the number of EPs increasesas the radius R grows, leading to a definite pattern ofquantization in the surface of our geometry. It can beseen that the order of each zero in the plot of Fig. 3(b)(from right to left) gives also the extent of the localiza-tion that the corresponding evanescent wave has in theradial direction, as illustrated in Fig. 4. The peak inthe probability distribution shifts to lower values of r asRe( k z ) decreases, reaching eventually the innermost re-gion of the surface. The number of evanescent states,besides growing linearly with the frequency Ω and theamplitude A , becomes actually proportional to the areaof the irradiated surface, at a rate of roughly one stateper (100 a ) (for ~ Ω = 0 . A = 0 . a − , a being atypical microscopic length scale in the material).Furthermore, the midgap evanescent states evolve intime with a rotation of their charge along the angularvariable θ . This can be shown by computing the an- (cid:144) r Χ ¤ FIG. 4. Left: Probability distribution r | χ | along the radialdirection for evanescent states corresponding to three decreas-ing values of Re( k z ) with Im( ε ) = 0 (starting with the outer-most evanescent state) for the same parameters as in Fig. 3and R = 400 a . Right: Time-dependent angular current rj θ Ω for the states in the upper part (top) and the middle part(bottom) of the left panel, represented for fixed time t = 0 atthe irradiated surface of the cylindrical geometry. gular component j θ of the probability current for theHamiltonian (6). This current has a static contribution, j θ static = − i ( v/r )( φ ∗ φ + φ ∗ φ )+h . c . , which leads to a verysmall intensity when integrated over the radial direction,as a consequence of the tendency of the two contribu-tions from j z = ± ~ / ~ Ω = 0 . A = 0 . a − , we get for instance val-ues of the intensity R dr rj θ static between ∼ − v/a and ∼ − v/a , when computing for evanescent states fromthe outermost to the innermost region of the surface.However, there is also a non-negligible time-dependentcontribution to j θ [28], given by j θ Ω = − i ( v/r )( φ ∗ φ e − i ( θ − Ω t ) + φ ∗ φ e i ( θ − Ω t ) ) + h . c . (10)which has a periodic dependence on θ − Ω t as shown inFig. 4 [33]. The intensity corresponding to j θ Ω has max-ima (in the angular variable) which turn out to be ingeneral about two orders of magnitude above the inten-sity obtained from j θ static , for every evanescent state[28].When introduced in the continuity equation, the formof j θ Ω implies that the charge of each state must have aperiodic modulation in the angular variable, and that itmust rotate with frequency Ω along the concentric ringswhere each evanescent state is confined[28]. From a prac-tical point of view, this leads to the formation of rotatingdipoles on the surface of the system, whose movement canbe controlled by tuning the parameters of the radiation.While we have referred here to the hybridization ofstates with j z = ± ~ /
2, the coupling of side bands withhigher values of J z results in surface states with similarproperties of localization and time evolution[28]. It canbe seen in particular that the hybridization of side bandswith values of j z differing by 2 ~ leads also to evanescentstates, which correspond in that case to the EPs foundat ǫ = 0 in the preceding section. Conclusion.—
An important practical consideration inthis study is that the infrared radiation may penetratesufficiently deep into the 3D semimetals, given the lim-ited screening in these materials. The penetration length l can be estimated as the inverse of the absorption co-efficient α , which is expressed in terms of the dielectricfunction ǫ (Ω) as α = 2ΩIm p ǫ (Ω) /c . We find for instancethat l ∼ µm in the near-infrared (Ω ∼
100 THz)[28],which is a large enough distance to afford the develop-ment of the evanescent states.The magnitude of the component j θ Ω of the currentoffers good perspectives to observe experimentally thenovel surface states. In our cylindrical geometry, the in-tensity of the current across the radial direction, I = e R dr rj θ Ω , gets maxima (in the angular variable) thatrange between ∼ − µA and ∼ µA , for individualstates taken from the outermost to the innermost regionin the top surface. The total intensity could be enhancedby a large additional factor, in a device able to measurethe contribution of a significant part of the surface states.A suitable experimental setup (i.e. two electrodes ontop of the surface of the semimetal) may be able to con-vert the rotation of the charge in the surface states intoan electrical current, if the device is made to work asa rectenna. This may benefit from recent developmentswhich make possible to rectify currents oscillating evenat the frequency of visible light[34]. In our case, it maygreatly help the fact that the rotation of all the surfacestates is synchronized with that of the radiation fields,making easier the observation of the macroscopic chiralcurrent that may develop at the irradiated surface.We acknowledge financial support from MICINN(Spain) through grant No. FIS2011-23713 and MINECO(Spain) through grants No. FIS2012-34479 and FIS2014-57432-P. [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov,Electric field effect in atomically thin carbon films, Sci-ence , 666 (2004).[2] M. Z. Hasan and C. L. Kane, Topological insulators, Rev.Mod. Phys. , 3045 (2010).[3] X.-L. Qi and S.-C. Zhang, Topological insulators and su-perconductors, Rev. Mod. Phys. , 1057 (2011).[4] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng,D. Prabhakaran, S.-K. 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B , 125427 (2013).[28] See supplemental material.[29] We set in what follows v z = 0 . v , taking as a guide therelative magnitudes observed in real 3D semimetals.[30] A. L´opez, A. Scholz, B. Santos, and J. Schliemann, Pho-toinduced pseudospin effects in silicene beyond the off-resonant condition, Phys. Rev. B , 125105 (2015).[31] We note that, in the semi-infinite geometry, two modeswith momenta k z and − k z are to be superposed, in orderto build each eigenstate vanishing at the top boundaryof the cylinder.[32] A typical magnitude of the imaginary part of the EPsis for instance Im( k i ) ∼ . a − , for ~ Ω = 0 . A = 0 . a − , where a stands for a typical microscopiclength scale in the material.[33] Our definition of the current in Eq. (10) (as well as thatof j θ static ) omits for simplicity its decay along the z -axis,which is inherited from the evanescence of the surfacestates.[34] A. Sharma, V. Singh, T. L. Bougher, and B. A. Cola, Acarbon nanotube optical rectenna, Nature Nanotech. ,1027 (2015). SUPPLEMENTAL MATERIALPenetration depth of the electromagnetic radiation in 3D Dirac and Weyl semimetals
We analyze here the penetration depth of the electromagnetic radiation in a 3D semimetal, focusing on absorptiondue to interband transitions between available electronic states. We start by noting that the refractive index n of thematerial can be expressed in terms of the dielectric function ǫ ( ω ) as n = p ǫ ( ω ) (11)According to this expression, n may get in general an imaginary part, which accounts for the absorption of theelectromagnetic radiation by the electron system.A measure of the exponential decay of the radiation energy in the material is given by the absorption coefficient α ,from which the penetration length l can be estimated as ∼ /α . The absorption coefficient can be obtained from therefractive index as α ( ω ) = (2 /c ) ω Im n ( ω ) (12)In the 3D semimetals, the imaginary part of the dielectric function ǫ ( ω ) has a smooth dependence on frequency, sothat the leading dependence of α on ω is linear. More specifically, the dielectric function in a 3D Dirac semimetalwith N Dirac points (or, equivalently, in a 3D Weyl semimetal with N pairs of Weyl points) is given at low energiesby ǫ ( ω ) = N e π ~ v log (cid:18) Λ | ω | (cid:19) + i N e π ~ v (13)where Λ is a high-frequency cutoff (related to the high-energy cutoff in the linear electronic dispersion) and v is theFermi velocity (assuming for simplicity that the dispersion is isotropic in all directions).Taking typical parameters for known 3D semimetals and keeping N ∼
1, we get from Eq. (12) a penetration length l ∼
100 nm for visible light, and l ∼ µ m in the near-infrared (with ω = 100 THz). This shows that the use ofinfrared radiation may become specially convenient, for the sake of observing the interaction between electromagneticradiation and the 3D semimetal proposed in the main text. High-frequency effective Hamiltonian for the 3D Dirac semimetal in the presence of circularlypolarized radiation
Using the following equations H mn = 1 T Z T dtH ( t ) e i ( m − n )Ω t , (14)we can compute the Floquet matrix elements of the model for a 3D Dirac semimetal. In our model the only non-zeromatrix-elements are H nn , H +1 = H n +1 n , H − = H n − n . The matrix elements H +2 = H n +2 n , H − = H n − n are zerodue to cancellations coming from the shape of the circularly polarized pulses. It is then, easy to see that the diagonalpart of the Floquet Hamiltonian is: H nn = H + ( n ~ Ω + c A ) I − m A σ z , (15)This is equivalent to a renormalization of the c and m parameters in the Hamiltonian H , which become c + c A and m − m A respectively. The non-diagonal parts are H +1 = i ~ v A ηζσ x − ~ v A σ y + [( c − m σ z ) i A ηk x + ( c − m σ z ) A k y ] . (16) H − = − i ~ v A ηζσ x − ~ v A σ y + [( m σ z − c ) i A ηk x + ( c − m σ z ) A k y ] . (17)The effective high-frequency Hamiltonian when only one-photon processes are taken into account can be written as H eff = H + [ H − , H +1 ] ~ Ω . (18)For the model of a 3D Dirac semimetal used in the paper: H eff = H − η ~ v A Ω σ z − m η v A Ω ( k x σ x − k y σ y ) . (19)The terms proportional to σ z change the effective mass terms so the new effective m is m eff0 = m − m A − η ~ v A Ω . (20)The terms proportional to σ x ( y ) renormalize the parameter vv eff = v − m η v A ~ Ω . (21)The Dirac crossing points are displaced decreasing or increasing the distance between them depending on thepolarization of the external field η . Choosing the parameters of the external field wisely, we can switch the bandinversion on or off by changing the sign of m eff0 . Although the high-frequency approximation fails to properly describethe details of the band structure except for very high-frequency fields or for very small amplitudes, it gives us goodestimates of the parameters needed for switching the band inversion. Examples of band structures in the first Floquet-Brillouin zone (quasi-energy ǫ ∈ [ − ~ Ω / , ~ Ω / k x , k y = 0. In the left panel, the original value of m is negative and there are two Dirac points. As we increasethe amplitude of the external field with right circular polarization η = 1 and ~ Ω = 0 . A = 0 . , . − there are four Dirac points. For A = 0 . − the four Dirac points havedisappeared and gaps have opened in symmetric positions around the origin. For A = 0 . − the two Dirac pointsreappear. In the right panel we show the case with positive value of m and how the Dirac points are switched onand off again as we increase A . If we would have η = −
1, the Dirac points would get closer to each other and tothe origin as we increase A , until they cancel each other at the origin and a gap opens. Note that the same bandstructure, including the Dirac nodes, is repeated periodically as a function of the quasi-energy with period ~ Ω. Alsofurther nodes appear at higher values of | k z | (not shown in the figure) corresponding to band crossings with Floquetindices | n | > FIG. 5. Band structure in the first Floquet-Brillouin zone as a function of k z with k x = k y = 0. The Hamiltonian parametersare c = c = c = 0 . , m = − . m = 0 . m = m = 1 . , ~ v = 1 . ζ = 1, and the externalfield parameters η = 1, ~ Ω = 0 . A indicated in the legend from 0 . . − . In the figure we haveeliminated contributions from higher bands at large values of k z for clarity of presentation. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 k z ( Å -1 ) -0.2-0.100.10.2 E ( e V ) -1.2 -0.8 -0.4 0 0.4 0.8 1.2 k z ( Å -1 ) -0.2-0.100.10.2 E ( e V ) -1.2 -0.8 -0.4 0 0.4 0.8 1.2 k z ( Å -1 ) -0.2-0.100.10.2 E ( e V ) FIG. 6. Band structure for a slab of width W = 100 ˚A in the x direction and infinite in both the y and the z direction as afunction of k z and with k y = 0. The Hamiltonian parameters are c = c = c = 0 . m = − . m = m = 1 . , ~ v = 1 . ζ = 1, and the external field parameters η = 1, ~ Ω = 0 . A = 0 . . − (top right), 0 . − (bottom left). In the right bottom plot, total electron density for states with k z = 0 for the different values of A showingthe exponential decay from the surface. Results for a finite slab
To explore the evolution of the midgap Fermi arc states as the external field is switched on, we perform calculationsin a slab of material with finite width W in the x direction but infinite in the y and z directions. We discretizethe model Hamiltonian with a standard tight-binding regularization including the Floquet subbands. To make theanalysis of the results easier, we present calculations including only one-photon processes, an approximation valid forsmall values of A and small values of k z . A more complete analysis of the Full Floquet results is outside the scopeof this work and will be presented elsewhere. In Fig. 6 we show different results of the band structure for a slab ofmaterial with the same parameters as the left panel of Fig. 5 with A = 0 . , . , . − . We see a band structureresembling the bulk bands calculated previously with midgap states in between the Dirac points. When there is nofield, the midgap states form the Fermi arc typical of Weyl semimetals. States in the Fermi arc are surface statesas shown in the right bottom panel of the figure. Without external field, there are two degenerate Fermi arc surfacestates, one in each surface. When the field is switched on but we still keep the Dirac points, the Fermi arc states loosethe degeneracy between the different surfaces (case with A = 0 . − ). For higher fields, when a gap is opened in theDirac cones, the surface states are not topologically protected anymore (bottom left panel with A = 0 . − ). In thebottom left panel we show the total electronic density as a function of the x coordinate for the states with k z = 0 forthe three values of A shown previously. There is an exponential decay of the surface states in the Fermi arcs whenthere is no field and similarly for small values of the field A = 0 . − . For larger fields A = 0 . − , the spatialdecay from the surface is very much reduced as the gap is opened in the Dirac cones and the topological protectionlost. Signatures of the exceptional points in the complex plane k z As mentioned in the main text, the exceptional points in the spectrum of the Hamiltonian e H W come in complexconjugate pairs k i , k i in the complex plane of the momentum k z . It may be actually seen that the lowest eigenvalue ε has along the line connecting each of these pairs the complex structure ε ( k z ) ∼ p k i − k z q k i − k z (22)The square root behavior means that k i and k i are branch points and that, for each pair, there are two differentbranches of the Riemann sheet going across the real axis of the complex plane. If we take for instance the lineRe( k z ) = Re( k i ), we get from (22) ε (Re( k i ) + iy ) ∼ p i Im( k i ) − iy p − i Im( k i ) − iy = p (Im( k i )) − y (23)This implies that there must be a segment of points connecting k i and k i (corresponding to | y | < | Im( k i ) | ) wherethe eigenvalue ε becomes purely real, with two possible opposite values from the two branches in the complex plane.On the other hand, when | Im( k z ) | > | Im( k i ) | , the above expression also implies the existence of a tail starting from k i (and another from k i ) where the real part of ε vanishes, while the imaginary part gets opposite values in the twodifferent branches.These features can be observed in the results obtained for the lowest eigenvalue ε after numerical diagonalizationof the Hamiltonian, showing the existence of the branch cuts in the spectrum. We have represented for instance inFig. 7 the absolute values of the real and imaginary parts of the eigenvalue ε for radiation parameters ~ Ω = 0 . A = 0 . a − , and radius R = 100 a ( a standing for a microscopic length scale in the material). One can clearlyobserve the signatures of four pairs of exceptional points for Re( k z ) >
0, which follow to a good approximation thebehavior expected from (22).
FIG. 7. Contour plots of the absolute values of the real part (left) and the imaginary part (right) of the lowest eigenvalue ε asa function of the momentum k z in the complex plane, from values close to zero drawn in dark blue and larger absolute valuesshown in light blue. The parameters taken for the radiation are ~ Ω = 0 . A = 0 . a − , and the radius of the cylindricalgeometry considered in the main text is R = 100 a (where a stands for a typical microscopic length scale in the material). As pointed out in the main text, the number of exceptional points grows linearly with the amplitude A of thevector potential as well as with the area of the surface exposed to the radiation. It can be seen in any case that eachpair of complex conjugate exceptional points is tied to the same kind of complex structure, which guarantees theirrobustness as no small perturbation can unfold the topology of the underlying Riemann surface. Evaluation of the angular component of the current and time evolution of the probability density
We review here the way in which the probability current j of the evanescent states can be computed from the spinorrepresentation χ defined in the main text. The formal expression of the current j can be found multiplying first the0evolution equation by the complex conjugate of the original Weyl spinor ψ , which leads to iψ † ∂ t ψ = ψ † [ − ivσ x ∂ x − ivσ y ∂ y − iv z σ z ∂ z + v A ( σ x cos(Ω t ) + σ y sin(Ω t ))] ψ (24)We can take then the complex conjugate of (24), summing the pair of expressions thus obtained to end up with thecontinuity equation ∂ t (cid:0) ψ † ψ (cid:1) + v∂ x (cid:0) ψ † σ x ψ (cid:1) + v∂ y (cid:0) ψ † σ y ψ (cid:1) + v z ∂ z (cid:0) ψ † σ z ψ (cid:1) = 0 (25)From this expression, we can read the current vector in cartesian coordinates j = ( vψ † σ x ψ, vψ † σ y ψ, v z ψ † σ z ψ ) (26)We are interested in particular in the angular component of the current (26), since we want to compute it forthe evanescent states which are confined to circular rings on the surface exposed to the radiation. Passing to polarcoordinates ( r, θ ), this angular component turns out to be j θ = − sin( θ ) r vψ † σ x ψ + cos( θ ) r vψ † σ y ψ (27)Our goal is to compute j θ from the expression of the eigenstates obtained in the spinor representation χ . To performthis operation, we must bear in mind that the spinors χ are related to the original Weyl fermions ψ by the consecutiveaction of the unitary transformations U and P defined in the main text, so that ψ = e − i (cid:16) − i∂ θ + 12 σ z (cid:17) Ω t e − i A r cos( θ ) χ (28)For the sake of facilitating the analytic evaluation of the current, we may assume at this point the condition A r ≪ j θ = − sin( θ ) r v (cid:0) e − Ω t∂ θ χ † (cid:1) e i σ z Ω t σ x e − i σ z Ω t (cid:0) e − Ω t∂ θ χ (cid:1) + cos( θ ) r v (cid:0) e − Ω t∂ θ χ † (cid:1) e i σ z Ω t σ y e − i σ z Ω t (cid:0) e − Ω t∂ θ χ (cid:1) = − sin( θ ) r v (cid:0) e − Ω t∂ θ χ † (cid:1) ( σ x cos(Ω t ) − σ y sin(Ω t )) (cid:0) e − Ω t∂ θ χ (cid:1) + cos( θ ) r v (cid:0) e − Ω t∂ θ χ † (cid:1) ( σ x sin(Ω t ) + σ y cos(Ω t )) (cid:0) e − Ω t∂ θ χ (cid:1) = − i vr (cid:16) e − i ( θ − Ω t ) (cid:0) e − Ω t∂ θ χ ∗ (cid:1) (cid:0) e − Ω t∂ θ χ (cid:1) − e i ( θ − Ω t ) (cid:0) e − Ω t∂ θ χ ∗ (cid:1) (cid:0) e − Ω t∂ θ χ (cid:1)(cid:17) (29)We may now work out the expression of the current for a spinor χ representing the hybridization between stateswith projection of the total angular momentum j z = ~ / − ~ / k z -dependencefor simplicity) χ ∼ (cid:18) φ ( r ) e iθ φ ( r ) (cid:19) + (cid:18) e − iθ φ ( r ) φ ( r ) (cid:19) (30)It can be seen that, if we compute j θ for each spinor contribution with well-defined j , the periodic time dependenceis canceled out in the expression (29). If instead we carry out the full calculation allowing for the mixing of the twospinor contributions, we get a piece in j θ that keeps the periodic time dependence as in Eq. (29). We obtain theresult j θ = j θ static + j θ Ω (31)with j θ static = − i vr ( φ ∗ φ − φ ∗ φ + φ ∗ φ − φ ∗ φ ) (32)and j θ Ω = − i vr ( e − i ( θ − Ω t ) φ ∗ φ − e i ( θ − Ω t ) φ ∗ φ + e i ( θ − Ω t ) φ ∗ φ − e − i ( θ − Ω t ) φ ∗ φ ) (33)1The static component j θ static is made of the contributions from the two spinors with j z = ~ / j z = − ~ /
2, whichtend to cancel each other when integrating the current to obtain the intensity across the radial direction. The twodifferent contributions are represented for instance in Fig. 8 for a typical evanescent state at the outermost region ofthe surface exposed to the radiation. The cancellation operating in the integral R dr rj θ static leads in general to rathersmall values of the intensity associated to j θ static . This does not apply however to the time-dependent component j θ Ω ,as shown in Fig. 8. The intensity obtained from this current turns out to have maxima (in the angular variable)that are about two orders of magnitude larger than that from j θ static , providing then the main observable signal of theevanescent states.We can express the time-dependent component as j θ Ω ( r, θ ) = vr f ( r, θ − Ω t ) (34)in terms of a function with the periodicity f ( r, θ + 2 π ) = f ( r, θ ). The continuity equation can be approximated thenby ∂ t (cid:0) ψ † ψ (cid:1) ≈ − ∂ θ j θ Ω = − vr ∂ θ f ( r, θ − Ω t ) (35)Eq. (35) can be integrated at once to give ψ † ψ ≈ v Ω r f ( r, θ − Ω t ) + const . (36)This shows that the probability density of the evanescent states has a time evolution that is inherited from the timedependence of the current. - ´ - ´ - rr j static Θ - ´ - ´ - rr j WΘ FIG. 8. Left: Plot of the two contributions to rj θ static (in units of v/a , where a stands for a microscopic length scale in thematerial) from the spinors with j z = ~ / j z = − ~ / rj θ Ω at t = 0, θ = 0 (in units of v/a ) from thefirst two terms (blue) and the last two terms (red) in Eq. (33), for the same state as in the left plot. The parameters of theradiation are ~ Ω = 0 . A = 0 . a − , and the corresponding radius of the cylindrical geometry is R = 400 a . Similar results are obtained when considering the evanescent states from the hybridization of bands with values ofthe projection of the total angular momentum differing by two units of ~ . This case is relevant since it correspondsto the appearance of evanescent states with quasi-energy ǫ = 0 in the conventional Floquet approach. We may startfor instance with a linear combination of states with j z = ~ / , − ~ / − ~ / χ ∼ (cid:18) φ ( r ) e iθ φ ( r ) (cid:19) + (cid:18) e − iθ φ ( r ) φ ( r ) (cid:19) + (cid:18) e − iθ φ ( r ) e − iθ φ ( r ) (cid:19) (37)Inserting (37) into (29), we observe that the angular current can be decomposed now into three different contributions j θ = j θ static + j θ Ω + j θ (38)with j θ static = − i vr ( φ ∗ φ + φ ∗ φ + φ ∗ φ ) + h . c . (39) j θ Ω = − i vr ( e − i ( θ − Ω t ) φ ∗ φ + e i ( θ − Ω t ) φ ∗ φ + e − i ( θ − Ω t ) φ ∗ φ + e i ( θ − Ω t ) φ ∗ φ ) + h . c . (40) j θ = − i vr ( e − i ( θ − Ω t ) φ ∗ φ + e i ( θ − Ω t ) φ ∗ φ ) + h . c . (41)2As shown with the Floquet approach in the main text, the hybridization of the bands differing by two units of ~ in the projection of the total angular momentum (energy difference equal to 2 ~ Ω) opens a gap along a circle inmomentum space. It can be seen that the resulting evanescent states are confined in concentric circular rings on thesurface exposed to the radiation, with currents j θ static and j θ Ω which have very small intensities when integrated alongthe radial direction. However, the component j θ leads to maxima of the intensity (in the angular variable) whichare about two orders of magnitude larger than those from j θ static and j θ Ω , producing a significant modulation of thecharge which rotates with frequency Ω. In this case we have j θ ( r, θ ) = vr g ( r, θ − Ω t ) (42)in terms of a periodic function with g ( r, θ + π ) = g ( r, θ ). The probability density of the evanescent states has now atime evolution given by ψ † ψ ≈ v Ω r g ( r, θ − Ω t ) + const ..