Electric Quantum Oscillation in Weyl Semimetals
EElectric Quantum Oscillation in Weyl Semimetals
Kyusung Hwang, Woo-Ram Lee,
1, 2 and Kwon Park School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA (Dated: March 1, 2021)Electronic transport in Weyl semimetals is quite extraordinary due to the topological propertyof the chiral anomaly generating the charge pumping between two distant Weyl nodes with oppo-site chiralities under parallel electric and magnetic fields. Here, we develop a full nonequilibriumquantum transport theory of the chiral anomaly, based on the fact that the chiral charge pumpingis essentially nothing but the Bloch oscillation. Specifically, by using the Keldysh nonequilibriumGreen function method, it is shown that there is a rich structure in the chiral anomaly transport,including the negative magnetoresistance, the non-Ohmic behavior, the Esaki-Tsu peak, and finallythe resonant oscillation of the DC electric current as a function of electric field, called the electricquantum oscillation. We argue that, going beyond the usual behavior of linear response, the non-Ohmic behavior observed in BiSb alloys can be regarded as a precursor to the occurrence of electricquantum oscillation, which is both topologically and energetically protected in Weyl semimetals.
Among the greatest mysteries in physics is the asym-metry between matter and antimatter in our known uni-verse. Baryogenesis is the hypothesized physical processproducing such an asymmetry, whose precise mechanismstill remains elusive. The chiral anomaly, also knownas the Adler-Bell-Jackiw anomaly [1–3], is generally re-garded as one of the most crucial elements in the mech-anism of baryogenesis.In this context, Weyl semimetals [5–20] have been re-cently attracting intense attention by providing a con-crete realization of the chiral anomaly in condensed mat-ter systems, whose parameters can be tuned in tabletopexperiments. This experimental tunability is highly use-ful to investigate various aspects of the chiral anomaly. Aspecific aspect of the chiral anomaly, which has attractedparticularly intense attention, is the negative magnetore-sistance (MR), i.e., the resistance decreases with strongermagnetic fields [3, 6]. While definitely important, how-ever, the negative MR is ultimately a semiclassical sig-nature of the chiral anomaly.Here, we propose a full quantum signature of the chi-ral anomaly, which is fundamentally due to the quanti-zation of the chiral charge pumping under strong electricfields. A main breakthrough in this work is the realiza-tion that the chiral charge pumping is essentially nothingbut the Bloch oscillation in the zeroth, or chiral Lan-dau level (LL), which is quantized to generate robustWannier-Stark ladder (WSL) eigenstates [21–23] topo-logically protected by the chiral anomaly. Albeit some-what less, robust WSL eigenstates can be also formedin nonchiral LLs due to the energetic protection of theBloch oscillation in Weyl semimetals.The formation of WSL eigenstates reveals an intriguingsimilarity between electricity and magnetism. The quan-tized cyclotron motion of electrons under strong magneticfields gives rise to well-known magnetic quantum oscil-lations [24] such as de Haas-van Alphen, Shubnikov-deHaas, and eventually the quantum Hall effects. Similarly, the quantized Bloch oscillation of electrons under strongelectric fields can give rise to an electric-field-induced os-cillation of the DC electric current, which we call theelectric quantum oscillation (EQO).Actually, the EQO brings out one of the most fun-damental differences between electricity and magnetism.That is, electric fields inevitably cause nonequilibrium,while magnetic fields do not, no matter how strong.This difference raises a pressing question. What is thenonequilibrium steady state induced by strong electricfields?In this work, we develop a full nonequilibrium quan-tum transport theory of the chiral anomaly by using theKeldysh nonequilibrium Green function formalism [25]in conjunction with the Lindblad quantum master equa-tion [26]. As a result, it is shown that there is a richstructure in the chiral anomaly transport, including thenegative MR, the non-Ohmic behavior, the Esaki-Tsupeak, and finally the EQO. Being the incipient nonlin-ear behavior characterizing the chiral anomaly transport,the non-Ohmic behavior observed in BiSb alloys [18] canbe regarded as a precursor to the occurrence of EQO,which can serve as the unmistakable quantum signatureof the chiral anomaly in Weyl semimetals. We emphasizethat the chiral anomaly provides a unique environmentfor the realization of WSL eigenstates in natural materi-als, which has been so far impossible except for syntheticsystems such as semiconductor superlattices [27, 28] andoptical lattices [29, 30].In the perspective of application, this work lays agroundwork to expand the frontier of nonequilibriumquantum transport and realize novel nonlinear electronicdevices by combining strong-field phenomena [31] withtopological matter. It is interesting to mention thatstrong-field phenomena have been also investigated incombination with various many-body correlation effectssuch as Mott transition [26, 32–38] and many-body lo-calization [39, 40]. a r X i v : . [ c ond - m a t . s t r- e l ] F e b FIG. 1.
Landau-Stark quantization. ( a ) Energy spectrum of a minimal tight-binding model for Weyl semimetals withoutexternal fields. χ denotes the chirality of each Weyl node. ( b ) Energy spectrum under a magnetic field applied in the z direction, showing the formation of Landau level (LL) eigenstates with the energy eigenvalue of (cid:15) n ( k z ). With each LL labeledby the LL index n , the n = 0, or chiral LL is plotted in red, while all other nonchiral LLs are in dark/light blue. C denotesthe Chern number of the 2D k z slices of the Brillouin zone before the application of magnetic fields. ( c ) Energy spectrumunder parallel electric and magnetic fields applied in the z direction, showing the formation of Wannier-Stark ladder (WSL)eigenstates in each LL, called Landau-Stark eigenstates, with the energy eigenvalue of (cid:15) nl = ¯ (cid:15) n + l Ω. The Landau-Starkquantization manifests itself as a series of discrete peaks in the local density of states (DOS) shown in right panels indicatingindividual contributions from various LLs. Here, the cyclotron and Bloch oscillation frequencies are set as ω c = 0 . . t , respectively. Landau-Stark quantization
To perform a concrete analysis of the chiral anomalytransport in the full quantum level, we consider a mini-mal tight-binding model for Weyl semimetals [41, 42]: H ( k ) = (cid:88) i = x,y,z h i ( k ) σ i , (1)where h x ( k ) = 2 t sin k x , h y ( k ) = − t sin k y , and h z ( k ) = 2 t [2 − cos( k x − k y ) − cos( k x + k y )] + 2 t cos k z with t , t , t being hopping amplitudes and σ x , σ y , σ z being Pauli matrices. With the time-reversal symmetrybroken, this model hosts a single pair of Weyl nodes at k = (0 , , ± π/
2) with zero energy. See Fig. 1 a for theillustration of the energy spectrum at t = 4, t = 2,and t = 1, which are to be used as hopping amplitudesthroughout this work. Note that all momenta are de-noted in units of corresponding inverse lattice constants.Also, unless specified otherwise, we set (cid:126) = c = 1 for sim-plicity. As elaborated later, in this work, we focus on halffilling by setting the chemical potential appropriately.Let us first investigate what happens to the energyspectrum of the model Hamiltonian with the applica-tion of magnetic fields in the z direction ( B = B ˆ z with B > x - y plane by replacing sin k i by k i and cos k i by 1 − k i / i = x, y in Eq. (1), while maintaining the full k z dis-persion.The application of magnetic fields can be implementedvia minimal coupling, i.e., k → − i ∇ + e A Landau with − e being the charge of electron and A Landau being the Landau-gauge vector potential. Consequently, the modelHamiltonian generates the following energy eigenvaluesunder magnetic fields: (cid:15) n ( k z ) = sgn( n ) (cid:113) (2 t cos k z + | n | ω c ) + 2 | n | ω c t /t , (2)where n is a nonzero integer, and ω c = 4 t a x a y /l B isthe cyclotron frequency with a x and a y being the lat-tice constants in the x and y directions, respectively, and l B = 1 / √ eB being the magnetic length. For simplicity,the zero-point energy ω c / (cid:15) n ( k z ). Note that the energy eigen-modes are composed of the usual LL eigenstates, whichare entirely dispersionless within the x - y plane, while dis-persive in the z direction.Now, an interesting thing happens if one tries to set n = 0 in Eq. (2). With the sign of zero undefined, therecould be two distinct energy eigenmodes correspondingto ± t | cos k z | . In reality, however, there exists only asingle energy eigenmode called the chiral LL with theenergy eigenvalue of (cid:15) ( k z ) = 2 t cos k z . Note that thissingleness of the chiral LL is a unique topological prop-erty of Weyl semimetals. See Methods for details. Also,see Fig. 1 b for the illustration of chiral versus nonchiralLLs.With the application of electric fields, each LL canbe further quantized into a series of WSL eigenstates.Usually, the formation of WSL eigenstates requires well-separated energy bands so that the Bloch oscillation cancomplete one full cycle without being interrupted by theLandau-Zener transition [23], which is unfortunately dif-ficult to achieve in natural materials. Fortunately, inWeyl semimetals, there is a nice protection of the Blochoscillation due to the aforementioned singleness of thechiral LL. Specifically, when electric fields are applied inthe z direction parallel to magnetic fields ( E = E ˆ z ), thereis absolutely no Landau-Zener transition between differ-ent LLs, unless they have the same | n | . This means thatthe Bloch oscillation in the chiral LL is completely im-mune from the Landau-Zener transition. While allowed,the Landau-Zener transition is also energetically sup-pressed between nonchiral LLs with the same | n | , whoseenergies can be well-separated across the zero-point en-ergy. Consequently, under strong magnetic fields, it issafe to assume that each LL is quantized into its ownindividual series of WSL eigenstates.Also, being so-called extended states, LL eigenstatesare generally known to be rather robust against disor-der [43]. This means that the k z dispersion of LL eigen-states and consequently the formation of WSL eigen-states can be also robust against disorder to certain ex-tents.Technically, the application of electric fields can be im-plemented in terms of either static scalar or temporal vec-tor potential gauge. In the static scalar potential gauge,the model Hamiltonian can be written in terms of theStark Hamiltonian for each individual LL: H Stark ,n = (cid:15) n ( k z ) + i Ω (cid:20) ∂∂k z + A n ( k z ) (cid:21) , (3)where Ω = eEa z is the Bloch oscillation frequency with a z being the lattice constant in the z direction. Here, A n ( k z ) is the Berry connection of the n -th LL, whichturns out to be zero regardless of n in our minimal model.The Stark Hamiltonian can be diagonalized via WSLeigenstates in each LL, called Landau-Stark eigenstates,i.e., H Stark ,n φ nl ( k z ) = (cid:15) nl φ nl ( k z ) with φ nl ( k z ) = e − i Ω (cid:82) kz dκ [ (cid:15) nl − (cid:15) n ( κ )] (4)and (cid:15) nl = ¯ (cid:15) n + l Ω, where ¯ (cid:15) n = (cid:82) π − π d k z π (cid:15) n ( k z ) is the meanenergy of the n -th LL and l is the WSL index. It is im-portant to note that WSL eigenstates are full quantumsolutions of the Stark Hamiltonian, while also obtainedas semiclassical solutions via the Bohr-Sommerfeld quan-tization [44]. See Fig. 1 c for the illustration of Landau-Stark energy levels, accompanied by the local density ofstates (DOS), whose details are given in Methods . Nonequilibrium quantum transport
Being standing waves, WSL eigenstates cannot gener-ate any nonzero net DC electric currents, unless thereis impurity scattering, which causes the transition be-tween different WSL eigenstates. Here, we develop a fullnonequilibrium quantum transport theory of the chiralanomaly by treating the process of impurity scatteringvia the Keldysh nonequilibrium Green function formal-ism [25]. Specifically, our nonequilibrium quantum trans-port theory is composed of three steps.
Temporal vector potential gauge.
The first step isto change the gauge and implement the application of
FIG. 2.
Schematic diagram of the Keldysh-Dyson self-consistency loop.
Our nonequilibrium quantum transporttheory is based on the Keldysh-Dyson self-consistency loopcomprising three parts; (i) the full Green function, G , is ob-tained by solving the Keldysh-Dyson equations with Σ be-ing the yet-to-be-determined self-energy, (ii) Σ is then relatedwith G via self-consistent Born approximation for impurityscattering, and (iii) the self-consistency loop is completedonce the noninteracting Green function, g , is fixed in termsof Landau-Stark eigenstates. Crucially, the noninteractinglesser Green function, g < , is constructed so that Landau-Stark eigenstates are appropriately thermalized according tothe WSL-wise thermalization scheme. electric fields via the temporal vector potential A Stark = − Et ˆ z with t being time, in which case the total vectorpotential is given as A = A Landau + A Stark . This par-ticular choice of gauge is made to preserve the spatialtranslation symmetry so that impurity scattering can betreated via the usual method of self-consistent Born ap-proximation (SCBA).In the temporal vector potential gauge, the modelHamiltonian can be written as follows: H n ( t ) = (cid:15) n ( k z − Ω t ) , (5)which is periodic in time with the period of 2 π/ Ω. Sucha time-dependent Hamiltonian can be analyzed by usingthe Keldysh nonequilibrium Green function method withnonequilibrium Green functions conveniently representedin the Floquet matrix form [33]. It is worthwhile to men-tion that WSL eigenstates in the static scalar potentialgauge are manifested as Floquet modes in the temporalvector potential gauge.
Keldysh-Dyson self-consistency loop.
The secondstep is to set up the Keldysh-Dyson self-consistency loopto capture the process of impurity scattering via SCBA.See Fig. 2 for the schematic diagram. Technically, thefull Green functions can be obtained by self-consistentlysolving the Keldysh-Dyson equations [25]:[ G r ] − = [ g r ] − − Σ r , (6) G < = G r { [ g r ] − g < [ g a ] − + Σ < } G a , (7)where G r ( g r ) and G < ( g < ) are the full (noninteracting)retarded and lesser Green functions, which contain theinformation about the DOS and occupation, respectively.The advanced Green functions, G a and g a , are relatedwith the retarded counterparts via complex conjugation.Meanwhile, Σ r and Σ < are the retarded and lesser self-energies, respectively, induced by impurity scattering. Inthe above expressions, we drop all the subscripts (LL andFloquet indices) and arguments ( k z and ω ) for simplicity.Importantly, the self-energies are related to the fullGreen functions via SCBA [45]:Σ r,< ( ω ) = V D (cid:90) π − π dk z (cid:88) n G r, 05 throughout this work.Note that the DC electric current would be net zero inthe presence of Γ alone [26]. Nonzero net DC electriccurrents can be only generated by the intricate interplayof both elastic and inelastic scattering.The noninteracting lesser Green function, g < , is givenin the Floquet matrix form as follows:[ g DC electric current density. The third and final stepis to compute the DC electric current density, J DC , fromthe full lesser Green function obtained as a convergedsolution of the Keldysh-Dyson self-consistency loop [26]: J DC = e D (cid:90) Ω / − Ω / dω π (cid:88) n,p,q p ¯ (cid:15) n ( p ) (cid:2) G 001 in this work. Also, LL indices aresummed up to | n | = 6, which is necessary for the rangeof magnetic fields studied in this work, except for theultra-quantum limit of strong magnetic fields, where it issufficient to consider only the chiral LL. Meanwhile, thenumber of summed Floquet indices is chosen adaptivelyto ensure that J DC is well converged at each given Ω. See Methods for details. ResultsElectric quantum oscillation via the generalLandau-Stark resonance. The DC electric current canoscillate via two different mechanisms. In this section, wefirst discuss the resonance between various Landau-Starkeigenstates with different LL indices, called the generalLandau-Stark resonance.Fig. 3 a shows that, in a general regime of electric andmagnetic fields, J DC oscillates as a function of both Ω and ω c , exhibiting a complicated, yet highly organized seriesof resonant peaks. Physically, the resonant behavior of J DC can be well understood in terms of the tunnelingformula between adjacent sites [26]: J tun ∝ (cid:90) ∞−∞ dερ loc ( ε ) ρ loc ( ε + Ω)[ f loc ( ε ) − f loc ( ε + Ω)] , (13)where ρ loc and f loc are the local DOS and distributionfunction, respectively. Specifically, ρ loc is given as thesum of individual contributions from various LLs, i.e., FIG. 3. Electric quantum oscillation via the general Landau-Stark resonance. ( a ) Color map of the DC electriccurrent density J DC as a function of the Bloch oscillation frequency Ω ( ∝ E ) and the cyclotron frequency ω c ( ∝ B ). Notethat J DC exhibits a complicated, yet highly organized series of resonant peaks, whose trajectories are accurately described bythe general Landau-Stark resonance condition, ¯ (cid:15) n − ¯ (cid:15) = ∆ l · Ω, for various cases of ( n, ∆ l ). The strongest resonant peaks areobtained along the trajectories of ( n = 1 , ∆ l = 1 , , , · · · ) plotted in dark blue, followed by progressively weaker resonant peaksalong those of n > b ) Illustrated mechanism of the general Landau-Stark resonance. Here, we take thecase of ( n = 1 , ∆ l = 3) as an example, marked by the grey dot in a , where the Landau-Stark energy levels coming from the n = 1 LL (dark blue) are perfectly aligned with those from the n = 0, or chiral LL (red). ( c ) Periodicity of J DC as a function of1 / Ω for various given ω c . Generally, the resonant peaks are equally spaced as a function of 1 / Ω with the period of 1 / (¯ (cid:15) n − ¯ (cid:15) ),as shown by the blue vertical lines indicating the locations of the strongest resonant peaks at n = 1. There is an exception tothis rule, as indicated by the red arrows here and also in a , revealing the existence of a different type of the electric quantumoscillation. Here, we set V imp = 1 and Γ = 0 . 05 in units of t . Finally, J DC is denoted in units of 10 − et / (cid:126) a z . ρ loc = D (cid:80) n ρ loc ,n with ρ loc ,n ( ε = ω + p Ω) = − π (cid:90) π − π dk z π Im[ G rn ( k z , ω )] pp , (14)where ε can cover the entire range of frequency bychanging the Floquet index p while ω ∈ [ − Ω / , Ω / f loc can be computed via f loc = N loc /ρ loc with the local occupation number N loc given by N loc = D (cid:80) n N loc ,n , where N loc ,n ( ε = ω + p Ω) = 12 π (cid:90) π − π dk z π Im[ G 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.2 0.4 0.6 Ω =0.3 ω c =1 2 3 4 J DC Ω imp =1 2 3 4 J DC Ω FIG. 4. Electric quantum oscillation via the chiral res-onance. ( a ) DC electric current density J DC as a function ofΩ at ω c = 1 with V imp = 1, covering four distinct regimes ofthe chiral anomaly transport; (i) negative magnetoresistance,(ii) non-Ohmic behavior, (iii) Esaki-Tsu peak (indicated bythe grey vertical line), and (iv) electric quantum oscillation(EQO). The inset highlights the periodicity of the EQO viathe chiral resonance as a function of 1 / Ω, which is simply π/ /t , being entirely independent of magnetic fields.( b ) Local DOS at Ω = 0 . 01, 0 . 06, and 0 . 3, indicated by theblack arrows in a . ( c ) Polynomial fitting of J DC at weak andweak-to-intermediate electric fields, showing the usual linearDrude conductivity (red straight line) and the non-Ohmic be-havior (blue curve), respectively. ( d ) Magnetic-field depen-dence of the linear Drude conductivity, σ , showing the behav-ior of negative magnetoresistance. ( e ) J DC as a function ofΩ for various given ω c , showing the overall increase of J DC with stronger magnetic fields. ( f ) J DC as a function of Ω forvarious given V imp , showing the Drude behavior, i.e., σ de-creases with stronger impurity scattering. As in Fig. 3, J DC is denoted in units of 10 − et / (cid:126) a z . / (¯ (cid:15) n − ¯ (cid:15) ), which is strongly reminiscent of the similarbehavior in magnetic quantum oscillation. Actually, thelow-electric-field data at ω c = 0 . Electric quantum oscillation via the chiral reso-nance. The general Landau-Stark resonance conditioncan be trivially satisfied with n = 0 and ∆ l = 0. Ifso, na¨ıvely, J DC could be always enhanced in the ultra- quantum limit of strong magnetic fields, where the chi-ral LL becomes the only transport channel with all othernonchiral LLs pushed far away from the Fermi level. Thisna¨ıve expectation, however, does not hold since the chi-ral LL alone cannot induce any actual electronic trans-port, at least via elastic impurity scattering alone. Inthis case, nonzero net DC electric current can be gener-ated with help of the broadening of Landau-Stark energylevels due to inelastic scattering processes.Fig. 4 a shows the behavior of J DC as a function of Ωranging from weak to strong electric fields in the ultra-quantum limit of strong magnetic fields, say, at ω c = 1,where it is sufficient to consider only the chiral LL solong as Ω (cid:46) . (cid:46) . . (cid:46) Ω (cid:46) . ET (cid:39) . (cid:38) . 05) electric fields,respectively.First, at strong electric fields, the EQO occurs via thechiral resonance, which is distinguished from the previ-ously described, general Landau-Stark resonance. In thecase of the chiral resonance, the DC electric current oscil-lates as a function of 1 / Ω with a constant period entirelyindependent of ω c , which is simply π/ /t inour minimal model for Weyl semimetals. Fundamentally,the mechanism of the chiral resonance can be understoodin terms of the wave function overlap between adjacentWSL eigenstates in the chiral LL, which oscillates asymp-totically as a function of electric field. See Methods fordetails.As Ω decreases, the EQO becomes less and less pro-nounced, finally merging into the Esaki-Tsu peak aroundΩ = Ω ET . Fig. 4 b shows that both Esaki-Tsu peakand subsequent EQO are closely correlated with the for-mation of well-separated WSL eigenstates. Note that,marking the onset of negative differential conductivity,the Esaki-Tsu peak [47] has been routinely observed insemiconductor superlattices [48].Fig. 4 c shows that, at Ω (cid:46) Ω ET , J DC increases as amonotonic, but in general nonlinear function of Ω, i.e., J DC = σ Ω + σ (cid:48) Ω + σ (cid:48)(cid:48) Ω + · · · , where σ denotes theusual linear Drude conductivity in the limit of weak elec-tric fields, while σ (cid:48) and σ (cid:48)(cid:48) are the two lowest-order coeffi-cients of the non-Ohmic behavior. It is important to notethat the non-Ohmic behavior is an inevitable crossoverphenomenon connecting between the linear Drude con-ductivity and Esaki-Tsu peak. Considering that bothEsaki-Tsu peak and subsequent EQO are closely corre-lated with the formation of well-separated WSL eigen-states, the non-Ohmic behavior can be regarded as aprecursor to the EQO. In this context, the non-Ohmicbehavior observed in BiSb alloys [18] suggests that theobservation of EQO might actually be within the reach ofexperiments since, in our results, the strength of electricfield necessary for the occurrence of EQO is only about10-20 times larger than that necessary for the non-Ohmicbehavior.Now, we would like to confirm if the linear Drude con-ductivity, σ , exhibits the expected behavior of negativeMR. Specifically, in the ultra-quantum limit of strongmagnetic fields, σ is expected to increase as a linear func-tion of magnetic field [3, 6]. Fig. 4 d confirms that thisis indeed exactly the case. Actually, Fig. 4 e shows that J DC increases as a whole with stronger magnetic fields.Finally, Fig. 4 f shows the behavior of J DC as a functionof Ω for various given V imp , confirming that σ decreaseswith stronger impurity scattering, as expected from theDrude behavior. Discussion In this work, it is shown that the chiral charge pumping isessentially nothing but the Bloch oscillation. Both topo-logically and energetically protected in Weyl semimetals,the Bloch oscillation can be quantized to generate ro-bust Landau-Stark eigenstates, eventually giving rise tothe resonant oscillation of the DC electric current as afunction of electric field.Called the EQO, this resonant oscillation of the DCelectric current can occur in Weyl semimetals via two dif-ferent mechanisms. First, the EQO can occur via the res-onance between various Landau-Stark eigenstates withdifferent LL indices. Second, in the ultra-quantum limitof strong magnetic fields, the EQO can also occur viaa form of the self-resonance within the chiral LL. Par-ticularly, in this limit, there are four distinct regimesof the chiral anomaly transport; (i) negative MR, (ii)non-Ohmic behavior, (iii) Esaki-Tsu peak, and (iv) EQOat weak, weak-to-intermediate, intermediate, and strongelectric fields, respectively. It is important to note thatboth negative MR and non-Ohmic behavior [18] havebeen already observed in Weyl semimetals, providing ex-perimental support for the occurrence of EQO in naturalmaterials.In broad perspective, understanding nonequilibriumsteady states of matter is among the foremost frontiersin physics. Induced by strong electric fields, the EQOwould be one of the most salient features of nonequilib-rium steady states realized in condensed matter. Usuallyachieved in synthetic systems such as semiconductor su-perlattices and optical lattices, a prerequisite for the oc-currence of EQO is the formation of robust WSL eigen-states. As emphasized in this work, the chiral anomalycan provide a unique environment for the formation ofrobust WSL eigenstates via the combination of strong-field phenomena with topological matter. Interestingly,Weyl semimetals can be also synthetically generated byfabricating a layered structure of alternating topologicaland magnetic insulators [5].Finally, there is a close analogy between the EQOstudied in this work and the radiation-induced quantum oscillation observed in quantum Hall systems [49, 50].It is interesting to mention that the radiation-inducedquantum oscillation has been analyzed via both Keldyshnonequilibrium Green function method and tunnelingformula [51–53], which are also two main theoretical toolsin this work. MethodsLandau quantization in Weyl semimetals. We be-gin by writing the continuum limit of the model Hamil-tonian in Eq. (1) within the x - y plane, which can beobtained by replacing sin k i by k i and cos k i by 1 − k i / i = x, y , while maintaining the full k z dispersion.Specifically, the model Hamiltonian can be written in thecontinuum limit as follows: H ( k ) = 2 t k x σ x − t k y σ y + (cid:2) t ( k x + k y ) + 2 t cos k z (cid:3) σ z , (17)where all momenta are denoted in units of correspondinginverse lattice constants.With the application of magnetic fields in the z di-rection, the model Hamiltonian is modified via mini-mal coupling, i.e., k → Π = − i ∇ + e A Landau with A Landau = B (0 , x, 0) being the Landau-gauge vector po-tential. At this moment, let us assume that B > 0. Thecase of B < B > 0, the model Hamiltonian can be written as H ( k ) = 2 √ t l B (cid:0) b † σ + + bσ − (cid:1) + (cid:20) ω c (cid:18) b † b + 12 (cid:19) + 2 t cos k z (cid:21) σ z , (18)where the LL raising and lowering operators, b † and b ,are defined, respectively, as follows: (cid:26) b † b (cid:27) = l B √ x ± i Π y ) (19)with l B = 1 / √ eB being the magnetic length. Similarly,the pseudospin raising and lowering operators, σ + and σ − , are defined, respectively, as follows: (cid:26) σ + σ − (cid:27) = 12 ( σ x ± iσ y ) . (20)Note that the cyclotron frequency is given by ω c =4 t /l B , and Π z is replaced back to its eigenvalue, k z .The Hamiltonian in Eq. (18) can be block-diagonalizedby using the convenient set of basis states, {| ν (cid:105) ⊗ | σ (cid:105)} ,which are composed of number eigenstates | ν (cid:105) (i.e., b † b | ν (cid:105) = ν | ν (cid:105) ) and the pseudospin up/down state | σ (cid:105) (i.e., | ↑ (cid:105) or | ↓ (cid:105) ). Now, by noting that | ν (cid:105) ⊗ | ↑ (cid:105) bσ − −−− (cid:42)(cid:41) −−− b † σ + | ν − (cid:105) ⊗ | ↓ (cid:105) , (21)one can obtain the block-diagonalized matrix form of theHamiltonian as follows: H ν ( k z ) = ω c I + (cid:34) t cos k z + νω c √ t l B √ ν √ t l B √ ν − t cos k z − νω c (cid:35) , (22)which is defined in the Hilbert space spanned by twobasis states, | ν (cid:105) ⊗ | ↑ (cid:105) and | ν − (cid:105) ⊗ | ↓ (cid:105) with ν ≥ H ν ( k z ) generates the energy eigenvalues ofnonchiral LLs as follows: (cid:15) ν, ± ( k z ) = ω c / ± (cid:113) (2 t cos k z + νω c ) + 2 νω c t /t , (23)which becomes identical to (cid:15) n ( k z ) in Eq. (2) after the LLindex is defined as n = ± ν , and the zero-point energy ω c / ν = 0: H ( k z ) = ω c / t cos k z , (24)which is defined in the Hilbert space spanned by the sin-gle basis state, | (cid:105) ⊗ | ↑ (cid:105) . Being diagonal, H ( k z ) itselfis the energy eigenvalue of the chiral LL, which equals to (cid:15) ( k z ) after the subtraction of the zero-point energy.It is important to note that the singleness of the chiralLL is a unique topological property of Weyl semimetals.To appreciate the origin of this topological property, it isbeneficial to consider what happens in the case of B < B as follows: H ( k ) = 2 √ t l B (cid:0) b † σ sgn( B ) + bσ − sgn( B ) (cid:1) + (cid:20) ω c (cid:18) b † b + 12 (cid:19) + 2 t cos k z (cid:21) σ z , (25)where the LL raising and lowering operators are now gen-eralized as follows: (cid:26) b † b (cid:27) = l B √ x ± i sgn( B )Π y ] (26)with l B = 1 / (cid:112) e | B | .After some algebra, one can show that the energyeigenvalues of nonchiral LLs are exactly the same as be-fore regardless of the sign of B except that the zero-pointenergy is now generalized as sgn(B) ω c / B : (cid:15) ( k z ) =2 t sgn( B ) cos k z . This sign dependence of the chiral LLis fundamentally due to the specific topological propertyof Weyl semimetals in our minimal model. Namely, the2D k z slices of the Brillouin zone form Chern or trivial in-sulators depending on whether k z is inside or outside theregion between two Weyl nodes with opposite chiralities. Noninteracting Green functions in the Floquetmatrix form. Here, we discuss how to construct thenoninteracting retarded and lesser Green functions in theFloquet matrix form. We begin by writing the nonin-teracting Hamiltonian in the temporal vector potentialgauge as follows: H = (cid:88) n,k z (cid:15) n ( k z − Ω t ) c † n,k z c n,k z , (27)where c † n,k z and c n,k z are the creation and annihilationoperators, respectively, for the n -th LL with k z .The noninteracting retarded Green function is definedas follows: g rn,k z ( t, t (cid:48) ) = − iθ ( t − t (cid:48) ) (cid:104){ c n,k z ( t ) , c † n,k z ( t (cid:48) ) }(cid:105) , (28)where c n,k z ( t ) = U n,k z ( t, t ) c n,k z ( t ) with the unitaryevolution operator, U n,k z ( t, t ), given by U n,k z ( t, t ) = exp (cid:20) − i (cid:90) tt dτ (cid:15) n ( k z − Ω τ ) (cid:21) , (29)where t is some arbitrary reference time.Now, noting that the unitary evolution operator inEq. (29) is essentially identical to the wave function ofLandau-Stark eigenstates in Eq. (4), U n,k z ( t, t ) can beexpressed in terms of φ nl as follows: U n,k z ( t, t ) = φ nl ( k z − Ω t ) φ nl ( k z − Ω t ) e − i(cid:15) nl ( t − t ) , (30)where l can be chosen arbitrarily. Then, pluggingEq. (30) into the anticommutation part in Eq. (28) leadsto the following result: (cid:104){ c n,k z ( t ) , c † n,k z ( t (cid:48) ) }(cid:105) = φ nl ( k z − Ω t ) φ ∗ nl ( k z − Ω t (cid:48) ) e − i(cid:15) nl ( t − t (cid:48) ) , (31)where it is used that { c n,k z ( t ) , c † n,k z ( t ) } = 1 and φ nl ( k z − Ω t ) φ ∗ nl ( k z − Ω t ) = 1.Next, by using the integral representation of the Heav-iside step function, − iθ ( t − t (cid:48) ) = (cid:90) ∞−∞ dε π e − iε ( t − t (cid:48) ) ε + iη , (32)one can express g rn,k z ( t, t (cid:48) ) as follows: g rn,k z ( t, t (cid:48) ) = φ nl ( k z − Ω t ) φ ∗ nl ( k z − Ω t (cid:48) ) × (cid:90) ∞−∞ dε π e − iε ( t − t (cid:48) ) ε − (cid:15) nl + iη , (33)which is obtained after an appropriate redefinition of theintegration valuable.Then, by using the Fourier transform of φ nl , φ nl ( k z − Ω t ) = (cid:88) p e − ip ( k z − Ω t ) ϕ nl ( p ) , (34)one can arrive at the final expression: g rn,k z ( t, t (cid:48) ) = (cid:90) Ω / − Ω / dω π (cid:88) p,q e − i ( ω + p Ω) t e i ( ω + q Ω) t (cid:48) × [ g rn ( k z , ω )] pq , (35)where[ g rn ( k z , ω )] pq = e ik z ( p − q ) (cid:88) j ϕ np ( j ) G rn ( ω + j Ω) ϕ ∗ nq ( j )(36)with G rn ( ε = ω + j Ω) = 1 / ( ε − ¯ (cid:15) n + iη ) being the re-duced retarded Green function of Landau-Stark eigen-states. Note that the l dependence completely disappearsin the final expression due to the translational symmetryof Landau-Stark eigenstates, i.e., ϕ np ( j ) = ϕ n,p + l ( j + l )for arbitrary l .The mathematical form of Eq. (35) indicates that[ g rn ( k z , ω )] pq is nothing but the Fourier transform of g rn,k z ( t, t (cid:48) ). Specifically, [ g rn ( k z , ω )] pq is the ( p, q )-th el-ement of the noninteracting retarded Green function inthe Floquet matrix form [33].Based on this realization, it is instructive to computethe noninteracting local DOS, ρ (0)loc = D (cid:80) n ρ (0)loc ,n , where ρ (0)loc ,n ( ε = ω + p Ω) = − π (cid:90) π − π dk z π Im[ g rn ( k z , ω )] pp = (cid:88) l | ϕ nl (0) | δ ( ε − (cid:15) nl ) , (37)which shows that the local DOS is composed of dis-crete peaks at ε = (cid:15) nl with their weights given bythe corresponding Landau-Stark eigenstates at a givensite, say, origin, | ϕ nl (0) | . Note that the broadeningof Landau-Stark energy levels can be implemented bysetting η = Γ / G rn ( ε ) with Γ being small, but fi-nite, in which case the delta function is replaced by theLorentzian: δ Γ ( ε − x ) = − π Im 1 ε − x + i Γ / g The DC electric current can be com-puted from the full lesser Green function obtained as aconverged solution of the Keldysh-Dyson self-consistencyloop.To begin with, whether DC or not, the electric currentdensity can be exactly expressed in terms of the full lesserGreen function as follows: J ( t ) = −D e (cid:90) π − π dk z π (cid:88) n ∂(cid:15) n ( k z − Ω t ) ∂k z (cid:104) c † n,k z ( t ) c n,k z ( t ) (cid:105) , (42)which can be understood as the sum of all contributionsfrom each conduction mode, whose individual contribu-tion is in turn given by the product between its groupvelocity and occupation number specified by the LL in-dex n and momentum k z . It is important to note thatthe above expression is in principle exact at arbitrarystrengths of electric and magnetic fields. As shown be-low, eventually, the electric current becomes strictly DCin our situation.First, by definition, the occupation number is equal tothe equal-time full lesser Green function, which can berelated to its Fourier transform as follows: (cid:104) c † n,k z ( t ) c n,k z ( t ) (cid:105) = − iG In the Floquet rep-resentation, Green functions are represented as infinite-dimensional Floquet matrices. For practical calculations,the dimension of Floquet matrices should be truncatedwith an appropriate cutoff limiting the range of Floquetindices. In other words, we would like to represent re-tarded and lesser Green functions as finite-dimensionalFloquet matrices, [ G r ( ω )] pq and [ G < ( ω )] pq , respectively,with p, q ∈ (0 , ± , · · · , ±L ). The cutoff L is determinedvia the following procedure.To begin with, we first estimate the cutoff by requiringthat the noninteracting local DOS is properly normalizedfor each individual LL. Specifically, it can be said thatthe noninteracting local DOS for the n -th LL is properlynormalized if (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L n (cid:88) p = −L n (cid:90) Ω / − Ω / dωρ (0)loc ,n ( ω + p Ω) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ tol , (46)where ρ (0)loc ,n is given in Eq. (37), L n is the cutoff for the n -th LL, and δ tol is a sufficiently small tolerance. In thiswork, we set δ tol to be 10 − .Finally, the overall cutoff L is chosen as the maximumof L n : L = max {L n } . As a general rule, the lower Ωbecomes, the higher L is required. Roughly speaking, L is of the order of 1 , 000 for Ω (cid:46) . 01 while typically lessthan 100 otherwise. Mechanism of the chiral resonance. To under-stand the mechanism of the chiral resonance, we beginby rewriting the tunneling formula as follows: J tun ∝ (cid:90) ∞−∞ dε [ ρ loc ( ε + Ω) N loc ( ε ) − ρ loc ( ε ) N loc ( ε + Ω)] , (47)where ρ loc and N loc are the local DOS and occupationnumber, respectively. Note that Eq. (47) is preciselyidentical to Eq. (13) since N loc = ρ loc f loc by definition. Now, assuming that WSL eigenstates are well sepa-rated in the chiral LL, the local DOS can be accuratelyapproximated as ρ loc ( ε ) ∼ (cid:88) l A l (Ω) δ Γ ( ε − l Ω) , (48)where A l (Ω) = | J l (2 t / Ω) | and δ Γ ( ε ) is the Lorentzianin Eq. (38). Similarly, the local occupation number canbe also accurately approximated as N loc ( ε ) ∼ (cid:88) l A l (Ω) δ Γ ( ε − l Ω) f FD ( ε − l Ω) , (49)which is obtained via the WSL-wise thermalizationscheme as explained in Eq. (41).After some rearrangements, Eq. 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