Quantum Hall Effect Induced by Chiral Landau Levels in Topological

Quantum Hall eﬀect induced by chiral Landau levels in topological semimetal ﬁlms

D.-H.-Minh Nguyen,1, ∗ Koji Kobayashi,1 Jan-Erik R. Wichmann,1 and Kentaro Nomura1, 2, † 1Institute for Materials Research, Tohoku University, Sendai, Miyagi, 980-8577, Japan 2Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan (Dated: May 14, 2021) Motivated by recent transport experiments, we theoretically study the quantum Hall effect in topological semimetal films. Owing to the confinement effect, the bulk subbands originating from the chiral Landau levels establish energy gaps that have quantized Hall conductance and can be observed in relatively thick films. We find that the quantum Hall state is strongly anisotropic for different confinement directions not only due to the presence of the surface states but also because of the bulk chiral Landau levels. As a result, we re-examine the quantum Hall effect from the surface Fermi arcs and chiral modes in Weyl semimetals and give a more general view into this problem. Besides, we also find that when a topological Dirac semimetal is confined in its rotational symmetry axis, it hosts both quantum Hall and quantum spin Hall states, in which the helical edge states are protected by the conservation of the spin-z component.

I. INTRODUCTION ment subbands of the chiral Landau levels in topological semimetal films. We study the QHE in Weyl semimetals Weyl orbits [1,2] in topological semimetals are unique (WSMs) in two cases of the confinement direction: with magnetic orbits that involve two Fermi arcs on oppo- and without the surface states. While in the former, the site surfaces connected via the chiral Landau levels run- energy levels agree with the results obtained from the ning through the bulk. Experiments studying the quan- semiclassical analysis as expected, we show that the chi- tum oscillation and quantum Hall effect (QHE) related ral Landau subbands behave differently in the latter. By to such orbits have been carried out in recent years [3–13] examining the difference between those two cases, we find and have sparked heated debates among research groups. a more general expression for the energy levels of Weyl For instance, as most of the experiments were conducted orbits in a WSM. Furthermore, we find that in DSM films with the topological Dirac semimetal (DSM) candidate confined in their rotational symmetry axis, the quantum Hall and quantum spin Hall states coexist due to the spin Cd3As2 [14–18], some works attribute the QHE in those conservation. Cd3As2 films to the surface Dirac cones instead of the Weyl orbits [11, 19]. This is because the surface states This paper is organized as follows. In Sec.II, we introduce the models describing our WSM and DSM. In Sec. of Cd3As2 are shown to be two-dimensional (2D) Dirac cones by the angle-resolved photoemission measurements III, we revisit the QHE based on Weyl orbits by con- [20–22] instead of the open Fermi arcs as theoretically sidering a WSM film with chiral surface states on their predicted [14]. In order to distinguish the two mecha- boundaries. We then carry out the same calculation but in a film confined in a direction so that it has no surface nisms of QHE in Cd3As2, it is proposed as a key that the QHE based on Weyl orbits depends on the film thickness states. In Sec.IV, we explain the results obtained in [2,3,7,8], which was observed in a wedge-shaped film the preceding section and examine the QHE in a WSM confined in an arbitrary direction. We also study the of Cd3As2 [3]. However, a recent study suggests that the QHE in such wedge-shaped films may come from a com- QHE in a DSM confined in its rotational symmetry axis pletely different mechanism [23], and thus the existence and consider a specific case of (001) Cd3As2 film to show of Weyl orbits remains elusive. As another example for that the effect is experimentally observable. Finally, our the debates, some experiments [3,4] find that the energy results will be summarized in Sec.V. levels of Weyl orbits are related to the quantum confinement subbands, which contradicts with some other works [6, 24, 25] distinguishing Weyl orbits with the con- II. MODELS finement effect. Hence, further studies are still needed in order to understand the nature of both Weyl orbits and In this work, we describe a WSM by the minimal model arXiv:2105.06171v1 [cond-mat.mes-hall] 13 May 2021 the surface states of Cd3As2. [28, 29] Motivated by the connection between Weyl orbits and the quantum confinement effect, and by the progress HW(k) =t(sin kxσx + sin kyσy) 0 in fabricating high-quality nanostructures of Cd3As2 + [M + t (cos kx + cos ky + cos kz)]σz (1) [26, 27], we investigate the QHE induced by confine- with σi being the Pauli matrices. Notice that the dimensionless ki denote the product qiai, where qi are the usual crystal momenta and ai lattice constants. This ∗ [email protected] cubic lattice model is centrosymmetric and breaks the † [email protected] time-reversal symmetry. Hereafter, we use t = t0 = 1 and 2

the parameters c0 = c1 = c2 = 0, t = m1 = m2 = 1 and m0 = −2.5 so that the material has a pair of Dirac points at kD = (0, 0, ±π/3) [Fig. 1(a)]. We will examine the WSM and DSM in a slab geometry confined in the i (i = x or z) direction, as shown in Figs. 1(b) and 1(c), with periodic boundary conditions (PBCs) in the other two directions. The slabs are subjected to a uniform magnetic field Bkî, whose magnitude can be written in terms of the magnetic flux Φ thread- FIG. 1. (a) WSM/DSM with two Weyl/Dirac nodes (orange ing a unit cell as B = Φ/(a a ) = Φ φ/(a a ). Here, points) aligned along the z axis. (b)/(c) A slab confined in j k 0 j k the x/z direction in a perpendicular magnetic field B. i 6= j 6= k,Φ0 = h/e is the magnetic flux quantum, and φ = Φ/Φ0.

M = −2.5 so that the Hamiltonian gives a pair of Weyl III. QUANTUM HALL EFFECT IN WEYL nodes at kW = (0, 0, ±π/3), as depicted in Fig. 1(a). If we confine this WSM in the x direction, the topologi- SEMIMETAL FILMS cal surface states called Fermi arcs will appear. In our minimal model, the arcs are straight lines connecting the A. Films confined in the x direction projections of the two bulk Weyl cones. When the Weyl points are located close to the Γ-point, First, we consider the WSM confined in the x direc- we can expand the Hamiltonian to obtain a continuum tion [Fig. 1(b)] with a magnetic field Bkxˆ represented model by vector potential A = (0, 0, Bay), where y is dimensionless. The Hamiltonian of our WSM is then modified 0 2 2 2 HW(k) = t(kxσx +kyσy)+[m−t (kx +ky +kz )/2]σz, (2) in accordance with the Peierls substitution as (Appendix A) where m = M + 3t0. We will use the lattice Hamiltonian for numerical calculations and its continuum counterpart X 1n H = d† [M + t0 cos(k + 2πφy)]σ d for obtaining the analytical expressions. W 2 xyk z xyk On the other hand, the DSM is described by an effec- x,y,k † tive Hamiltonian of Cd3As2 and Na3Bi [14, 30] 0 + dxyk(t σz − itσx)d(x+1)yk o M˜ (q) Aq 0 0  † 0 + + dxyk(t σz − itσy)dx(y+1)k + h.c. (6) Aq −M˜ (q) 0 0 H (q) = E˜ (q) +  −  D 0  ˜  †  0 0 M(q) −Aq−  with d (dxyk) being creation (annihilation) operator, ˜ xyk 0 0 −Ak+ −M(q) k momentum in the z direction. This Hamiltonian gives (3) an energy spectrum [Fig. 2(a)] that was shown in Ref. ˜ 2 2 2 with q± = qx ± iqy, E(q) = C0 + C1qz + C2(qx + qy), and [31] with a similar model to have the energy levels in ˜ 2 2 2 agreement with those of Weyl orbits [1,2] M(q) = M0 + M1qz + M2(qx + qy). We transform this into a tetragonal lattice model using the substitution πt n = (n + γ), n = 0, 1,... (7) 1 2 2 ka/(2πφ) + Nx + 1 qi → sin ki, qi → 2 (1 − cos ki). (4) ai ai Here, ka is the length of the surface Fermi arcs, Nx the Here, ax = ay = a and az = c are the tetragonal lattice number of lattice sites in the x direction, and γ the phase constants. The Hamiltonian then becomes offset that depends on the distance between the Weyl points [32]. Additionally, to confirm the quantum Hall HD(k) =c0 + c1 cos kz + c2(cos kx + cos ky) state of this system, we compute the Chern numbers of + t(sin kxσzτx − sin kyτy) some energy gaps formed by the discrete levels of Weyl orbits using the Streda formula [33] (AppendixB). The + [m0 + m1 cos kz + m2(cos kx + cos ky)]τz, (5) nonzero Chern numbers indicate the existence of a quantized Hall conductance and are equal to the number of where the Pauli matrices σi and τi represent the spin chiral modes at the edges of our system. Moreover, such and orbital degrees of freedom, respectively. These mod- a quantum Hall state depends on the film thickness as els of DSMs preserve both time-reversal and inversion the energy levels do [Eq. (7)], which is regarded as a sig- symmetries, and have a rotational symmetry axis par- nature of the quantum Hall effect based on Weyl orbits allel to the z axis. If we confine the DSM in any di- [3,7,8]. rection not parallel to z, each of its surfaces has two As mentioned in some experiments [3,4], the Weyl disconnected Fermi arcs with opposite spin and chiral- orbit levels are related to the quantum confinement effect ity. From now on, unless stated otherwise, we choose in topological semimetal films. In order to illustrate this 3

3 2 2 1 1 0 0 -1 -2 -1 -2

(a) (b) (c)

FIG. 2. Energy spectra against the magnetic flux φ for (a) a WSM slab confined in the x direction [Eq. (6)], and (b) a WSM bulk [Eq. (9)] with kx = πζ/(Nx + 1). The green lines represent semiclassical result determined by Eq. (7) for (a) ka = 2π/3 and γ = 0.7, (b) ka = 0 and γ = 1. (c) The bulk Landau bands of a WSM confined in the x direction and subjected to a magnetic flux φ = 0.01. The green dots indicate confinement subbands formed from the chiral Landau bands. The Chern numbers of some energy gaps are shown in (a) and (b). All the results are computed with Nx = 20. idea, we consider those levels [Eq. (7)] in the limit where The spectrum is shown in Fig. 2(b), where we see that the Fermi arcs vanish, ka = 0, and consider only the distinct energy gaps still exist even in the absence of sur- tunneling process of the quasi-particles via the bulk chiral face states. The energy levels also agree well with the Landau levels. In this case, we see that the energy levels semiclassical result given by Eq. (7) for ka = 0, as de- 0 take the form n = πt(n+1)/(Nx +1) and do not change noted by the green lines. To find their origin, we show with the magnetic field. On the other hand, we know the Landau bands obtained from Hamiltonian (9) for a that in the bulk WSM the linear dispersion of the chiral fixed magnetic flux [Fig. 2(c)]. Then, by adding the con- Landau levels at the Weyl points is given by finement effect [Eq. (10)], each continuous Landau band becomes a set of discrete confinement subbands. We see Ec = ±tkx. (8) that the energy levels in Fig. 2(b) are actually the confinement subbands formed from the chiral Landau levels, A comparison between 0 and E indicates that the mo- n c as we have predicted. In the presence of boundaries, these mentum k is quantized in a similar way to an infinite x chiral Landau subbands hybridize with the surface Fermi quantum well problem, i.e., k = (n+1)π/(N +1). This x x arcs and bend towards zero energy in the low field regime, indicates that the energy levels of Weyl orbits in the limit giving rise to the Weyl orbit levels shown in Fig. 2(a). As k = 0 are confinement subbands stemming from the chi- a a result, we can conclude that the quantization of Weyl ral Landau levels. To further substantiate this argument, orbits gives energy levels that are chiral Landau subbands we carry out some numerical calculations as follows. We hybridizing with the surface Fermi arcs. impose an additional PBC on the Hamiltonian (6) in the x direction to eliminate the surface states and preserve From this interpretation of the Weyl orbit levels, we only the bulk spectrum. The Hamiltonian is then given gain two new perspectives about the QHE in topologi- by cal semimetal films. First, the QHE induced by Weyl orbits is intrinsically two-dimensional instead of three- X 1 † 0 dimensional as being claimed before [6, 24, 25] since the HW = d t sin kxσx + [M + t cos kx 2 kxykz energy levels originate from the quantum confinement ef- kx,y,kz fect. Further evidence for this 2D nature is that the Hall 0 + t cos(kz + 2πφy)]σz dkxykz resistance of our WSM film is a factor of the Klitzing 2 † 0 constant RK = h/e , in agreement with the experiments, + d (t σz − itσy)dk (y+1)k + h.c.. (9) kxykz x z whereas in a so-claimed 3D QHE induced by the charge- density wave [34], the Hall resistance is much smaller In order to obtain only the bulk spectrum of our WSM than RK . Second, if our material somehow has the bulk slab, i.e., to add the effect of quantum confinement, we chiral Landau levels but with no open Fermi arcs on its apply the particle-in-a-box method (PiBM) (Appendix surfaces, e.g. the arcs are combined into a closed Fermi C) by diagonalizing this Hamiltonian only at the mo- loop [1], the gaps between the chiral Landau subbands menta still remain. Hence, a thickness-dependent QHE or quan- πζ tum oscillation in relatively thick topological semimetal kx = , ζ = 1, 2 ...,Nx. (10) Nx + 1 films is not conclusive evidence for observing either Weyl 4

† the momenta kx and ky with ladder operators l, l as

p † p † kx → πφ(l + l), ky → −i πφ(l − l). (12)

7 The Hamiltonian then becomes 6 √ 5 M(kz) 2 πφtl 4 HW(kz) = √ † , (13) 2 πφtl −M(kz) 0 1 with M(k ) = m − t0k2/2 − 2πφt0 l†l + . Then, us- z z 2 T ing the trial wavefunctions (α1 |ν − 1i , α2 |νi) for ν = 1, 2,... and (0, |0i)T for ν = 0, where ν is the band index, (a) (b) we can obtain the spectrum of HW(kz) from the secular equations as √ √ FIG. 3. (a) Energy spectrum against the magnetic flux φ K + πt0φ − E 2 πφt ν for a WSM slab confined in the z direction. The green lines det νkz√ √ = 0, (14) 2 πφt ν −K + πt0φ − E represent energy levels determined by Eq. (19). The Chern νkz numbers of some energy gaps are shown. (b) The bulk Landau bands of a WSM confined in the z axis and subjected to a for ν = 1, 2,..., and magnetic flux φ = 0.01. The green dots indicate confinement − K + πt0φ − E = 0, for ν = 0. (15) subbands formed from the chiral Landau band. All the results (ν=0)kz are computed with Nz = 20. t0 2 0 Here, Kνkz = m − 2 kz − 2πt φν. The 1D Landau bands of the WSM are then given by orbits or surface Fermi arcs, in contrast to the usual ex- 0 pectation [3,7,8, 13]. Moreover, since such a QHE can t 2 0 E0(kz) = −m + kz + πt φ, (16) take place even without the surface Fermi arcs, a ques- 2 q tion then naturally arises; is the QHE observable in our 2 2 0 Eν (kz) = ± Kνk + 4πt φν + πt φ. (17) WSM when it is confined in the z direction, which has z no nontrivial states on its boundaries? Now, we can transform the dispersion of the zeroth level 2 into a lattice version by substituting kz → 2(1 − cos kz), which yields B. Films confined in the z direction 0 0 0 E0(kz) = −M − 2t − t cos kz + πt φ. (18) We now consider a WSM confined in the z direction [Fig. 1(c)] with a magnetic field Bkzˆ given by vector po- Finally, we take into account the effect of quantum con- tential A = (0, Bax, 0). Similar to the previous case, the finement by employing the PiBM, i.e., replacing kz = bulk spectrum of our WSM decomposes into 1D Lan- πζ/(Nz + 1) with ζ = 1, 2,...,Nz. The subbands of chi- dau bands dispersing along the z direction, as shown in ral Landau level read Fig. 3(b), and the chiral level still evolves differently from πζ (ζ) = −M − 2t0 − t0 cos + πt0φ, (19) other bands. If we introduce confinement in z, we expect 0 N + 1 that the chiral Landau subbands also form energy gaps z distinct from others, and the QHE in relatively thick films and they evolve linearly with respect to the magnetic flux will thus be observable. The Hamiltonian of our system φ, as shown by the green lines in Fig. 3(a). On the other now reads hand, the Landau bands with ν > 0 move away from 0 and make the gaps between the chiral Landau subbands X 1 H = d† [M + t0 cos(k + 2πφx)]σ observable. These gaps also have nonzero Chern num- W 2 xkz z x,k,z bers, indicating the existence of the QHE. † 0 + t sin(k + 2πφx)σy dxkz + dxkzt σzdxk(z+1) † 0 IV. DISCUSSION + dxkz(t σz − itσx)d(x+1)kz + h.c. (11) with k being momentum in the y direction. The energy Based on the spectrum in Fig. 3(a), we make two in- spectrum of this Hamiltonian is shown in Fig. 3(a). Since ferences: this slab does not have topological surface states, we can First, we see that when our WSM is confined in the z also obtain its spectrum analytically by using the PiBM. direction, the dependence of its chiral Landau subbands We start with adding the effect of magnetic field B on on φ deviates considerably from that of the slab perpen- the continuum model [Eq. (2)], which is done by replacing dicular to the x axis. In particular, if confinement is in 5 the x direction, and we neglect the surface Fermi arcs, points isotropic. In the vicinity of the first Weyl point the subbands stay constant as the field strength increases (−kW sin θ, 0, kW cos θ), the Hamiltonian reads [Fig. 2(b)], which agrees with the semiclassical equation. 0 0 On the other hand, if the WSM is confined in z, the HW(k ) = tk1(cos θσx + η sin θσz) + tk2σy subbands depend linearly on the flux and are unevenly t0 + tk0 (sin θσ − η cos θσ ) − (k02 + k2 + k02)σ , (22) spaced, and thus behave differently from Eq. (7). From 3 x z 2 1 2 3 z Eq. (19), we know that such a difference stems from the 0 0 second-order terms in k of Hamiltonian (2) while Eq. (7) where k1 = k1 + kW sin θ and k3 = k3 − kW cos θ. When was obtained by using a linear dispersion of the chiral a magnetic field B is applied, the momenta k1 and k2 Landau levels [1]. However, another problem then comes are quantized in terms of the ladder operators similar to up: those quadratic terms in k contribute substantially Eq. (12). We then have HW(kz) = H1(kz) + H2(kz), to the spectrum in Fig. 3(a) but does not affect the one where the first-order term is in Fig. 2(b). We investigate this problem and revisit the −η cos θ sin θ H (k ) = tk0 semiclassical Weyl orbits in subsectionIVA. 1 z 3 sin θ η cos θ Second, since a DSM can be regarded as a combina- † † † p η sin θ(l + l) cos θ(l + l) − (l − l) tion of two WSMs with opposite spin and chirality, we + t πφ † † † , expect that a (001) film of DSM will have a spectrum cos θ(l + l) + (l − l) −η sin θ(l + l) with nonzero spin Hall conductance. We demonstrate (23) this idea in subsectionIVB, and show that such a quan- and the second-order one reads tum spin Hall effect (QSHE) is observable in the DSM candidate Cd As . t0 3 2 H (k ) = − 2πφ(2l†l + 1) + k02 σ . (24) 2 z 2 3 z We now make an approximation by solving the A. Dependence of the chiral Landau levels on the 2 chiral Landau level from H1 and treating H2 as k -terms in HW(k) a perturbation. With the eigenvector |Lc+i = [1/p2(1 + cos θ)](η sin θ, 1 + cos θ)T |0i, the chiral Lan- 2 We study how the k -terms in Hamiltonian (2) af- (0) dau level at the first Weyl point is given by E (k0 ) = fect the dispersion of the zeroth Landau bands by find- + 3 ηtk0 . The first-order correction from the perturbation ing their analytical expressions when the magnetic field 3 H is B is applied along an arbitrary direction. We trans- 2 form the vectors (k , k , k ) of the crystal frame into the 02 x y z (1) 0 0 k3 ˆ E+ (k3) = hLc1|H2|Lc1i = t πφ + cos θ. (25) (k1, k2, k3) of the magnetic frame (k3kB) using a 3D rota- 2 tion matrix. For simplicity and without loss of generality, ˆ The dispersion of the zeroth Landau level is then ex- we assume that B ⊥ ky and rotate the vectors about the pressed as y axis (k2 ≡ ky) as       E+(k3) = ηt(k3 − kW cos θ) k1 cos θ 0 − sin θ kx 2 0 (k3 − kW cos θ) k2 =  0 1 0  ky (20) +t πφ + cos θ. (26) 2 k3 sin θ 0 cos θ kz The first-order correction from H2 allows us to repro- with θ being the angle between k3 and kz. The continuum duce both Eq. (8) and Eq. (18), and hence we neglect all Hamiltonian of our WSM then reads higher-order corrections. A similar calculation yields the chiral Landau level HW(k) = t(cos θk1 + sin θk3)σx + k2σy + M(k)σz, (21) crossing the second Weyl point (− sin kW , 0, cos kW ) as

0 2 2 2 E−(k3) = −ηt(k3 + kW cos θ) where M(k) = m − t (k1 + k2 + k3)/2. The 2 eigenvalues of this Hamiltonian are given by E = 0 (k3 + kW cos θ) p 2 2 2 2 +t πφ + cos θ. (27) ± M + t [(cos θk1 + sin θk3) + k2], and the Weyl 2 points are located at (−kW sin θ, 0, kW cos θ) and p 0 These expressions demonstrate that the contribution (kW sin θ, 0, −kW cos θ), kW = 2m/t . from H2 to the chiral Landau levels vanishes when the magnetic field is perpendicular to the line connecting the two Weyl points, i.e., Bkxˆ. Besides, the effect of H2 be- 1. Bulk chiral Landau levels comes more significant when the magnetic field direction approaches the z axis. To assess the reliability of these To find the chiral Landau bands analytically, we con- results, we compare Eqs. (26) and (27) with the Landau sider the parameters {t, t0, m} that satisfy the constraint bands obtained from the lattice model of Eq. (21), as 0 t kW /t = η = ±1, which keeps the velocity at the Weyl shown in Fig.4. 6

휃 = 60° 휃 = 45° 휃 = 30°

(a) (b) (c)

FIG. 4. Landau bands of our WSM bulk in a magnetic ﬁeld with (a) θ = 60◦, (b) θ = 45◦, and (c) θ = 30◦. The Landau bands (magenta lines) are obtained from the lattice model of Hamiltonian (21). The green lines represent the analytical results given by Eqs. (26) and (27). The blue lines are also those equations but being modiﬁed to a lattice version by Eq. (4). The green and blue lines approach each other as the Weyl points come close to the Γ-point.

2. QHE in a WSM slab conﬁned in an arbitrary direction dicular magnetic ﬁeld is described by

X 1 † n HW = a t sin(k + 2πφx1)σy After getting a more general expression for the chi- 2 x1kx3 ral Landau bands to explain the difference between x1,k,x3 o Figs. 2(b) and 3(a), it is interesting to seek an expres- 0 + M + t cos(k + 2πφx1) σz ax1kx3 sion for the Weyl orbit levels taking into account the 2 + a† (t0σ − it cos θσ ) a effect of the k -terms. The Onsager - Bohr - Sommerfeld x1kx3 z x (x1+1)kx3 quantization for a classical orbit reads + a† (t0σ − it sin θσ ) a + h.c.. x1kx3 z x x1k(x3+1) (31) I p · dr = 2π (n + γ). (28) ~ The energy spectra of this WSM are shown in Fig.5 for different tilting angles, which shows a transition from Fig. 2(a) to Fig. 3(a), and agree well with Eq. (29). When After some calculations, we get the confinement direction deviates from the x axis, both the spacing and the dependence on magnetic flux φ of the chiral Landau subbands change. Interestingly, we can B h p i see how the subbands evolve by computing the Chern = A + B − 2AB + B2 + t2 − 2πφt02 cos2 θ n t0 cos θ numbers of the gaps between them. (29) In DSMs, we assume that the physics is somewhat similar. As the experiments about Weyl orbits are often conducted in (112) films of Cd3As2, this result may give with better explanations to those magneto-transport studies, e.g. the high values of the Landau indices in the quantum 0 Hall measurements. 0 π(n + γ) 2 Lz A = t 0 cos θ + kW sin θ and B = 2πφt . Lz ka (30) 0 B. Quantum spin Hall effect in topological Dirac Here, Lz = (Nz + 1)/ cos(Θ − θ) is dimensionless, ka = semimetal thin films 2kW sin Θ, θ and Θ determine the directions of magnetic field and confinement, respectively. In the limits Θ = θ → 90◦ and Θ = θ → 0◦, this equation reproduces Finally, we show how the QHE and QSHE induced Eq. (7) and Eq. (19). by chiral Landau subbands take place in the DSM films. For our DSM [Eq. (5)], a magnetic field Bkzˆ preserves the We now compare Eq. (29) with the results obtained spin-z component (Sz) and lifts the spin degeneracy even from the lattice model of Hamiltonian (21). A WSM in the absence of Zeeman coupling. A DSM film grown confined in the k3 direction and subjected to a perpen- along its rotational symmetry axis (z) and subjected to 7

Θ = 휃 = 60° Θ = 휃 = 45° Θ = 휃 = 30°

6 3 4 5 2 3 4 1 2 3 0 -1 1 0 0

(a) (b) (c)

◦ FIG. 5. Energy spectra against the magnetic ﬂux φ for a WSM slab conﬁned in the k3 direction determined by (a) Θ = 60 , (b) Θ = 45◦, and (c) Θ = 30◦. The green lines represent semiclassical result determined by Eq. (29) with γ = 0.7. The Chern numbers of some energy gaps are shown. All the results are computed with a thickness corresponding to N3 = 20.

2 B is described by 0↓(qz) = C0 − M0 + (C1 − M1)qz eB µ g B X 1n † B p H = d m τ + m cos(k + 2πφx)τ + (C2 − M2) − . (35b) D 2 xkz 0 z 2 z ~ 2 x,k,z To make a comparison with the lattice model spectrum − t sin(k + 2πφx)τy + λzφσzτg dxkz [Fig. 6(a)], we let Ci = 0 and transform the equations † using Eq. (4), which gives + dxkz(m2τz − itσzτx)d(x+1)kz † o + d m1τzdxk(z+1) + h.c., (32) πζ xkz 0↑(ζ) = m0 + 2m2 + m1 cos − πm2φ + λzgsφ, Nz + 1 h2 (36a) where λz = 2 and 8πmea πζ 0↓(ζ) = −m0 − 2m2 − m1 cos + πm2φ − λzgpφ. gs 0 Nz + 1 τg = , (33) 0 gp (36b) These expressions show that the presence of Zeeman in- gs and gp are the effective g-factors for s and p orbitals, teraction still keeps the chiral Landau subbands evolving respectively. For simplicity, we choose λzgs = 2 and linearly with the magnetic flux but changes their slopes. λzgp = 1. Solving the eigenvalues of HD numerically Specifically, the gs-factor modifies the slope of the spin- gives an energy spectrum as shown in Fig. 6(a), which up chiral Landau subbands whereas gp affects the spin- is composed of two sets of chiral Landau subbands with down ones, which reflects the band inversion of DSMs. opposite chirality and spin polarization. They evolve in different directions with respect to the magnetic field and Due to our choice of parameters, the spin-up chiral thus cross each other to form the energy gaps that can be Landau subbands are hole-like, and the spin-down ones seen as an overlap of two separate gaps. This spectrum are electron-like. As a result, when our material has is also explained by the quantum confinement picture as an additional boundary, the spin-up subbands are bent shown in Fig. 6(b). According to this figure, we should downward and give right-handed edge states while those notice that the two sets of subbands cross if π/(Nz +1) < with spin-down disperse upward giving left-handed edge |kD|, i.e., states. The combination of these edge modes may give π rise to the coexistence of chiral and helical edge states. Lz > . (34) |qD| For example, we consider an energy gap of Chern number C = 1, which is a combination of C↓ = 7 and C↑ = −6 To find how the chiral Landau subbands depend on the gaps. When the Fermi level lies within this gap, the spin magnetic field, we also follow the same calculation as Hall conductance is quantized as [35, 36] presented in Sec.IIIB and get an analytical expression of those subbands e σs = −Cs (37) 2 2π 0↑(qz) = C0 + M0 + (C1 + M1)qz eB µBgsB with Cs = (C↑ − C↓)/2 = −6.5. At the boundary, energy + (C2 + M2) + , (35a) levels are bent and form 7 left-handed spin-down and ~ 2 8

2 4 1 2 0 1 -1 0 -1 -2 -2 4 -3 3 2 0 0 0 -1

(a) (b) (a) (b)

FIG. 6. (a) Energy spectrum against the magnetic flux φ FIG. 7. Energy spectra against the magnetic flux φ for a for a DSM slab. The red lines denote spin-up chiral Landau (001) film of Cd3As2 described by two different models: (a) subbands given by Eq. (36a). The blue lines denote spin- Hamiltonian (3) with parameters given by Ref. [42], the film down chiral Landau subbands given by Eq. (36b). The Chern thickness is 50 nm, and (b) the hyperbolic model proposed by numbers of some energy gaps are shown. (b) Bulk Landau Ref. [44], the film thickness is 120 nm. Some Chern numbers bands of the DSM confined in the z axis and subjected to are shown in both figures by comparing with Fig. 6(a). a magnetic flux φ = 0.01. The red and blue dots indicate confinement subbands formed from the chiral Landau bands. All the results are computed with Nz = 20. formed by the chiral Landau subbands: those with nonzero spin Chern number, those with nonzero Chern number and zero spin Chern number, and one with van- 6 right-handed spin-up edge states, which would result ishing Chern and spin Chern numbers. We are interested in a total of 1 chiral edge state and 6 helical edge states. in the gaps with nonzero spin Chern number, which do Hence, both quantum Hall and quantum spin Hall phases not exist if the film is too thin but is not observable coexist in our system whose time-reversal symmetry is if the film is too thick. If we assume an energy gap broken by the magnetic field. In this case, the helical ∆ ≥ 10 K ∼ 1 meV to be observable, the majority edge states exist because S is approximately conserved z of gaps in Fig. 7(a) and all gaps in Fig. 7(b) satisfy the [37]. This scenario is well-known in graphene [38–40], condition. Hence, the QHE induced by the chiral Lan- where the helical edge states are protected by an addi- dau subbands can be observed in relatively thick films tional symmetry instead of the time-reversal symmetry compared to the conventional quantum wells (5-20 nm). as in topological insulators. If we use this simple model [Fig. 7(a)] to explain the QHE observed in Ref. [19], it may capture two features of the experiment, i.e., the absence of ν = 1 plateau and Estimation for Cd As 3 2 the high resistance at strong magnetic field. Neverthe- less, just like the explanation given in that reference, it is Since a quantized Hall conductance has recently been also unable to clarify the difference in activation energy observed in (001) films of Cd3As2 [19] under a strong between the states at even and odd filling factors. On magnetic field, we roughly estimate whether the QHE the other hand, if the actual distance between the Dirac and QSHE induced by the chiral Landau subbands can nodes agrees with the Landau level spectroscopy mea- be observed in such films. Depending on the growth con- surements, the QHE induced by chiral Landau subbands dition, Cd3As2 can be either centrosymmetric or non- may not be observable in such a quantum Hall experi- centrosymmetric [41]. Here, we consider a centrosym- ment since the film thickness does not satisfy Eq. (34). metric Cd3As2, which is more popular and can be described by Hamiltonian (3) with parameters obtained by fitting with the ab initio calculation [42]. Additionally, V. CONCLUSION as the distance between the Dirac points of Cd3As2 obtained by the Landau level spectroscopy measurements [43–46] is one order smaller than the ab initio calcula- Our study gives a generic and simplified picture of the tions [14, 47, 48], we also consider the hyperbolic model QHE induced by confinement subbands stemming from proposed by Ref. [44]. A detailed procedure is presented the chiral Landau levels in topological semimetal films. in AppendixE. The spectra computed for these two mod- Using a minimal model of WSM, we have demonstrated els are shown in Fig.7, where, in both cases, we use that the energy levels of Weyl orbits originate from the gs = 18.6 [44] and choose gp = 10 for simplicity. confinement subbands of the chiral Landau levels that In general, both spectra have three types of gaps hybridize with the surface Fermi arcs. We have then 9 studied how the k2-terms in a WSM Hamiltonian affect be written explicitly as the evolution of the chiral Landau bands with respect to T the magnetic field. We have found a general expression  †      Ξ1k ∆ Ω 0 ··· 0 Ξ1k for the Weyl orbit levels and shown how the QHE takes † † X  Ξ2k  Ω ∆ Ω ··· 0   Ξ2k  place in a WSM film confined in an arbitrary direction. H =       , W  .  ......   .  Furthermore, when examining a DSM confined in its ro- k  .   . . . . .   .  † † tational symmetry axis, we may have not only explained Ξ 0 0 ··· Ω ∆ ΞNxk Nxk the QHE recently observed but also predicted the coexis- (A3) tence of both quantum Hall and quantum spin Hall states where Ξxk are 2Ny-spinors, and the two 2Ny × 2Ny ma- in such a system. trices ∆ and Ω are given by

 D O 0 ···O†  1 1 Ny  †  ACKNOWLEDGMENTS  O1 D2 O2 ··· 0  ∆ =  . . . . .  (A4)  ......   .  This work was supported by the Japan Society † ON 0 ···O DN for the Promotion of Science KAKENHI (Grant Nos. y Ny −1 y JP19K14607 and JP20H01830) and by CREST, Japan with Dy = [M + cos(k + 2πφy)]σz, Oy = (σz + iσy)/2, Science and Technology Agency (Grant No. JP- and Ω = 1Ny ×Ny ⊗(σz −iσx)/2. Notice that we keep the MJCR18T2). periodic boundary condition in the y direction, which requires 2πφNy = 2nπ or Ny = n/φ with n = 1, 2,.... On the other hand, as we can write the magnetic flux as Appendix A: Lattice Hamiltonian in a magnetic field a ratio of two integers, i.e., φ = p/q, we choose n = p and thus q = Ny for simplicity. This choice corresponds to a magnetic Brillouin zone that has only one k-point in We present how to obtain the Hamiltonian in Eq. (6) the y direction. and write its explicit form. First, we take the Fourier transform of the lattice model (1) into real space as Appendix B: Computing Chern numbers 1 X h H = − d† (t0σ − itσ )d + d† (t0σ + W 2 m z x m+ˆx m+ˆx z m To illustrate the quantum Hall effect induced by chiral † 0 † 0 itσx)dm + dm(t σz − itσy)dm+ˆx + dm+1 (t σz + itσy)dm Landau levels, we compute Hall conductance of the gaps y between them using the Streda formula [33] for a 3D † † 0 † 0 i − 4dmMσzdm + dmt σzdm+ˆz + dm+ˆzt σzdm , (A1) system of sizes Lx × Ly × Lz. For instance, when the Fermi energy lies in a gap, the Hall conductivity of films where m = (x, y, z) is a dimensionless position vector. perpendicular to the x axis can be determined by In the presence of a magnetic field B, the Hamiltonian ∂n(EF ) is modified by the Peierls substitution so that it remains σyz = −e (B1) invariant under the U(1) gauge transformation. In par- ∂Bx ticular, the hopping integrals are changed, such as with n(EF ) being the particle density. We transform this formula as † † (y) d tdm −→ d t exp(iϑ )dm (A2) m+ˆy m+ˆy m 2 ∆n(EF ) e ∆N(EF ) qay az σyz = −e = − . (B2) (y) 2π R m+ˆy ∆Bx h Lx∆p Ly Lz with ϑm = − A · yadyˆ . Using the vector po- Φ0 m (x) (y) tential A = (0, 0, Bay), we get ϑm = ϑm = 0 and Here, N(EF ) is the number of states below Fermi energy, (z) Φ /Φ = p/q, and the magnetic field is varied by chang- ϑm = −2πφy. Inserting these phases into Eq. (A1) and yz 0 taking Fourier transform along the z direction, we obtain ing p and keeping q constant. The Hall conductance is Eq. (6). then given by Since the vector potential breaks the translation sym- 2 e ∆N(EF ) 1 metry along the y direction, the momentum ky is no Gyz = σyzLx = − , (B3) longer a good quantum number, and we are not able h ∆p Nyz to take the Fourier transform in this direction. We can where Nyz is the number of k-points in the 2D magnetic retain the translation symmetry by introducing the so- Brillouin zone. Similarly, for slabs perpendicular to z, called magnetic unit cells and the corresponding mag- the Hall conductance reads netic Brillouin zone. Nevertheless, keeping the Hamilto- 2 nian in the real space representation along the y direction e ∆N(EF ) 1 Gxy = σxyLz = − . (B4) is a simpler choice for our work. The operator can then h ∆p Nxy 10

However, in order to show a connection with the edge Nevertheless, if nontrivial edge states are present, e.g., states forming in such QHE, instead of finding the Hall the WSM confined along x, this method becomes a rough conductance we compute the Chern numbers defined by approximation as it is unable to demonstrate the local- ized edge states, but it is still sufficient to demonstrate ∆N 1 ∆N 1 C(xy) = and C(yz) = . (B5) the formation of bulk subbands due to the quantum con- ∆p Nxy ∆p Nyz finement. These Chern numbers give the number of chiral edge modes in our quantum Hall system. Appendix D: Semiclassical quantization

Appendix C: Particle-in-a-box Method Chiral Landau levels at the two Weyl points are given by When solving the slab geometry of a lattice Hamilto- nian which does not have surface states, we can obtain Ec(k3) = ηt(k3 − kW cos θ) its spectrum simply by quantizing the momentum along 1 +t0 (k − k cos θ)2 + πφ cos θ, (D1) the confinement axis as in the particle-in-a-box problem. 2 3 W To illustrate this idea, we consider a particular instance, i.e., the WSM given by Eq. (1) confined in the z direction with thickness Lz. To obtain its spectrum, we Ec(k3) = −ηt(k3 + kW cos θ) often Fourier transform Eq. (1) along z into real space 1 +t0 (k + k cos θ)2 + πφ cos θ. (D2) and get the Hamiltonian 2 3 W X n † HW = dkz (2 − cos kx − cos ky) σz − sin kxσx To obtain the semiclassical quantization of Weyl orbits, k,z we apply the Bohr - Sommerfeld - Onsager quantization σ σ o condition, i.e., H p · r = 2π (n + γ). We split the line − sin k σ d − d† z d − d† z d . (C1) ~ y y kz kz 2 k(z+1) k(z+1) 2 kz integral into two parts This matrix has a size of 2Nz × 2Nz with Nz being • The integration along the Fermi arcs: the number of lattice sites, z = 1, 2,...Nz. The open Z 2 2 boundary condition, or hard-wall boundary condition, is ~ ~ E ~ kaE imposed by setting the hopping terms between sites 1 p · dr = 2 Sk ≈ ka2 = , (D3) arcs eBa eΦ t πφ t and Nz to be zero, which implies that the wavefunctions always vanish at sites 0 and (Nz + 1). Hence, where Sk is dimensionless, Θ is the angle determin- the width of our quantum well, or the film thickness, ing the confinement direction, and ka = 2kW sin Θ. is Lz = (Nz + 1)az. This boundary condition is an ap- propriate approximation since we are mainly interested • The integration along the bulk chiral Landau levels in the low-energy limit of our model. Diagonalizing H √ gives the energy spectrum of our WSM film. Z 0 2 ! 0 0 −t kW sin θ + ∆ A more convenient way to get the same result is the p · dr = ~k32Lz = 2~Lz 0 lls t cos θ aforementioned PiBM, which is to set kz → πζ/(Nz + 1) with ζ = 1, 2,... in our lattice Hamiltonian Eq. (1). (D4) 0 Nz + 1 Notice that the periodicity of our lattice restricts 0 < Here, Lz = is dimensionless, θ is the ζπ/(N + 1) < π [49], and thus we have the integers cos(Θ − θ) z angle for the direction of magnetic field, and ∆ = ζ ∈ [1,Nz]. Using this substitution, we can obtain the 2 0 0 exact energy spectrum of Hamiltonian Eq. (C1) just by t − 2t cos θ(πt φ cos θ − E). diagonalizing the matrices Using these equations, we get the energy levels of Weyl orbits as given in Eq. (29). hζ (k) = − sin kxσx − sin kyσy πζ + M − cos kx − cos ky − cos σz. (C2) Nz + 1 Appendix E: Laudau quantization of topological The eigenvalues are simply Dirac semimetals " 2 2 We consider the continuum model of DSM given by εζ (k) = ± sin kx + sin ky Eq. (3). In the presence of a magnetic field Bkzˆ, the

1 momenta (qx, qy) are replaced by the ladder operators as # 2 πζ 2 + M − cos kx − cos ky − cos . (C3) 1 † −i † Nz + 1 qx → √ (l + l), qy → √ (l − l). (E1) 2lB 2lB 11 and the Zeeman interaction is added. The Hamiltonian and then becomes 2 ↑ 0↓(qz) = C0 − M0 + (C1 − M1)qz HD(kz) 0 HD(kz) = ↓ (E2) eB µBgpB 0 HD(kz) + (C2 − M2) − . (E4) ~ 2 with  √  On the other hand, the hyperbolic model of Cd3As3 µB gsB 2A † proposed by Ref. [42] can be obtained simply by replacing ↑ M(qz) + l 2 lB HD(qz) = Eqz +  √  the function M˜ q) with 2A µB gpB ( l −M(qz) + lB 2 q and ˜ (j) (j) (j) 2 (j) 2 (j) 2 2 M (q) = M0 + (M3 ) + (M1 qz) +M2 (qx +qy).  √  µB gsB 2A (E5) M(qz) − − l (j) ˚ (j) H↓ (q ) = E +  √ 2 lB  . Here, the parameters are A = 2.75 eVA, M0 = D z qz (j) (j) (j)  2A † µB gpB  ˚2 ˚2 − l −M(qz) − −0.06 eV, M1 = 96 eVA , M2 = 18 eVA , M3 = lB 2 ˚2 (j) (j) ˚2 0.05 eVA , C0 = −0.219 eV, C1 = −30 eVA , 2 −2 † (j) ˚2 Here, we have E(qz) = C0 + C1qz + C2lB 2l l + 1 and C2 = −16 eVA . Following the same calculation as 2 −2 † M(qz) = M0 + M1qz + M2lB 2l l + 1 . According to before, we get the spectrum in Fig. 7(b) and the analyt- Ref. [42], the parameters are A = 0.889 eVA,˚ M0 = ical expressions 2 2 −0.0205 eV, M1 = 18.77 eVA˚ , M2 = 13.5 eVA˚ , C0 = ˚2 ˚2 q −0.0145 eV, C1 = 10.59 eVA , and C2 = 11.5 eVA .A ˜ 2 2 ˜ 2 0↑(qz) = C0 + M0 + C1qz + M3 + M1kz numerical diagonalization of HD(kz) yields the spectrum in Fig. 7(a). Besides, using the eigenvectors (|0i , 0, 0, 0)T eB µBgsB T + (C2 + M2) + , (E6) and (0, 0, 0, |0i) , we can analytically obtain the chiral ~ 2 q Landau bands ˜ 2 2 ˜ 2 0↓(qz) = C0 − M0 + C1qz − M3 + M1kz 2 0↑(qz) = C0 + M0 + (C1 + M1)qz eB µBgpB + (C2 − M2) − . (E7) eB µBgsB ~ 2 + (C2 + M2) + (E3) ~ 2

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