Finite-Size Effects in the Dynamic Conductivity and Faraday Effect Of

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Our calculations insulator as generally as possible, we use the no-spill current also show that optical dipole transitions between edge states condition at the boundary, which is more suitable to be used contributes to an almost constant value of e2/h in the dynamic with a continuum Hamiltonian model. We will discuss this Hall conductivity that remains constant even for frequencies boundary condition in more details. exceeding the band gap. Combining all three types of optical transitions among the edge and bulk states, the total dynamic conductivities of the finite disk are found to agree with the main features calculated from the massive Dirac model with an additional parabolic dispersion term that breaks electron- hole symmetry in an extended system. The finite-size effects in the conductivities are also seen in the magneto-opticalFara- day rotations. Here, we also study the ‘finite-size effects’ along the out-of-plane direction by considering a substrate in- terfacing the quantum anomalous Hall insulator. The finite FIG. 1. (Color online) Schematic of the system under investigation. thickness of the substrate gives rise to Fabry-Pérot oscillations The cyan (dark) area is a quantum anomalous Hall insulator nanodisk of the Faraday effect, which are found to exert a stronger in- with a radius R and a positive finite mass in the Dirac model Eq. (1), fluence on the Faraday rotation spectrum than the effect of the surrounded by a yellow (light) domain with a mass going to negative finite disk size. infinity, to ensure a topologically nontrivial domain wall. The outer This article is organized as follows. In Sec. II we describe region is infinitely large. The linear response dynamics of this system the model Hamiltonian and boundary condition of the quan- is studied in Sec. III by initializing a weak perpendicular incident tum anomalous Hall insulator disk and derive semi-analytic light with energy ω. expressions for the eigenfunctions and energies. Sec. III de- scribes our calculations and results for the dynamic longitudinal and Hall conductivities using the Kubo formula. A flow The wavefunctions and energy eigenvalues are obtained by diagram plotting the real parts of the longitudinal and Hall solving the eigenvalue problem hDψ = ǫψ, where ψ is a two- conductivities is discussed in Sec. IV, which approaches the component spinor. In view of the rotational symmetry of the behavior of a Chern insulator with increasing disk radius. In system, the eigenvalue problem can be facilitated by using the 5 Sec. V, we provideresults for the magneto-opticalFaraday ro- following wavefunction ansatz in polar coordinates , tation of the quantum anomalous Hall insulator disk and study the effects of finite disk radius and the presence of an under- ψ ilφ Al(ρ) ψ(ρ, φ) = ↑ = e iφ , (2) ψ ! e Bl(ρ)! lying substrate. Sec. VI summarizes our work. ↓ where ρ is the radial position, φ the azimuthal angle, and l is an integer corresponding to the orbital angular momentum II. THEORETICAL MODEL quantum number. Combining Eq. (1) and Eq. (2), one obtains the wavefunctions for states lying both outside and inside of As depicted in Fig. 1, we consider a quantum anomalous the bulk energy gap Hall insulator in a disk geometry with a radius R. The low- energy physics of a quantum anomalous Hall insulator is de- eilφ (ǫ + M) J (β ρ˜) Ψ (ρ, φ) = l ln , ǫ > M scribed by the massive Dirac Hamiltonian, ln iφ √ 2 2  √Nln +e ǫ M Jl+1(βlnρ˜)! | |  −  hD = v (σ p) zˆ + Mσz, (1)  × ·  ilφ  e (M + ǫ)Il(blnρ˜) Φ (ρ, φ) = , ǫ < M where v is the band velocity of the Dirac model, M > 0 is the  ln iφ √ 2 2  √ ln e M ǫ Il+1(blnρ˜)! | | Zeeman interaction that breaks time-reversal symmetry and  N − −  (3) provides a bulk band gap, and σ = (σx, σy, σz) is the 3-vector  formed by the Pauli matrices corresponding to the spin degree where we have defined the system’s characteristic fre- of freedom. Under periodic boundary conditions, the massive quency ωR v/R, the dimensionless wavenumbers β = Dirac Hamiltonian gives a Chern number 1/2. To incorporate ≡ √ǫ2 M2/(~ω ) and b = √M2 ǫ2/(~ω ), as well as the finite-size effects, we need to specify the appropriateboundary R R scaled− dimensionless radial distance− ρ ˜ ρ/R [0, 1]. Here condition at the edge of the disk. While a vanishing wavefunc- J (x) is the Bessel function and I (x)≡ the modified∈ Bessel tion boundary condition is suitable for graphene with zigzag l l function, both of the first kind. The normalization coef- edge termination, there is no reason to assume that such a con- ficients Nln and ln follow from the normalization condi- dition also applies in our case without appealing to atomistic N 2π R tion of the wavefunctions 1 = dφ dρρ ψ(ρ, φ) 2 = details at the boundary. In particular, since we aim to providea 0 0 1 | | treatment of the finite-size effects of quantum anomalous Hall 2πR2 d˜ρρ˜ ψ(ρ, φ) 2, and are consequentlyR R given by 0 | | R 3

2 2 2 2l + 1 M/ǫln Nln = 2πR ǫln(ǫln + M) J (βln) + J (βln) − Jl(βln)Jl+1(βln) , ǫ > M, " l l+1 − β # | |  ln (4)   2 2 2 2l + 1 M/ǫln  ln = 2πR (M + ǫln)ǫln Il (bln) Il+1(bln) − Il(bln)Il+1(bln) , ǫ < M. N " − − bln # | |  

In a finite-sized geometry, a clear delineation between bulk fers from that in the original Berry’s paper27, which addressed and edge states does not exist because there is always some a situation without a topological domain wall. For this reason, degree of mixing between bulk and edge states. However, for we refer to the boundaryconditionwe use here as the topolog- the sake of terminology, we shall label Ψln (having energies ical infinite mass boundary condition. Armed with the above, beyondthe bulk band gap) and Φln (having energies within the one has the following equations for the eigenenergies ǫln from bulk band gap) as the bulk and edge states, respectively. This Eq. (3) terminology is supported by examining the radial probability density profile as a function of the scaled dimensionless radius 2 2 (ǫln + M) Jl(βln) = ǫln M Jl+1(βln), ǫ > M, ρ˜ in Fig. 2. It is seen that the in-gap states Φln displays a  − q − | | (5)  monotonically increasing profile towards the edge of the disk  2 2 (M + ǫln)Il(bln) =+ M ǫln Il+1(bln), ǫ < M. and indeed behaves like edge states. On the other hand, the  q − | |  out-of-gap states Ψln exhibits oscillations as a function of the Equation (5) is transcendental equations with multiple roots radial positionρ ˜ and can therefore be associated with bulk ǫln for each given l, where n = 1, 2,... indicates the mul- states. tiplicity. While a number of candidates for Chern insulators have been proposed in the literature6,29–33, we choose the band parameters corresponding to the quantum anomalous Hall in- 5 1 sulator of a Bi2Se3 thin film with v = 5.0 10 m s− and a Zeeman energy 2M = 0.1eV. As a low-energy× · effective theory, the massive Dirac model is valid up to a certain energy cutoff ǫc. For concreteness we use ǫc = 0.3eV as the energy cutoff, noting that the main findings of our work are not dependent on the precise value of ǫc with ǫc M. We therefore only seek the numerical roots of Eq. (5)≪ within the range [ ǫc, ǫc]. Figures− 3 (a), (c) and (e) show the obtained energy spectrum for ǫ as a function of l for different values of R. One can identify features that correspond predominantly to bulk states and edge states. The bulk state spectrum contains a gap in which a one-way chiral edge dispersion runs across. As R FIG. 2. (Color online) Radial probability density as a function of the increases from 10nm to 100nm, the delineation between the bulk and edge spectra becomes more evident. In the limit R scaled dimensionless radiusρ ˜ (with R = 30 nm) for several typical → values of angular momentum l and energies ǫ . Hollow-shape-dotted that can be achieved physically when R ~v/M, the chiral ln ∞ ≫ curves indicate the edge states Φln and solid-shape-dotted curves the edge state dispersion can be obtained by expandingthe second bulk states Ψln. equation in Eq. (5) using a large R expansion:

ǫl ǫ (l + 1/2) , (6) To determine the energy eigenvalues, we impose the no- ≃− 0 spill boundary condition that the outgoing radial component with ǫ ~ωR[1 + ~ωR/(2M)]. We note that the 1/2 on the of the current vanishes at the edge of the disk. Within the 0 ≡ massive Dirac model, this condition corresponds to taking the right hand side originates from the spin angular momentum of mass term M outside the disk region to infinity and is some- the electron. Gratifyingly, this approximate analytic disper- times called the infinite mass boundary condition5. We note sion is in excellent agreement with the exact numerical results that in principle there can be two choices for the sign of the as shown by the red solid line in Fig. 3. Using Eq. (6), the number of the in-gap states can be estimated as 2M/(~ωR) , mass term, with M(ρ > R) . Since we assume M(ρ ⌊ ⌋ R) > 0, the choice M(ρ > R→) ±∞ is appropriate here for≤ a where ... denotes the floor function. Figures 3 (b), (d) and → −∞ (f) show⌊ the⌋ density of states of the calculated spectrum from topologically nontrivial domain wall, ensuring a change in the 34 topological character across the disk boundary25,26. This re- the expression sults in the following constraint between the two components 1 1 of the wavefunction27,28 ψ /ψ = αeiφ, with α = 1, (cf. Ap- D(ǫ) = Im , (7) ↓ ↑ − π ǫν ǫ iη pendix A for more information). This choice of M(ρ> R) dif- A Xν − − 4

throughout this work. The nonvanishing peaks of the density of states in the gap indicate the existence of the in-gap chiral edge states. When the radius increases, the DOS profile ex- pectedly becomes smoother as more states are introduced into the system. On the other hand, a small radius (R . ~v/M) en- hances the quantum confinement effect as seen from the more prominent resonances from the individual quantum states.

III. DYNAMIC CONDUCTIVITY

We now introduce into the system a weak, linearly polarized alternating current probe ﬁeld that is normally incident on the quantum anomalous Hall insulator disk. With the obtained energy spectrum and wavefunctions, we proceed to cal- culate the longitudinal and Hall optical conductivities using the Kubo formula35 in the real space representation (derivation is provided in Appendix C):

2 e f ′ f0 m xi m′ m′ x j m σ (ω) = 2iω 0 − h | | i h | | i , (8) i j h ∆ǫ ω iη R2 Xmm′ − − where i, j x,y , m(m ) is a collective label for the relevant ∈ { } ′ quantum numbers, f0 is the Fermi-Dirac distribution function, and ∆ǫ = ǫ′ ǫ is the energy difference between the final (primed) and the− initial (unprimed) states in a transition, and ω is the photon energy of the incident light. The matrix elements in Eq. (8) capture the transition pro- cesses among the bulk states Ψln and edge states Φln and there are three types of transitions, i.e., edge-to-edge (E-E), edge- to-bulk (B-E), and bulk-to-bulk (B-B). Using the expressions of the wavefunctions Eq. (3) together with their normalization FIG. 3. (Color online) Energy spectrum (energy ǫ versus angular mo- coefficients Eq. (4), we obtain the following matrix elements mentum quantum number l) and the corresponding density of states 5 1 for the three types of transitions (DOS) for v = 5.0 10 m s− and M = 0.05 eV. The region shaded in light blue (gray)× indicates· the bulk energy gap, within which the red (darker) line shows the analytic dispersion Eq. (6) of the chiral S ′ x S S ′S 1 1 δl,l′+1 ψl n ψln = R l n ,ln , (9) edge state. * ′ ′ y! + I ′ ′ i i! δl,l 1! − ′−

where S ′, S B, E stand for bulk and edge states, δl,l′ is the where ‘Im’ stands for imaginary part, is the area of the Kronecker delta∈ { symbol,} and S ′S is a dimensionless radial l′n′,ln disk and the broadening parameter is setA as η = 2.4 10 3 eV integral deﬁned by I × −

+ + + 2 2 2 2 (M ǫ′)(M ǫ)Il′ (bl′n′ ρ˜)Il(blnρ˜) (M ǫ′ )(M ǫ )Il′+1(bl′n′ ρ˜)Il+1(blnρ˜) − − , S ′ = S = E,  √ p /(πR2)  l′n′ ln  N N 1 (ǫ + M)(M + ǫ) J (β ρ˜)I (b ρ˜) (ǫ 2 M2)(M2 ǫ2) J (β ρ˜)I (b ρ˜) S ′S 2  ′ l′ l′n′ l ln ′ l′ +1 l′n′ l+1 ln = d˜ρρ˜  − − − , S ′ = B, S = E, (10) l′n′,ln  p 2 I Z0  √Nl n ln/(πR )  ′ ′ N 2 2 2 2 (ǫ + M)(ǫ + M) J (β ρ˜) J (β ρ˜) + (ǫ M )(ǫ M ) J + (β ρ˜) J + (β ρ˜)  ′ l′ l′n′ l ln ′ l′ 1 l′n′ l 1 ln  − − , S ′ = S = B.  √N pN /(πR2)  l′n′ ln  

Equation (9) expresses an angular momentum selection rule: angular momenta ∆l = 1. The remaining radial integrations transitions are allowed only between states with a change in ( S ′S ) in Eq. (10) are computed± numerically. Il′n′,ln 5

Figure 4 shows our results for the real (blue) and imaginary whether our system behaves as a Chern insulator in response (red) parts of the longitudinal conductivity σxx(ω) when the to an a.c. field as well. To address this question, we compare Fermi level ǫF = 0 for different values of R separated into the the calculated conductivities of our finite-size system for large three contributions: E-E (first row), B-E (second row) and B- R with the conductivities from the low-energy Dirac model B (third row). First, we note that the finite size of the disk has H = d(k) σ, defined on an extended system. Here we con- a different effect on the E-E conductivity contribution com- sider two cases· and take the spin vector d(k) to be linear in pared to the other two types of contributions involving the momentum with d (k) = Aky, Akx, M , and quadratic in 1 − bulk. As R is increased, the E-E conductivity [Figs. 4 (a1)- 2 2 momentum with d (k) = Aky, Akx, M B(k + k ) , where (a3)] remains approximately the same while the B-E and B-B 2 − − x y A, B > , M contributions [Figs. 4 (b1)-(b3) and (c1)-(c3)] display consid- ( 0) and are band parameters independent of mo- d k erable changes. According to the selection rule in Eq. (9), mentum. The winding number spanned by the vector ( ) is there can only be one E-E transition along the chiral edge evaluated by this integral state dispersion from below to above the Fermi level, there- 1 d (∂k d ∂k d ) fore there is always only one peak in the E-E conductivity = 2k · x × y . d 3 (11) regardless of the size of the disk. The peak’s position is seen C 4π Z d to shift toward ω = 0 with increasing R, because the edge Accordingly, the quantized Hall response of the linear model, states become more closely spaced with their energy separa- as applicable for the surface states of three-dimensional topo- tion ǫ0 1/R [Eq. (6), to first order]. In contrast, for the 2 36 logical insulators, is σxy = sgn(M)e /2h , and that of the B-E≈ [Figs.∝ 4 (b1)-(b3)] and B-B contributions [Figs. 4 (c1)- second model, as applicable for Chern insulators, is σxy = 0 (c3)], since the number of possible transitions is directly pro- 2 when M < 0 and σxy = e /h when M > 0. portional to the number of bulk states, the number of peaks For extended systems, momentum is a good quantum num- increases and the conductivity approaches a smooth contin- ber and the dynamic conductivity for the above models can be uous curve as R increases. The threshold beyond which the calculated from the Kubo formula37 in the momentum repre- B-B contribution becomes finite corresponds to the bulk en- sentation as usual ergy gap 2M. 2 d k f f n jα m m jβ n The corresponding dynamic Hall conductivity σxy(ω) is = ~ m n h | | i h | | i σαβ(ω) i 2 − (12) shown in Fig. 5. The above description for the longitudi- Z (2π) ǫm ǫn ǫm ǫn (ω + iη) Xmn − − − nal conductivity is also applicable here, if we note that the 2 ˜ + ˜+ ˜ + ˜+ ∗ iG d k jα− jβ− jα− jβ− roles of the reactive and dissipative components are played = 0 + , by the real and imaginary parts of σ (ω) respectively. For 4π d ω 2d + iη ω + 2d +iη xy Z   both the longitudinal and Hall conductivities, the E-E con-  −    tribution displays a smooth profile across all values of fre- ˜   where jα = ∂kα H. Analytic results of the dynamic con- quency, which is the result of only one possible edge-to-edge ductivity tensor for the linear Dirac model can be obtained transition. The B-E contribution is smaller than both E-E and and is available38. For the quadratic Dirac model, we com- B-B contributions by an order of magnitude for the smallest pute the dynamic conductivity numerically from Eq. (12) with radius R = 30nm and is further suppressed with increasing 2 parameters A = ~v 0.3291eV nm, B = ~ /(2 me) radius. For frequency within the bulk gap 2M, its profile ex- 0.0381eV nm2 and M≃ = 0.0500eV.· Interestingly, our re-≃ hibits many closely spaced sharp peaks corresponding to the sults show· that not only electronic wavefunctions and disper- many possible edge-to-bulk transitions, and is smooth for fre- sions, but boundaryconditionscan also change the topological quency beyond the gap. The opposite behavior is seen in the property of a system. B-B contribution. Its profile exhibits wild fluctuations due to Figure 6 shows the dynamic longitudinal and Hall con- even more possible bulk-to-bulk transitions for frequency be- ductivities for the three cases of finite-sized disk with R = yond the bulk gap, which are suppressed when the disk ra- 150nm, linear and quadratic Dirac models defined on an ex- dius is increased. There is an important distinction between tended system. For the longitudinal conductivity σxx, we see Fig. 4 and Fig. 5. As shown in Fig. 4, the E-E contribution of from panel (a) that the qualitative behavior of all three sets Im[σxx(ω)] quickly drops to zero with increasing ω. However, of results resemble each other closely for frequencies beyond the corresponding reactive contribution in the Hall conductiv- the band gap. Within the gap, there is a peak in Re(σxx) anda ity, Re[σ (ω)], becomes a constant e2/h when ω exceeds the xy corresponding zero-crossing in Im(σxx) near ω = 0 in the case value of the bulk gap 2M. We find that the flatness of this of finite-sized disk, which are absent in the extended systems. plateau as a function of ω is not sensitive to the system size, These features originate from the E-E transition [Fig. 4 (a3)]. as seen by comparing Figs. 5 (a1)-(a3). Adding the three E- For the Hall conductivity σxy, we see from panel (b) that the E, B-E and B-B contributions, we conclude that the total Hall qualitative behavior the finite-size result matches closely with conductivity in the direct current limit within our finite-size the extended quadratic Dirac model result, while the extended 2 2 model is e /h, rather than e /2h as would have been expected linear Dirac model result displays a similar trend with increas- for massive Dirac electrons in an extended system. ing frequency but the overall profile is shifted upward. This Therefore, in the direct current limit, our system behaves as shift is consistent with the direct current limit of the Hall con- a quantum anomalous Hall insulator with an (integer) Chern ductivity for the three cases. Close to ω = 0, the finite-size and number, i.e., a Chern insulator. This motivates the question extended quadratic Dirac models both give a Hall conductivity 6

2 FIG. 4. (Color online) Plots of the longitudinal conductivity σxx (in unit of G0 = e /h) versus frequency ω for diﬀerent radius R and for contributions arising from diﬀerent types of transitions. Blue (dark) indicates the real part and red (light) the imaginary part. The light blue (gray) regions indicate the extent of the bulk energy gap. Panels (a1)-(a3) depict the contribution from edge-to-edge (E-E) transitions, (b1)-(b3) the contribution from edge-to-bulk (B-E) transitions, and (c1)-(c3) the contribution from bulk-to-bulk (B-B) transitions. Values of v and M are the same as in Fig. 3.

2 FIG. 5. (Color online) Hall conductivity σxy (in unit of G0 = e /h) versus frequency ω for diﬀerent radius R and for contributions arising from diﬀerent types of transitions. Blue (dark) indicates the real part and red (light) the imaginary part. 7

2 2 Re(σxy) of e /h whereas the linear Dirac model gives e /2h. tation angle is given by In addition, a strong peak is apparent near ω = 0 in Im(σxy) for finite-sized disk due to E-E transition [Fig. 5 (a3)]. 1 t t θF = arg E+ arg E . (14) 2 − − Our results for the Faraday angle are shown in Fig. 8, com- IV. FLOW DIAGRAM puted with different values of disk radius R and substrate thickness a. θF shows both finite-size effects due to a finite R Further insights can be obtained by mapping σxx and σxy and Fabry-Pérot resonances46 due to the presence of an under- onto a parametric plot using ω as the parameter. In the di- lying substrate, which serves as an optical cavity. Let us first rect current regime, such a σxx-σxy plot generates a flow dia- focus on the first row (a1)-(a4) with a small radius R = 30nm. gram and was studied for quantum anomalous Hall insulators The effect of discrete energy eigenstates in a finite disk is re- 39 40,41 theoretically and experimentally using the system’s size, flected in θF as fluctuations with frequency [panel (a1)], sim- temperature and gate voltage as parameters. The flow diagram ilar to the behavior seen in the dynamic conductivities. As of a quantum Hall insulator consists of two semicircles with the substrate thickness is increased [panels (a2)-(a4)], these a unit diameter located at ( e2/2h, 0)42,43 [dashed black semi- fluctuations become overwhelmed by the periodic oscillations ± circles in Fig. 7], while the flow diagram of the topological resulting from the Fabry-Pérot resonance. For larger radius insulator surface states forms a semicircle centered at the ori- values across the rows, the fluctuations due to finite disk ef- gin [dot-dashed black semicircle in Fig. 7]. To study the flow fect become smoothened and only Fabry-Pérot oscillations re- diagram for our finite quantum anomalous Hall insulator disk, main. Since the lowest Fabry-Pérot resonance frequency is we focus on only the E-E contribution of Re(σxx) and Re(σxy) c/(2a √εSi), more resonances appear for a thicker substrate. for frequency ω< 2M. In this frequency range, contributions When the frequency value is equal to the band gap, we notice involving bulk states are almost suppressed and thus will not a kinkin θF that becomes more apparent for thinner substrates, show in the flow diagram. The various shape-dotted lines in resulting from the onset and peak features in σxx and σxy. An- Fig. 7 show the flow diagram for different values of disk radius other interesting feature is the non-monotonic increase in the from 10nm to 150nm. The diagram resembles large circular envelope of the Fabry-Pérot oscillations, which become ap- arcs for small Rs, and gradually approaches a semicircle cen- parent when both a and R are large [panels (b4), (c4), and tered at ( e2/2h, 0)as R is increased (contrast the purple curve (d4)]. The envelopeof oscillations is seen to increase with fre- − for 10nm and the blue curve for 150nm). This is consistent quency when the frequency is increased towards the gap and with our finding that the finite disk subjected to the topological then decreases when the frequency is further increased outside infinite mass boundary condition becomes a Chern insulator. the gap. For large disks, the low-frequency Faraday rotation is closest to the universal value arctan α 7.3mrad [indi- cated by the lower dashed line in− panels (c1)≈ and− (d1)] as pre- V. FARADAY EFFECT dicted for extended systems20 when the substrate thickness a is small, where α = 1/137 is the fine structure constant. Here The dynamic Hall conductivity and its quantized value near the value θF arctan α is consistent with the low-frequency ≃− 2 the direct current limit can be probed with magneto-optical limit of the dynamic Hall conductivity Re(σxy) e /h. For ≃ Faraday effect in topological systems20,38,44. We consider a small radius [panel (a1)], θF deviates from this universal value setup consisting of a quantum anomalous Hall insulator nan- due to finite size effects. odisk on top of a dielectric substrate, surrounded by vacuum. For concreteness, we choose silicon45 as the substrate, which VI. CONCLUSION has a dielectric constant εSi = 11.68. Using the scattering matrix formalism, the transmitted electric field can be expressed as In conclusion, we have studied the finite-size effects in the dynamical conductivities and magneto-optical Faraday rota- t 1 0 E = t¯ I r¯′ r¯ − t¯ E , (13) tion of a quantum anomalous Hall insulator disk. We find that B − T B T the continuum massive Dirac model subjected to the topolog- 0 where E is the incident field,r ¯, r¯′ and t¯, t¯′ are 2 2 matrices ical infinite mass boundary condition becomes a Chern insu- accounting for single-interface reflection and transmissi× on, re- lator with a unit Chern number. In addition, the overall fre- spectively. The subscript ‘T’ labels the top interface at the quency dependence of both the longitudinal and Hall conduc- quantum anomalous Hall insulator between vacuum and the tivities matches those calculated from an infinitely extended substrate while ‘B’ labels the bottom interface between the Chern insulator model described by the massive Dirac Hamil- substrate and vacuum. Equation (13) expresses the transmit- tonian with an additional parabolic term. A flow diagram ted field in the form of geometric series resulting from Fabry- plotting the edge state contribution to Re(σxx) versus Re(σxy) Pérot–like repeated scattering at the top and bottom interfaces. shows a semicircle in agreement with that expected for a in- The incident light is normally illuminated onto the setup and teger quantum Hall insulator. Our numerical results further is linearly polarized. show that the edge-to-edge transitions constitute the domi- Resolving the transmitted electric field into its positive and nant contribution in the Hall conductivity for frequencies not t t t negative helicity components E = Ex iEy, the Faraday ro- only within but also beyond the bulk band gap. Studies on the ± ± 8

FIG. 6. (Color online) Total longitudinal conductivity σxx [panel (a)] and Hall conductivity σxy [panel (b)] as a function of frequency, blue (dark) for real part and red (light) for imaginary part. Results from the ﬁnite disk (with R = 150 nm) are shown by the thick solid lines, those from the linear Dirac model by the dashed lines and those from the quadratic Dirac model by dot-dashed lines. Conductivities are expressed 2 in unit of G0 = e /h.

size quantization in the Faraday rotation spectrum. Our findings highlight the importance of finite-size effects in optical measurements of dynamic Hall conductivity and Faraday effect in which the laser spot coverage exceeds the sample size.

VII. ACKNOWLEDGMENT

Work at USTC was ﬁnancially supported by the Na- tional Key Research and Development Program (Grant No.s: 2017YFB0405703 and 2016YFA0301700), the National Nat- ural Science Foundation of China (Grant No.s: 11474265 and 11704366), and the China Government Youth 1000-Plan Tal- ent Program. Work at Alabama was supported by startup funds from the University of Alabama and the U.S. Depart- ment of Energy, Oﬃce of Science, Basic Energy Sciences under Early Career Award #DE-SC0019326. We are grateful to the supercomputing service of AM-HPC and the Supercom- puting Center of USTC for providing the high-performance computing resources.

FIG. 7. (Color online) Scaling behavior [Re(σxx) versus Re(σxy)] for the E-E contribution. The black dot-dashed line shows the behavior Appendix A: Infinite mass boundary condition of the topological insulator surface states and the black dashed line the integer quantum Hall state. The various shape-dotted lines show For completeness, here we include the derivation for the in- the behavior for the quantum anomalous Hall insulator disk with dif- finite mass boundary condition following Ref. 27. Rather than ferent radii, which approaches the behavior of the integer quantum vanishing wavefunction, this boundary condition requires that Hall state as R increases. Conductivities are expressed in unit of 2 the normal component of the current at each point on the G0 = e /h. boundary vanishes, and for our case this means eρ j(R) = 0 along the radial direction. For a general two-dimensional· massive Dirac model defined on a domain in the real space, magneto-opticalFaraday rotation of a small disk show fluctua- D H = CI + Aσ ( i ) + Mσz, tional features as a function of frequency. These features arise · − ∇ due to optical transitions between discrete bulk states and is 2 we consider the total energy E = d x Ψ†HΨ, which can be present only for frequencies beyond the band gap. The direct written as RD current limit of the Faraday angle shows noticeable deviations α 2 with decreasing disk radius from its theoretical value arctan E = d x Ψ†(CI + Mσz)Ψ (where α = 1/137) based on extended model. In the presence Z D h i of an underlying substrate, we find an interplay between the 2 iA d x (Ψ†σΨ) Ψ† σΨ Fabry-Pérot resonances and the finite-size effects due to the − Z ∇· − ∇ · D h i 9

FIG. 8. (Color online) Faraday rotation angle θF (in milliradian) as a function of ω for diﬀerent values of radius R and substrate thickness a. The light blue (gray) region indicates the extent of the bulk energy gap. The two blue dashed horizontal lines indicate the values of 3 3 arctan(α/2) 3.649 10− and arctan α 7.289 10− , with α being the ﬁne structure constant. − ≈− × − ≈− ×

∗ justification for applying the same boundary condition to two- = d2 x Ψ (CI + Mσ )Ψ iA d2 x Ψ σ Ψ † z † dimensional quantum anomalous Hall insulators, here we also Z h i − Z h · ∇ i! D D show explicitly that in the massive Dirac model with a finite iA dl en j. (A1) M such a boundary condition cannot be applied. − I∂ · D If we try to find edge states by setting one component, say In the last equality, Gauss’ theorem in two dimensions is used the first one, of the spinor wavefunction in Eq. (3) to vanish to rewrite the last term into a line integral. Equation (A1) on the boundary (˜ρ = 1) just as what was done in Ref.5, that implies that E = E iA dl e j, where j = Ψ σΨ is the means for l , 0, we have ǫ = M. However, this makes the ∗ ∂ n † ± current operator. Since− E isD real, we· have e j = 0. Next we second component also vanish, so states with ǫ = M do not H n ± show how this condition leads to the constraint· between the exist. two components of the wavefunction Ψ= (Ψ , Ψ )⊤: Then how about the states with ǫ = 0? To that end, we ↑ ↓ need to first set ǫ = 0 and then solve the Schrödinger equa-

0 = eρ j(R) = Ψ†eρ σΨ = Ψ†(σx cos φ + σy sin φ)Ψ tion which results in a modified Bessel equation as the radial · · iφ equation. The wavefunction takes the form below [up to a 0 e− Ψ iφ ~ = Ψ∗ Ψ∗ iφ ↑ = 2Re Ψ∗Ψ e . normalization coefficient and κ = M/( v) > 0] ↑ ↓ e 0 ! Ψ ! ↓ ↑ ↓ n o ilφ Il(κρ) Therefore, on the boundary, the two components of the Φln(ρ, φ) e iφ . (B1) ∼ ie Il+1(κρ)! wavefunction satisfy Ψ /Ψ = iαeiφ, with α = 1 for − 27↓ ↑ ± sgnMin sgnMout = 1 , where Min/out represents the mass For this wavefunction, neither of the component can vanish ± term for the region of interest (the nanodisk)or the region sur- on the boundary with a finite M. rounding it.

Appendix C: Derivation of the Kubo Formula in the Real-Space Appendix B: Vanishing wavefunction boundary condition Representation

The zigzag boundary condition is used in graphene along a Here we provide a derivation of the real-space Kubo for- zigzag edge where one of the pseudospin components is taken mula used in this paper based on density matrix. Let us start to vanish. In addition to the fact that there is no microscopic from the Liouville-von Neumann equation i~∂tρ = [H, ρ] and 10 consider a system subjected to a time-dependent perturbation magnetic contributions37, can now be obtained as H = H + H (t). In linear response, the density matrix is ex- 0 ′ p p panded to first order ρ = ρ0 + δρ, with δρ satisfying i~∂tδρ = Πab(ω) Πab(0) σab(ω) = − [H0, δρ]+[H′, ρ0]. Assuming a sinusoidal time dependenceof iω H (t) e iωt, δρ therefore satisfies ~ωδρ = [H , δρ]+[H , ρ ]. ′ − 0 ′ 0 2πi ( f0′ f0)∆ω ∝ = G − m x m′ m′ x m Solving for the matrix element of δρ from above and using the 0 ∆ω ω η a b mm i h | | i h | | i relations H0 m = ǫ m , ρ0 m = f0(ǫ) m gives A X′ − − | i | i | i | i 2πi ω = G0 ( f0′ f0) + 1 f0(ǫ′) f0(ǫ) − ∆ω ω iη = Xmm ! m′ δρ m ~− m′ H′ m , (C1) A ′ − − h | | i ǫ′ ǫ (ω + iη) h | | i − − m xa m′ m′ xb m × h | | i h | | i where f0 is the Fermi-Dirac distribution. Using the sum- 2πi ( f0′ f0)ω = G − m xa m′ m′ xb m mation convention and denoting the elementary charge as 0 ∆ω ω iη h | | i h | | i Xmm e = e > 0, the perturbation is H = eA p /m = ev E /(iω). A ′ − − ′ b b b b 2πi | | = Using the Heisenberg equation of motion, m′ va m + G0 ( f0′ f0) m xa m′ m′ xb m , (C4) m x m = m x , H m / ~ m x m ǫ hǫ /| |~ i − h | | i h | | i ′ ˙a ′ [ a ] (i ) ′ a ( ′) (i ). For A Xmm′ ah two-dimensional| | i h | system| i of area≈ h ,| the| i average− paramag- p A where ∆ω (ǫ′ ǫ)/~. netic current density is Ja = Tr δρ ja where ja = ( e)va/ is ≡ − the single-particle current density{ operator.} This gives− A The second term vanishes identically because of the com- mutativity among the components of the coordinate operator, p Ja = m′ δρ m m ja m′ (C2) h | | i h | | i ( f0′ f0) m xa m′ m′ xb m Xmm′ − h | | i h | | i Xmm ( f f )∆ǫ2 ′ 2π 0′ 0 m xa m′ m′ xb m = f0′ m′ xb xa m′ f0 m xa xb m = G0 − h | | i h | | i Ab, h | | i− h | | i − ~ ∆ǫ ~(ω + iη) Xm′ Xm Xmm′ − A = f0 m [xb, xa] m = 0, (C5) 2 m h | | i where G0 e /h is the conductance quantum and ∆ǫ ǫ′ ǫ X ≡ ff ≡ − is the energy di erence between two transition states. The where in deriving the second line we have used the com- Π paramagnetic current-current correlation function ab(ω), de- pleteness of states. Therefore substituting Eq. (C5) back into p =Π fined through Ja ab(ω)Ab, is therefore Eq. (C4), we obtain the final expression of the conductivity Eq. (8) in the main text with area = πR2 for a disk of radius ∆ǫ2 2π ( f0′ f0) m xa m′ m′ xb m R, A Πab(ω) = G − h | | i h | | i. − ~ 0 ∆ǫ ~(ω + iη) Xmm′ − A ( f ′ f0)ω m xa m′ m′ xb m (C3) σ˜ (ω) = 2i 0 − h | | i h | | i. (C6) ab ∆ω ω iη R2 Xmm′ − − The conductivity, consisting of both paramagnetic and dia-

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