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Light-induced phase crossovers in a quantum spin Hall system

In this work, we theoretically investigate the light-induced topological phases and finite-size crossovers in a paradigmatic quantum spin Hall (QSH) system with high-frequency pumping optics. Taking the HgTe quantum well for example, our numerical results show that circularly polarized light can break time-reversal symmetry and induce the quantum anomalous Hall (QAH) phase. In particular, the coupling between the edge states is spin dependent, and it is related not only to the size of the system but also to the strength of the polarized pumping optics. By tuning the two parameters (system width and optical pumping strength), we obtain four transport regimes, namely QSH, QAH, edge conducting, and normal insulator. These four different transport regimes have contrasting edge conducting properties which will feature prominently in transport experiments on various topological materials.

The quantum anomalous Hall (QAH) insulator, which was first proposed by Haldane [29], is another key member of the family of quantum Hall insulators. A QAH insulator can have a quantized Hall conductivity even without an external magnetic field that breaks the timereversal symmetry [29][30][31][32][33][34][35]. The QAH effect was first experimentally observed in Ref. [30]; in another work on finite-size effects in QAH systems [13], they adopted the magnetic doping (Mn doping) to break the time-reversal symmetry and introduce the QAH phase in the HgTe quantum well. By appropriate tuning of the doping concentration and the system size, they have obtained a variety of topological phases [13]. Yet, this experimental success [13] in realizing QAH states proved difficult to follow up upon, since the doping concentration is difficult to tune continuously in experiments. Therefore, departing from previous works [13], we shall provide a more experimentally accessible alternative to realizing QAH states, using circularly polarized light instead of magnetic doping to break the time-reversal symmetry in HgTe quantum well. This hinges on the fact that the intensity of circularly polarized light can be easily tuned continuously in experiments, and that Floquet driving can induce a host of exotic phases in various media, such as disorder-induced Floquet topological insulators [36], quenched topological boundary modes induced by Floquet engineering [37], Floquet semimetal with exotic topological linkages [38], Floquet mechanism for non-Abelian fractional quantum Hall states [39], photoinduced half-integer quantized conductance plateaus in topological insulators [40]. In this work, based on numerical calculations with high-frequency Floquet expansion, we find that circularly polarized light can break the timereversal symmetry and introduce the QAH phase in HgTe quantum well. Our findings complement other works that utilizes circularly polarized light to induce new topological phases in Graphene [41][42][43][44][45][46][47][48][49][50][51], but with significant differences. In particular, the coupling between the edge states is spin dependent, and it is related not only to the width of the system but also to the strength of the polarized pumping optics. By appropriately tuning of the system size and strength of the pumping optics, we obtain four transport regimes including QSH, QAH, edge conducting (EC), and normal insulator (NI).
The paper is organized as follows. In Section II, we introduce the model Hamiltonian which we use. In Section III, we derive the corresponding Floquet effective Hamiltonian by using the high frequency expansions. In Section IV, we calculate the energy dispersions under the open boundary condition along y direction and periodic boundary conditions along x direction with fixed light intensity and different system size by numerically diagonalizing the corresponding tight-binding Hamiltonian. In Section V, we give and discuss our phase diagram. Finally, we summarize our results in Section VI.
In particular, the h(k) and h * (−k) in the matrix diagonal blocks of Eq. (1) provide the simplest ansatz that ensures time inversion symmetry in the model Hamiltonian. Besides, h(k) and h * (−k) are equivalent to the two-dimensional Dirac model or Qi-Wu-Zhang (QWZ) model [53][54][55], which has a quantized Hall conductance without Landau levels. Thus, h(k) and h * (−k) correspond to two equal but opposite QAH Hamiltonians. Therefore, the net Hall conductance of the Hamiltonian (1) is zero, but the spin Hall conductance for h(k) or h * (−k) is nonzero. This phenomenon corresponds to QSH state. The BHZ model can be viewed as two copies of QWZ models with opposite Hall conductance.

III. FLOQUET HAMILTONIAN
To realize QAH states in the Hamiltonian (1), circularly polarized light can be used to break the timereversal symmetry between the two spin blocks of the matrix (1) to result in nonzero net Hall conductance.
The optical driving field can be expressed as E(t) = ∂A(t)/∂t = E(cos(ωt), cos(ωt + ϕ), 0), where A(t) = A(t + T ) is the time periodic vector potential with period T = 2π/ω, ω is the frequency of the optical field, and E is the amplitude of the optical field. The optical field is circularly polarized with ϕ = ±π/2 and A(t) = A a (sin(ωt), ± cos(ωt), 0), where A a = E/ω and ± corresponds to right-and left-handed circularly polarized waves, respectively. To guarantee the off-resonant regime in which the central Floquet band is far away from other replicas, the driven frequency can be set as ω = 150 meV, which is much larger than the bandwidth [22,[25][26][27]56].
After irradiation with light, the photon-dressed effective Hamiltonian is given by Within the Floquet theory [19,20,22,41,[57][58][59][60] and A 0 = eA a / which denotes the light intensity, the effective static Hamiltonian in high-frequency regime can be expanded as where we use H n,m = 1 T T 0 H(k, t)e i(n−m)ωt dt with n and m are integers, and the concrete analytical expressions for H 0,0 , H 0,−1 , and H 0,1 can be found in Appendix A.
From Eq.
The energy dispersions of the Floquet Hamiltonian (4) are given by To estimate the validity of the high-frequency expansions developed here quantitatively, we evaluate the maximum instantaneous energy of the time-dependent Hamiltonian (2) averaged over a period of the field < ω at the Γ point (k x = k y = 0). The optical field parameters have to satisfy the condition A 0 A/( ω) < 1. In the high-frequency regime ω ∼ 2.2789 × 10 2 THz ( ω = 150 meV), one can obtain A 0 0.41 nm −1 . In the following numerical calculations, we have regularized the continuous model Hamiltonian (4) into the corresponding lattice Hamiltonian where topological invariants are well-defined (the detailed derivations can be found in Appendix C). All the numerical results are based on the lattice regularized Hamiltonian, even though they all pertain to the region around the Γ point. Here, the definitions of E ↑± and E ↓± are the energies of two spin-up and spin-down edge bands under open boundary condition along y direction. Here, QSH is topological protected for both spin-up and spin-down edge states, QAH is topological protected for one of them only, and EC for none of them, although they still exist unless broken by disorder. The threshold of the energy gap for calculating the phase boundaries is 0.01 meV. The other parameters are given as A0 = 0.06 nm −1 , ax = ay = 1 nm, ϕ = π/2, and ω = 150 meV. Here, ax and ay are the lattice constants along x and y directions respectively. Notice that we have mapped the continuous model Hamiltonian (4) into the lattice Hamiltonian (the detailed derivations can be found in Appendix C).

IV. ENERGY DISPERSIONS
In order to explore how sufficiently small system sizes can lead to Floquet phase transitions in QSH system, we calculate the tight-binding energy spectra of HgTe/(Hg,Cd)Te quantum well with fixed light intensity in a stripe geometry with sample width L y along y direction, and the system length in x direction is infinite. The energy gap opening and closing in the energy disper- sions of the edge states are the most important feature of finite-size effects with fixed light intensity. As shown in Fig. 1, we calculate the energy spectra of the HgTe/(Hg,Cd)Te with fixed light intensity A 0 = 0.06 nm −1 and different width L y under open boundary condition along y direction and periodic boundary conditions along x direction (the detailed derivations can be found in Appendix C). Besides, we also give the detailed derivations of the tight-binding model under open boundary condition along x direction and periodic boundary conditions along y direction in Appendix D. In Fig. 1, the red thick line indicates the upper band E ↑+ of the spinup edge states, the red thin line indicates the lower band E ↑− of the spin-up edge states, the blue thick line indicates the upper band E ↓+ of the spin-down edge states, and the blue thin line indicates the lower band E ↓− of spin-down edge states.
To illustrate the physical picture for the four phases with different system width in Fig. 1, we provide a schematic diagram for the four phases with the clockwise (red lines) and anti-clockwise (blue lines) charge currents of the spin-up and spin-down edge states respectively as shown in Fig. 2.
In Fig. 1(a), the system width is L y = 850 nm which is sufficiently large that there is no coupling between edge states. It is found that the energy dispersions for both spin-up and spin-down edge states are gapless and each spin state provides a charge conductance e 2 /h as shown in Fig. 2(a). This indicates that the system is in the QSH state which is time-reversal symmetry broken due to circularly polarized light, the corresponding charge conductance along x direction is 2e 2 /h as shown in Fig. 2(a) with the contributions from both spin-up and spin-down charge currents flowing from higher potential to lower potential, and it is robust against the disorder.
In Fig. 1(b), the system width is decreased to L y = 400 nm, where the coupling between the spin-down edge states is strong enough to open an energy gap in the corresponding energy dispersions, while the energy dispersions of the spin-up edge states remain gapless which provides a charge conductance e 2 /h. As a result, the spin-down edge states can easily be broken by the disorder. Because, the transport electrons in the spin-down edge channels can possibly be backscattered by impurities. Hence, the charge conductance along x direction in this state ranges from e 2 /h to 2e 2 /h with disorder as shown in Fig. 2

(b). This state is called QAH state [13].
In Fig. 1(c), when the system width decreases to L y = 150 nm, all the energy dispersions of the edge states become gapped. Particularly, the lower band of spin-up edge states is higher than the upper band of spin-down edge states near k x = 0 point. Therefore, there are always conducting edge channels near the Fermi level which is in the bulk gap. This state is called edge-current (EC) state [13], in which the charge currents through the edge can be killed by disorder. So the corresponding charge conductance along x direction ranges from 0 to 2e 2 /h as shown in Fig. 2(c). In Fig. 1(d), by further decreasing the system width to L y = 80 nm, there is a bulk energy gap in the energy dispersions and we get a NI state which corresponds to Fig. 2(d) and it is induced by finite-size effects. Meanwhile, along the y direction, the edge states have a probability distribution in the sense of the wave function of edge states. As shown in Fig. 3, one can obtain a clear idea how well-located edge sates are forced to be mixed by changing L y . Here, the red solid curve denotes the probability distribution for spin-↑ edge state and the blue solid curve denotes the one for spin-↓ edge state. It is indicated from Fig. 3(a) that the wave functions for both spin-↑ and spin-↓ edge states are well localized at the left and right edges of the system with large system width L y = 850 nm where there is no coupling between edge states. When the system width L y becomes smaller and smaller as shown in Figs We furthermore investigate phase crossover due to adjusting the finite system width, with light intensity fixed. In Fig. 4, the definitions of E ↑± and E ↓± are the energies of two spin-up and spin-down edge bands at the zero momentum point k x = 0 under open boundary condition along y direction. Meanwhile, the red thick line indicates the upper band E ↑+ of the spin-up edge states, the red thin line indicates the lower band E ↑− of the spin-up edge states, the blue thick line indicates the upper band E ↓+ of the spin-down edge states, and the blue thin line indicates the lower band E ↓− of spin-down edge states.
We calculate the energies (E ↑± and E ↓± ) of the spinup and spin-down edge states at k x = 0 as a function of the system width L y as shown in Fig. 4(a) with the same parameters as those in Fig. 1. When L y is very small, the energy gap for both bulk and edge states is large and the system is insulating in NI state. By gradually increasing the width L y , once the lower energy band of the spin-up edge states touches the upper energy band of the spin-down edge states, the system enters into EC state. Particularly, we notice that there is a critical point which separates EC state from NI state as shown in Fig. 4. In EC state, the lower energy band of the spin-up edge states is higher than the upper energy band of the spindown edge states, and the energy dispersions for both spin-up and spin-down edge states are gapped. By further increasing the width L y , the energy gap of the spinup edge states vanishes but the energy dispersion for the spin-down edge states is still gapped. At this time, the system enters into QAH state. When L y is very large, the system enters into QSH state where the energy dispersions for both spin-up and spin-down edge states are gapless.
Different phases are determined by edge states and energy gaps. Here, we consider the finite-size effect on the edge states and energy gaps. However, Chern number and Bott index are used to describe the topological properties of an infinite-size system under periodic boundary conditions both along x and y directions. Therefore, for a small-size system under open boundary condition along y direction here, both of Chern number and Bott index are inapplicable. Concretely, when the system size is very small, the coupling between edge states becomes very strong and the edge states are destroyed by opening a gap as shown in Fig. 3(d). Besides,Ref. [11] also mentioned that the finite-size effect can destroy the quantum spin Hall effect phase without a topological index transition. Next, we will reply to the applicability of our approach to the phase transition regime. For finite small sizes, our phase transitions belong to continuous phase transitions which are in general crossovers, and we can only use the energy gaps of edge states to determine the phase transitions quantitatively. Meanwhile, as mentioned in Ref.
[13], another applicable method is to calculate the transport or conductance of the finite-size system by using the Landauer-Büttiker formalism [64,65]. For example, Ref.
[18] has calculated the transport properties of a finite-size system in HgTe/CdTe quantum wells without optical pumping. We will investigate the transport properties of finite-size Floquet system in the future.

V. OVERALL PHASE DIAGRAM
In this section, we will investigate the overall phase diagram of the system. limit. There are four kinds of phase regimes, i.e., NI, EC, QAH, and QSH, are shown. The white zone denotes the NI regime, the green zone denotes the EC regime, the yellow zone denotes the QAH regime, and the blue zone denotes the QAH regime. The threshold of the energy gap for calculating the phase boundaries is 0.01 meV, which is also used as a criterion of the energy gap opening. The other parameters are the same as those in Fig. 1.
Based on the energy gap for spin-up and spin-down edge states, we plot the whole phase diagram with two tunable parameters: light intensity A 0 and system width L y as shown in Fig. 5. There are four kinds of phase regimes, i.e., NI, EC, QAH, and QSH, as shown in Fig. 5. An important feature is that each transport regime has its own permitted width range with fixed light intensity. For example, the width for the EC state is in the range from 116 nm to 391 nm with fixed A 0 = 0.06 nm −1 . The permitted width range for different transport states is important because it can help us to find the desired transport regime by choosing appropriate device width and intensity of light.

VI. SUMMARY
Even though Floquet engineering had been extensively used to engineer various topological phases that are elusive in the static regime, our proposal is the first that engineered QAH and other phases in a realistic QSH setup that takes into account of the hybridization effects due to the finite size of realistic samples. We numerically in-vestigate the light-induced topological phases and finitesize topological phase crossovers in a QSH system with high-frequency pumping optics, taking the HgTe quantum well as the paradigmatic realistic model. We find that the coupling of edge states depends on both the system width and the light intensity. Furthermore, we can get four different topological transport regimes by choosing proper width and light intensity. Since the intensity of light can easily be continuously tuned in experiments, our proposal that uses circularly polarized light instead of magnetic doping provides a feasible route towards realizing QAH states in experiments.
With state of art of nano-Fabrication, it is possible to engineer realistic quantum materials to reduce its size in the length scale presented in our figures. For example, in Ref. [66], it is reported that the quantum anomalous Hall (QAH) devices can be fabricated miniaturized with channel widths down to 600 nm in six quintuple layers of Cr 0.12 (Bi 0.26 Sb 0.62 ) 2 Te 3 film which were epitaxially deposited onto GaAs (111) substrates. Furthermore, they measured the longitudinal and transverse resistivities in four devices with a channel width of 0.6, 1.5, 3, and 5 µm [66]. Meanwhile, Ref.
[67] presents the QAH effect persists in a Hall bar device with width 72 nm in three quintuple layers of Cr-doped (Bi,Sb) 2 Te 3 , four quintuple layers of (Bi,Sb) 2 Te 3 , and three quintuple layers of Cr-doped (Bi,Sb) 2 Te 3 [68, 69], and they measured the transverse resistance and longitudinal resistance in Hall bar devices with width from 300 nm to 10 µm. Technologically, first, by using photolithography, the samples can be patterned into a Hall bar with width 10 µm [66,67]. Then by using another experimental technology: electron-beam lithography, the realization of the QAH bar devices with width from 10 µm down to 72 nm are further fabricated [67].      The motivation of Appendix A is to show the concrete analytical expressions for H 0,0 , H 0,−1 , and H 0,1 in the Floquet Hamiltonian (3) of the main text. The concrete analytical expressions for H 0,0 , H 0,−1 , and H 0,1 in the Floquet Hamiltonian (3) are given as Appendix B: Time-reversal symmetry The motivation of Appendix B is to prove that when ϕ = 0, the Floquet Hamiltonian (4) in the main text still satisfies the time-reversal symmetry. However, when ϕ = 0, the time-reversal symmetry of the Hamiltonian (4) is broken.
The Floquet Hamiltonian (4) under time-reversal transformation becomes where T = iτ y ⊗σ 0 K [61] is the time-reversal operator with the complex conjugate operator K, we have KH (F ) (k)K −1 = H (F ) * (k), and we use In a lattice, one makes the following replacements [70] where j = x, y, z, a j is the lattice constant along j direction, sin(k j a j ) = e ik j a j −e −ik j a j 2i and cos(k j a j ) = e ik j a j +e −ik j a j 2 .
where k 2 = k 2 x + k 2 y . In this way the hopping terms in the lattice model only exist between the nearest neighbor sites. With this mapping, one obtains the following tight-binding model with high-frequency pumping in the basis (c k,+,↑ , c k,−,↑ , c k,+,↓ , c k,−,↓ ) T as where x cos(k x a x ) C † kx,jy,↑ σ 0 C kx,jy,↑ + D a 2 y kx,jy C † kx,jy,↑ σ 0 C kx,jy+1,↑ +h.c.
In this way the hopping terms in the lattice model only exist between the nearest neighbor sites. With this mapping, one obtains the following tight-binding model with high-frequency pumping in the basis (c k,+,↑ , c k,−,↑ , c k,+,↓ , c k,−,↓ ) T as where Performing the Fourier transformation, one obtains x jx,ky C † jx,ky,↑ σ 0 C jx+1,ky,↑ +h.c.
Therefore, the tight-binding Hamiltonian under y-PBCs and x-OBCs in the basis (C jx,ky,↑ , C jx,ky,↓ , C jx+1,ky,↑ , C jx+1,ky,↓ , · · · , C jx+Nx−1,ky,↑ , C jx+Nx−1,ky,↓ ) T is given by where we use σ + +σ − = σ x . In addition, we can also get the tight-binding Hamiltonian with pseudo-spin-↑ ("+") under y-PBCs and x-OBCs in the basis (C jx,ky,↑ , C jx+1,ky,↑ , · · · , C jx+Nx−1,ky,↑ ) T as where Further, we can get the tight-binding Hamiltonian with pseudo-spin-↓ ("-") under y-PBCs and x-OBCs in the basis (C jx,ky,↓ , C jx+1,ky,↓ , · · · , C jx+Nx−1,ky,↓ ) T as where Appendix E: Probability distributions of the edge states for Eσ− The motivation of Appendix E is to give the probability distributions of the edge states for E σ− as a supplementary for Fig. 3 in the main text. It is indicated from Fig. 6 that the probability distributions of the edge states for the "-" bands E σ− have similar properties as those for the "+" bands E σ+ with changing system size L y .