Floquet Chern insulators with C > 1

Chapter III: Periodically driven Topological systems

3.4 Floquet Chern insulators with C > 1

Figure 3.3: The bandstructure for the model Hamiltonian (defined in Eq. (3.22) ) in different parameter regimes. (a) The original time-independent band-structure for the Hamiltonian for the parameter regimes: A/M = 0.2, B/M = −0.2. The band is topologically trivial. (b) The quasienergy bandstructure in the presence of driving for the same system parameters as (a), withV/M = (0,0,1)andΩ/M = 3.

This band-structure is clearly non-trivial with edge states in the gap at the resonant quasienergy, =Ω/2. (a) and (b) are plotted for the system in cylindrical geometry : periodic boundary conditions in thexdirection and open boundary conditions with L =60 sites in theydirection. (c) The spin texture of the lower band of the effective two-band model,H_eff^F, as defined in Eq. (3.27), defined in a torus geometry. This is the mapping of the unit vector ˆn(k)from the Brilluoin zone to the unit sphere. The parameters are chosen to be the same as (b). Clearly it wraps around the full sphere, indicating a phase with non-zero Chern number.

circle to winds around the north pole [69], where

n(k)ˆ = V⊥(k)/|V⊥(k)|, (3.29)

As a consequence, when ˆnhas zero winding along the resonance circle, the induced phase is trivial. Therefore, in the presence of driving, it is possible to obtain trivial Floquet insulators. For, example, when the radiation potential is alongz direction, i.e. V = Vzz, the driven phase is always topological. In contrast, forˆ V = Vxx, theˆ driven system is trivial. In Fig. 3.3(c) we plot the mapping of the unit vector ˆnon the unit sphere, ˆn(k) :T² → S². Clearly, this wraps around the sphere indicating a topological phase.

case of using elliptically polarized radiation on the BHZ model. We show that a topological phase with Chern number,C= 2 is induced.

Floquet Haldane model with a resonance

Let us consider the tight-binding model on a honeycomb lattice with a energy bandwidth, W. This scenario corresponds to a driving frequency in the regime W/2 < Ω < W. In this case, the quasienergy band-structure has two gaps at (i) = 0, and (ii) = ±Ω/2, where the topologically non-trivial features may be measured. This is shown in Fig. 3.1 (c). Clearly the band-structure has two radiation induced bulk gaps at these quasienergies. The gap at = 0 is the same as that discussed in Section3.2and is equal to∆. We incorporate the effect of off- resonant processes on the quasienergy band-structure by making the replacement H0→ Heffin the Floquet Hamiltonian defined in Eq. (3.9). For quasi-energies close to resonance, ∼ Ω/2, adjacent diagonal Floquet blocks, H_eff, and H_eff−Ω, are nearly degenerate. Just like in Section3.3,HFmust be diagonalized in this subspace of two adjacent Floquet blocks, to obtain the correction to the quasi-energies. The effective two band Hamiltonian is given by

(Hr^F)eff= PΩ* ,

H_eff−Ω V₊ V− Heff

+ -

PΩ, (3.30)

wherePΩis the projector onto the bands with quasi-energies in the range 0< < Ω.

This is exactly the same as degenerate first order perturbation theory, and therefore, the gap exactly at resonance, = Ω/2, is proportional to|V±|.

The quasi-energies ofH_r^Fare periodic inΩ. As discussed in Section3.2, to properly define the Chern number for a band (C), we must specify its upper (u) and lower (l) bound in quasi-energies. An alternative is to measure the Chern number (C^trunc) of all bands below a particular quasienergy, for a truncated H_r^F as defined in Eq.

(3.24). This means to set,l = −∞. It has been shown [94] that the Chern number for the truncated Hamiltonian,C_n^trunc, corresponds to the number of edge states that will be observed at that particular quasienergy irrespective of chirality. For the case of single resonance, the Chern number of the truncatedH^F for M < ∆is

C^trun = 







1 if = 0

2 if = ±Ω/2, (3.31)

and forM > ∆

C^trun = 









0 if =0

2 if =±Ω/2. (3.32)

The Chern number of the bands (when l = ±Ω/2 and u = 0) are C = ±3 when M <∆, andC= ±2 when M >∆.

Floquet Chern insulator : BHZ model under elliptically polarized light

Let us now discuss an example of a Floquet Chern Insulator obtained in the Trivial BHZ model, in the presence of elliptically polarized radiation. This method uses a periodically modulated planar electric field, that is introduced through a spatially uniform gauge potential,A= (Axsin(Ωt), A_ycos(Ωt)). In this case,Vis dependent on the momentum vector,k. As a consequence, the topological phase has a Chern number, C = 2 with co-propagating edge-modes.In the following, we outline the key steps to show that the quasienergy bandstructure is non-trivial.

In the presence of elliptically polarized light, the time reversal symmetry is explicitly broken. Through the rest of the section, we will only focus on only one block (upper) of the BHZ model as defined in Eq. (3.22) (with V = 0). The results can be generalized to the lower block by an appropriate time-reversal operation on the lower block. Let us introduce polarized light through a Peierls substitution of a time-dependent gauge fieldA ≡ (φx, φ_y) =

Axsin(Ωt), A_ycos(Ωt)

, where Ax

and Ayare in general different, indicating an elliptic polarization. The Hamiltonian transforms under this substitution,d(k)·σ →d(k−A)·σ≡ d˜·σ, and the individual components of the vector, ˜d≡

d˜x,d˜y,d˜z

, are d˜x = A

sinkxcosφx−coskxsinφx, d˜y = Af

sinkycosφy−coskysinφy

g, (3.33)

d˜z = M−2B[2−(coskxcosφx+sinkxsinφx, +coskycosφ_y+sinkysinφ_y)].

In the perturbative limit for the radiation field, A_x,y → 0, we can set, cos(φ_x,y) ≈ 1 and sin(φx,y)≈ φx,y. This simplifies the Hamiltonian,H(t),

H(t) = d·σ+V ·σe^iΩt +V^†·σe^−iΩt, (3.34) where

V = i AAx

2 coskx,−AA_y

2 cosky,B(Aysinky−i Axsinkx)

!

. (3.35)

-1.0 -0.5

0.0 0.5

1.0 -1.0

-0.5 0.0 0.5

0 1 2 3 4

0.0 0.5 1.0 1.5

ΘHarctan@AyAxDL

ChernnumberFloquet

Figure 3.4: (a)Winding ofV⊥(k) on the unit sphere [See Eqs. (3.39-3.41)] as the momentum vector,kvaries along the resonance circle [defined in Eq. (3.25)]. The black point indicates the north pole,(0,0,1)and the red point indicates ˆV⊥(k₀)with k₀ = (1,0).(b) We show the dependence of the Chern number and the gap in the quasienergy bandstructure of the upper block on polarization of light parameterized by,θ_pol =arctan(A_y/Ax).

As discussed in Section 3.3, the quasienergy gap at resonance is governed by the matrix element,|hψ₊|V ·σ|ψ−i|, where|ψ±iare the eigenstates of the unperturbed Hamiltonian. In the original basis, the eigenstates are,

|ψ₊i = cosθ

2| ↑i+sinθ

2e^iφ| ↓i, (3.36)

|ψ−i = sinθ

2| ↑i −cosθ

2e^iφ| ↓i, (3.37)

whered(k) ≡ |d|(sinθcosφ,sinθsinφ,cosθ), and the original basis is defined as, {| ↑i,| ↓i}. As discussed in Section3.3, the winding around the north pole,V⊥(see Eq. 3.27) is necessarily related to the Chern number of the Floquet bands. In the basis of the eigenstates of the time-independent Hamiltonian,V⊥ is defined as,

V⊥·σ ≡ hψ₊|V ·σ|ψ−i|ψ₊ihψ−|+h.c. (3.38) Rewriting the vector, hψ₊|V |ψ−i = V_R + iVI in the original pseudospin basis,

{| ↑i,| ↓i},

V⊥ = V_R(−cosθcosφ,−cosθsinφ,sinθ)

+V_I(−sinφ,cosφ,0), (3.39)

with,

V_R = AAx

2 sinφcoskx+ AAy

2 cosθsinφcosk_y

+B Aysinkysinθ, (3.40)

V_I = −AAx

2 cosθcosφcoskx− AAy

2 cosφcosk_y

−B Axsinkxsinθ. (3.41)

Note that the angles θ and φ are obtained from the definitions of d(k). We are interested inV⊥ along the resonance circle given by|d(k)| = ^Ω₂.

The Chern number of the bands obtained from irradiation of the quantum wells with circularly polarized light is ±2. Figure 3.4 (a) shows the winding of the vector V⊥ along the resonance circle. Clearly, it winds twice around the north pole, which is consistent with a Chern number = ±2. The Chern number of the band also depends on the polarization of the incident light as shown in Fig. 3.4 (b). The approximate value of the Chern number for a given polarization angle, θ_pol = arctan(A_y/Ax), is obtained by computing C defined in Eq. (3.28). The quasienergy gap in the band structure depends on the incident polarization of the radiation, given by|V⊥|. Clearly, the gap closes as function ofθ_pol, when a transition happens from a topological (C =2) to the trivial (C =0) phase.