Edge states of Floquet–Dirac semimetal in a laser-driven semiconductor quantum-well

Band crossings observed in a wide range of condensed matter systems are recognized as a key to understand low-energy fermionic excitations that behave as massless Dirac particles. Despite rapid progress in this field, the exploration of non-equilibrium topological states remains scarce and it has potential ability of providing a new platform to create unexpected massless Dirac states. Here we show that in a semiconductor quantum-well driven by a cw-laser with linear polarization, the optical Stark effect conducts bulk-band crossing, and the resulting Floquet-Dirac semimetallic phase supports an unconventional edge state in the projected one-dimensional Brillouin zone under a boundary condition that an electron is confined in the direction perpendicular to that of the laser polarization. Further, we reveal that this edge state mediates a transition between topological and non-topological edge states that is caused by tuning the laser intensity. We also show that the properties of the edge states are strikingly changed under a different boundary condition. It is found that such difference originates from that nearly fourfold-degenerate points exist in a certain intermediate region of the bulk Brillouin zone between high-symmetry points.

The boundary condition A is set along the edge in the x-direction, where an electron is confined in the y-direction perpendicular to the electric field of lase F (t) (the x-direction), and the system of concern, namely, the quantum wire, is spatially periodic in the x-direction. (b) The boundary condition B is set along the edge in the y-direction, where an electron is confined in the x-direction parallel to F (t), and the system of concern is spatially periodic in the y-direction. In these figures, an electron is confined in a well layer of the quantum well (HgTe) with zinc blende structure the unit cell of which is depicted. Here, shown are schematic contour-maps ofD(k) in the (k x , k y )-plane, where regions with D(k) > 0 andD(k) < 0 are indicated by red and blue colors, respectively, a transient region between the two is indicated by a white color, and the boundary ofD(k) = 0 (the zero contour) is indicated by a black solid line. Further, edge states revealed in one-dimensional energy dispersions E(k x ) (under the boundary condition A) and E(k y ) (under the boundary condition B) are schematically shown by purple solid lines; parts of edges states embedded in a bulk band continuum are indicated by a purple dashed line. In addition, when a Dirac crossing is caused, the resulting Dirac cone projected onto the k x and k y axes is shown by intersected filled triangles. (a)D(k) is shown at F Γ x . Here edge states are absent.
The black dashed lines show the projection of the zero contour onto the k x and k y axes, and these delimit the edge states appearing in E(k x ) and E(k y ), respectively. Here topological edge states manifest themselves. (c) The same as the panel (b) but at F X 2 x . Here Dirac-Tamm states manifest themselves just under the boundary condition A. (d) The same as the panel Here Tamm states manifest themselves just under the boundary condition A. (e) The same as the panel (b) but at F X 1 x . Here Dirac-Tamm states manifest themselves just under the boundary condition A. (f ) The same as the panel x . Here topological edge states manifest themselves.

Supplementary Note 1: Derivation of the modified Bernevig-Hughes-Zhang (BHZ) Hamiltonian
We begin with the Schrödinger equationĤ with the total Hamiltonian asĤ(t) =Ĥ 0 +Ĥ ′ (t) at time t, whereĤ 0 andĤ ′ (t) represent a material Hamiltonian and a Hamiltonian of a laser-electron electric-dipole interaction, respectively. These are given in terms of a Wannier state |bR⟩ with b and R a band index and a two-dimensional site vector of the system of concern (a semiconductor quantum well) as follows: andĤ in terms of the corresponding matrix elements of In the first equality, F (t) represents an electric field vector of an irradiated laser and r represents the position vector of electron. In the second equality, X is defined as the displacement of electron from the position of an ionic core of the quantum well, namely, X = r − R, and an electric dipole matrix element at each R is approximately given by ⟨bR|X|b To do this, the property has been used that a Wannier function ⟨r|bR⟩ is just a function of X, and mostly localized inside a cell situated at R. Thus, ⟨bR|X|b ′ R ′ ⟩ is considered almost independent of R. Therefore, Eq. (1) is cast into Now, the Peierls phase transform defined by is introduced to the above equation, where A(t) represents the vector potential given by . This brings Eq. (5) into the following equation for Φ bR ≡ ⟨bR|Φ⟩ as where in terms of the HamiltonianĤ(t) =Ĥ 0 + F (t) · X. Next, Eq. (7) is rewritten in the Bloch representation, where the Wannier state |bR⟩ is represented in terms of the Bloch state |bk⟩ as with k as the Bloch momentum and N the number of the sites of quantum well. In light of the fact that Eq. (7) is expressed as one obtains the following equation as with are employed. Here, the Hamiltonian matrix element of Eq. (11) is cast into Employing the BHZ model for the material HamiltonianĤ 0 , Eq. (12) becomes the BHZ Hamiltonian modified by the application of the laser-electron interaction, that is, where a matrix representation is employed with H BHZ (K(t)) as the BHZ Hamiltonian for the Bloch momentum Finally, a remark is made on the scheme of calculating edge states supported by bulk bands, when an electron is confined in one direction of the two-dimensional plane of the quantum-well and the other direction remains periodic; such a confined system is equivalent to a semiconductor quantum wire, as seen in Supplementary Figure 1. The Hamiltonian suitable for such a system is obtained by transforming Eq. (7) into a Hamiltonian in terms of a mixed representation using the Wannier representation in one direction and the Bloch representation in the other direction. The resulting wavefunction is represented as ⟨bk x(y) R y(x) |Φ⟩ with k x(y) and R y(x) the x(y)-and y(x)-components of k and R, respectively. The edge states and the associated band structure with respect to k x(y) are calculated under a boundary condition A(B).

Supplementary Note 2: Fourier-Floquet matrix elements
The Fourier-Floquet expansion is employed for evaluating the eigenvalue equation given in Eq. (4) of the text. Here, the Floquet Hamiltonian L(k, t) is of the form where (2) of the text, and K(t) = (k x − Fx ω sin ωt, k y ). Below, we evaluate the following Floquet matrix (Eq. (5) of the text) as: with ∆n = n − n ′ . This is expressed in terms of the N th-order Bessel function of the first kind with z = F x a/ω (a: a lattice constant). Therefore, we obtaiñ Let a matrix element with component . Given Eq. (15) and the above expressions ofD i,nn ′ (k, ω) (i = 3 ∼ 6), it is readily seen that L b(n)b ′ (n ′ ) (k, ω) vanishes if either ∆n is an odd number and b = b ′ , or ∆n is an even number and b ̸ = b ′ .

Supplementary Note 3: Additional condition of quasienergy band crossing
The origin of quasienergy band crossings is examined. It is expected that because of the conservation of both T-and pseudo-I-symmetries, the band crossings occur at the high-symmetry points [1]. Below, we show that an additional condition is required that is pertinent to the difference of the photon numbers between two bands participating to the crossing.

A σ z -conserving interaction
Since different spin states are decoupled, namely, [L(k, t), I 2 ⊗ σ z ] = 0, just an upper-spin diagonal block-matrix ofL nn ′ (k, ω), denoted asL ↑↑ nn ′ (k, ω), suffices for this purpose. Equation (5) of the text is solved to seek an eigenvalues E α (k j ) at a high-symmetry point j = Γ, X 2 , and X 1 . Here we consider how one eigenvalue E b(n) (k j ) of state b(n) is affected by another state b ′ (n ′ ). The state b(n) is coupled with b ′ (n ′ ) through a set of sequential interactions composed of products of matrix element V ω). Let the mth-order sequential interaction for couplings between the two states b(n) and b ′ (n ′ ) be represented as a matrix element U where

and a net change of photon numbers carried by
Below, it is inductively proved that U evidently it vanishes for ∆n even, and so does U (1) s(n)p(n−∆n) . Further, since this also vanishes for ∆n odd. Second, bearing in mind that U if ∆n is even, either U vanishes, if ∆n is odd.
Next, suppose for k ≥ 3 that U

p(n)s(n−∆n) is considered, which is read as
Here, if ∆n is even, either U (l) b 2 (n 2 )s(n−∆n) vanishes for any set of indices of b 2 and n 2 , because l, k + 1 − l ≤ k for l = 1 ∼ k. Thus, U s(−1)s(−1) , respectively. Thus, it is found that the crossings between these two bands occur at F j x ensuring a single algebraic equation E p(1) (k j ) = E s(−1) (k j ). In view of the spin degree of freedom, p(1) and s(−1) are four-fold degenerate at k j 's. This result holds correct for any interband coupling, in general, between p(n) and s(n ′ ) if (n − n ′ ) is even, while for odd numbers of (n − n ′ ), s-and p-bands are gapped out; especially, p(1) and s(0) are anticrossed due to the leading interaction V

A σ z -non-conserving interaction
Next, we consider the effect of a σ z -non-conserving interaction on the obtained results, which is given by where D i (k, t) = d i (K(t)), and d i (k)'s are odd functions with respect to k to keep v z (k, t) both Tand pseudo-I-symmetric. Based on the two-component model, when ∆n is even, it is evident that a gap remains closed because ofD i,n 0 n ′ 0 (k j , ω) = 0 with i = 1, 2. Similarly to the above case of the σ z -conserving interactions, this still hold in general, as is verified in Supplementary Note 2.
Next, we examine the effect of a σ z -non-conserving interaction on the obtained results, the interaction of which is given by Eq. (6) of the text as with D j (k, t) = d j (K(t)). Here, d 1 (k) and d 2 (k) are odd function with respect to k to keep v z (k, t) both T-and pseudo-I-symmetric. This coupling between the same spin states of p(1) and s(−1) is conducted by series of interactions composed of a spin inversion due tõ and the σ z -conserving coupling U . It is evident thatṽ z,nn ′ (k j , ω) ̸ = 0 if (n − n ′ ) is odd, otherwise it vanishes, and thusṽ z,nn ′ (k j , ω) is subject to the property similar to V Therefore, the coupling of concern between p(1) and s(−1) is also provided by the form of the sequential interaction modified from U . This leads us to the conclusion that these two bands remain gapless.

Summary
For the realization of band crossings at k j 's in the concerned Floqeut system, we require the additional condition that the difference of photon numbers between participating bands be an even number with the condition of the conservation of both T-and pseudo-I-symmetries Here, we examine the variance of the degree of localization of the edge states in the vicinity of thē X 2 point (k x = 0), as F x changes around F X 2 x ; the results shown right below are also applicable to the edge states emerging at F x in the vicinity of F X 1 x . To do this, the electron density is defined as where Ψ kxα (t) represents the αth eigenvector of Eq. (4) of the text under the boundary condition A, and |bk x R y ⟩ is a basis in the mixed representation, as mentioned in Supplementary Note 1. It is obvious that ∑ Ry ρ kx (R y ) = 1. Here, α is set an edge state emerging at each F x given below. The calculated results of ρ kx (R y ) are shown in Fig. 4 of the present Note alongside the associated E α (k x ). As shown in the panel (a), the topological edge state (shown in the panel (d)) remains localized in the edge region irrespective of the change of k x . As shown in the panel (b), the Dirac-Tamm state (shown in the panel (e)) has the same pattern of change as the above topological edge state, because both of the edge states originate from the band inversion at the Γ point; see Figs. 5d and 5e of the text. However, there is an exceptional case that the Dirac-Tamm state becomes delocalized at k x /π = 0.025 just close to theX 2 point. Such a property is caused by an interaction of this state with the Floquet DSM continuum phase at the X 2 point. To be more specific, this interaction is attributed to an almost energetically degenerate coupling between the Γ and X 2 points, because k y is no longer a good quantum number in this 1D confined system, and states with different k y 's are likely coupled.
As regards the Tamm state (shown in the panel (f)), it is shown in the panel (c) that this is delocalized in the whole R y -region and disappears in a local region of k x where the anticrossing around the X 2 point occurs (k x /π = 0.025 and 0.05), though this is still localized in a region where |k x | is greater (k x /π = 0.1). In fact, due to this anticrossing, the parity hybridization is strongly reduced, leading to the absence of an edge state. It is evident from Fig. 5f of the text that the anticrossing formation around the X 2 -point gives rise to the sign inversion ofD(k) -from D(k) > 0 seen in Figs. 5d and 5e of the text -and the zero contour vanishes in the whole k y region with k x assigned around theX 2 point. Thus, an edge state does not exist in this local k x region.
To summarize, all edge states (a topological edge state, a Dirac-Tamm state, and a Tamm state) extending over k x regions provided by the projection of the zero contours are localized in the y direction of the quantum well, and properties pertinent to the localization seem unaltered with the change of F x around F X 2 x , whether there is a Dirac node or not in an edge state. However, it is noted that the Dirac-Tamm state is made delocalized in the close vicinity ofX 2 due to the interaction with the associated Floquet DSM phase.