Supplementary Material

The Hamiltonian which describes the Dynamics of atoms in the bichromatic lattice is:

Figure 1 shoes the band structure of such a Fourier-synthesized lattice for and , where .

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The eigenvalue equation for this Hamiltonian will be

where is the Bloch wavefunction and could be written in the form

Here is the quasimomentum that is conventionally restricted to the reduced Brillouin zone Inserting the wavefunction intoSchrodinger equation we get

Equating the coefficients of each and harmonics, we get a system of three differential equations:

or eigenvalue equation

where

,

and

Taking into account that (we work in the center of the Brillouin zone) for M we get

Now let`s find the eigenvalues of this matrix.

We are especially interested for the eigenvalues ​​in the second and third Bloch band for a quasi-momentum . This should be in the range of , so that it is useful to introduce a new variable . The determinant will give the following equation for coefficients:

Since the Bloch bands are in the vicinity of (the crossing point on the Figure1. Is near ), the term could be neglected, so that the resulting quadratic equation by the coefficient can be resolved

In the next step, the difference between the two eigenvalues ​​is then calculated to obtain information about the energy split between the second and third Bloch-Bands.

(When ). Thus,

From Eq. (1.12)

Here the term could be neglected, since . The last term is the energy difference between the 2nd and 3rd bands, which is . So, the eigenvalue will be:

From the second equation of the (1.9) system

Adiabatic elimination of the ground state () leads to . At the same time

thus, . Refering back to Shrodinger equations for and we write

Recalling that for the reduced M matrix we get

Applying a rotation to Eq.(1.21) with a unitary () transformation matrix

and replacing q by the corresponding operator finally gives an effective Hamiltonian

Where and are Pauli matrixes:

Consequently,

For a derivation of the full effectively relativistic wave equation Hamiltonian with an external potential which acts on spinors with and corresponding to course grain atomic wavefunctions in the upper and lower bands, respectively. The corresponding time-dependent wave-equation will be

For each spinor the time-dependent waveequation will be

Taking time-derivative from (1.4) and replacing in it expression for from (1.5) one can get

Next, we find an expression for from (1.4) and replace it in (1.6). So, we get

Here we multiply both sides by :

So,

or,

Dispersion relation can be obtained with the help of plane wave solution

The corresponding group velocities will be

and effective masses: