II. FORMULATION Our system is a modulated two-dimensional electron gas (M2DEG) in the presence of an external magnetic field and RSOI. The magnetic field is applied perpendicular to the x − y plane in which electrons with unmodulated areal density n$_{D}$ , effective mass m ∗ and charge − e are confined. The Hamiltonian of this system is written as H = H$_{0}$ + H ′ , where H$_{0}$ is the unmodulated and H ′ the modulation term. We employ the Landau gauge and write the vector potential as A = (0 , Bx, 0). The two-dimensional Schrodinger equation in the Landau gauge is[13, 19]; H$_{0}$ = ( ∧ p + e ∧ A ) 2 2 m ∗ + α $_{ℏ}$[ ∧ σ × ( ∧ p + e ∧ A ]$_{z}$ + 1 2 gµ$_{B}$Bσ$_{z}$, (1) H ′ = V$_{0}$ cos( Kx ) . (2) φ$_{nk}$$_{y}$ (¯ x ) = 1 √ L$_{y}$ e ik$_{y}$y ∞ ∑ n =0 u$_{n,k}$$_{y}$ ( x + x$_{0}$ ) ( C + n C − n ) , (3) ε + 0 = ε$_{0}$ = 1 / 2 ℏ ω$_{c}$ − gµ$_{B}$B/ 2 (4) where ∧ p is the momentum operator of the electrons, g the Zeeman factor, µ$_{B}$ the Bohr magneton, ∧ σ = ( σ$_{x}$, σ$_{y}$, σ$_{z}$ ) the Pauli spin matrices, α the strength of the RSOI with K = 2 π/a , with a the period of modulation and V$_{0}$ the amplitude of modulation. Since the Hamiltonian does not depend on the y coordinate, the unperturbed wavefunctions are plane waves in the y -direction. This allows us to write for the wavefunctions, with L$_{y}$ being the normalization length in the y -direction and ¯ x a 2D position vector on the x - y plane, x$_{0}$ = l $^{2}$k$_{y}$ = ℏ k$_{y}$ m ∗ $_{ω$_{c}$}$, is the coordinate of cyclotron orbit center and l = √ ℏ eB is the magnetic length, m ∗ is the effective mass. In the x -direction, the Hamiltonian has the form of a harmonic oscillator Hamiltonian. Hence, we can write the unmodulated eigenstates in the form φ$_{nk}$$_{y}$ (¯ x ) = 1 √ $_{L$_{y}$}$e ik$_{y}$$^{y}$u$_{nk}$$_{y}$ ( x ; x$_{0}$ ) , with u$_{nk}$$_{y}$ ( x ; x$_{0}$ $_{) = (}$√$_{π}$$_{2}$ $^{n}$n ! l ) − 1 2 exp( − 1 2 l 2 ( x − x$_{0}$ ) $^{2}$) H$_{n}$ ( $^{x}$−x 0 l ) , where u$_{nk}$$_{y}$ ( x ; x$_{0}$ ) is a normalized harmonic oscillator wavefunction centered at x$_{0}$ and H$_{n}$ ( x ) are Hermite polynomials with n the Landau level quantum number. The energy of the lowest Landau level is with wave function φ + 0 ( k$_{y}$ ) = 1 √ $_{L$_{y}$}$e ik$_{y}$$^{y}$u$_{0}$ ( x + x$_{0}$ ) ( 0 1 ) . For n = 1 , 2 , 3 , .... , there are two branches of levels, denoted by + and − , with energies 4