# Spin Dependent Transport through Driven Magnetic System with Aubry-Andre-Harper Modulation
## 2 Magnetic Junction, TB Hamiltonian and Theoretical Formulation  
### Junction Setup and the Hamiltonian  
Let us begin with the spin polarized setup, shown in Figure 1, where a magnetic chain having \(N\) lattice sites (filled red balls) is clamped between two non-magnetic (NM) electrodes, commonly referred as source (S) and drain (D). Each site of the magnetic chain contains a finite magnetic moment which interacts with the injected spin \(\sigma\) (\(\sigma=\uparrow,\downarrow\)). The magnetic chain is subjected to light irradiation (yellow arrows) that plays the central role for engineering spin polarization in our analysis. The Hamiltonian of the magnetic nanojunction can be written as  
\[H=H_{mag}+H_{S}+H_{D}+H_{tun} \tag{1}\]  
where different sub-Hamiltonians are associated with different parts of the junction. We describe explicitly all these terms one by one in TB framework as follows. The sub-Hamiltonian \(H_{mag}\) reads as  
\[H_{mag}=\sum_{i}c_{i}^{\dagger}\left(\epsilon_{i}-\vec{h}_{i}.\vec{\sigma} \right)c_{i}+\sum_{i}\left(c_{i+1}^{\dagger}\vec{i}c_{i}+h.c.\right) \tag{2}\]  
where \(c_{i}^{\dagger}=\left(c_{i\uparrow}^{\dagger}\quad c_{i\downarrow}^{\dagger }\right)\). \(c_{i\sigma}^{\dagger}\) (\(c_{i\sigma}\)) is the usual fermionic creation (annihilation) operator. \(\epsilon_{i}=\text{diag}(e_{i\uparrow},\epsilon_{i\downarrow})\) where \(e_{i\sigma}\) represents the site energy. In the presence of AAH modulation, the site energies are expressed as [71; 72; 73]\(e_{i\uparrow}=e_{i\downarrow}=W\cos(2\pi bi+\phi_{v})\), where \(W\) measures the strength of the cosine modulation, \(b\) is an irrational number and \(\phi_{v}\) is the AAH phase factor. In our calculations, we set \(b=(1+\sqrt{5})/2\) without loss of any generality. With a suitable setup, one can regulate the phase factor \(\phi_{v}\), and here we discuss its effect on SP. The term \(\vec{h}_{i}.\vec{\sigma}\) is responsible for spin dependent scattering [64; 65; 66; 67; 68; 69], where \(\vec{h}_{i}\) is the spin-flip scattering factor and \(\vec{\sigma}\) denotes the Pauli spin vector. It is a well known scattering phenomenon and has been elaborately studied in literature (see Refs. [64; 65; 66; 67; 68; 69], and the references therein). The strength \(h_{i}\) is usually very large and in some cases it becomes an order of magnitude higher than the SO coupling [64]. Because of this fact, we get large mismatch between the two spin channels. The orientation of the spin flip vector is described by the conventional polar and azimuthal angles, \(\theta_{i}\) and \(\varphi_{i}\), respectively. Here, it is relevant to note that, in the present formulation we ignore the effect of interaction among the neighboring magnetic moments. It is well established that the moment-moment interaction can be expreesed as an effective Zeeman like term which represents the interaction of localized magnetic moments with an effective \(B\)-field (commonly referred as the \"molecular field\"). Compared to the interaction of itinerant electrons with local moments, as the Zeeman coupling is too weak, due to the existance of the factor \(\mu_{B}\), no appreciable change is expected in SP even when the magnetic field is too high. The rest part of \(H_{mag}\) is associated with the hopping of an electron from one site to its neighboring sites. \(\vec{t}=\text{diag}(\vec{t},\vec{t})\). In the presence of light irradiation, the nearest-neighbor hopping (NNH) strength \(t\) gets renormalized and it takes the form [59; 60]  
\[\vec{t} = t\frac{1}{\mathcal{T}}\int\limits_{0}^{\mathcal{T}}e^{i(p-q) \Omega\tau}e^{\vec{t}.\vec{x}}\,d\tau \tag{3}\] \[= t_{(p-q)}(\Lambda)\]  
where \(J_{(p-q)}\) is the \((p-q)\)th order Bessel function of the first kind. To include the effect of irradiation we follow the Floquet ansatz.