# Extended SSH Model in Non-Hermitian Waveguides with Alternating Real and Imaginary Couplings
## 2 Geometry and Theoretical Model  
The general SSH model depicts topological phenomenon in a one-dimensional dimer chain. Each unit cell has two lattices. Hopping between neighboring lattices in different unit cells is, \(c_{1}\) and hopping between adjacent lattices in the same unit cell is \(c_{2}\), which are real numbers. The SSH model has two energy bands under periodic boundary conditions, which degenerate at the Dirac point and experience band inversion as \(c_{1}=c_{2}\). According to topological theory, this critical point is the topological phase transition point. The topological invariant of the SSH model can be characterized by the Berry phase \(\varphi_{B}\), which is \(\varphi_{B}=0\) for \(c_{1}<c_{2}\) and for \(\varphi_{B}=\pi\) for \(c_{1}>c_{2}\). The system is topologically nontrivial for nonzero Berry phase and is able to sustain robust edge stats with zero energy at its boundary for open boundary conditions. In this work, we extended SSH by changing the intra-cell hopping to be imaginary, while the inter-cell hopping remained unchanged. Thus, the system becomes non-Hermitian, and the two hopping values are denoted \(ic_{2}\) and \(c_{1}\), respectively. Hopping, which is also called coupling in optics, typically uses real numbers in weakly coupled waveguides. We used adiabatic elimination to modify the coupling to be imaginary. Adiabatic elimination is an efficient method to control coupling, and it requires two lattices indirectly coupled via an intermediate decoupled lattice. Details about the procedure of adiabatic elimination can be found in [41]. As shown in Figure 1a, we integrated an extra lattice C into the bare lattices A and B. The center lattice C serves as intermediate site and has strongly detuned onsite energy (also called propagation constants in waveguide systems). We set the imaginary part of onsite energies to be  
\[\delta_{\text{A,B}}=-ic_{2},\delta_{\text{C}}=i\j^{2}/c_{2} \tag{1}\]  
where \(J\) denotes the original coupling strength between lattice C and A (B). The coupling coefficient \(J\) is assumed to be a real number, as the dielectric waveguides are utilized in the following study. When detuning \(\delta_{\text{c}}\) is much larger than coupling \(J\), then site C can be adiabatically eliminated. Effective coupling between two sublattices A and B is determined to be \(ic_{2}\). Consequently, imaginary hoping is employed through adiabatic elimination and appropriately fixes the onsite gain and loss in different lattices. More details about this technique can be found in recent studies [33]. Considering the condition of adiabatic elimination (\(\delta_{\rm c}>>J\)), the original coupling \(J\) should remarkably exceed the effective coupling strength \(|ic_{2}|\). Figure 1b shows the effective binary chain corresponding to Figure 1a after adiabatic elimination. The coupling in the whole system is alternatively real and imaginary, that is,...... Im-Re-Im-Re-Im-Re.........  
We now investigate the band structure and the topological invariant in this extended SSH model. For the periodic boundary condition we can apply the Bloch theorem, and the Hamiltonian system is derived as  
\[\mathbf{H}=\left( \begin{array}{cc}0&ic_{2}+c_{1}e^{i\varphi}\\ ic_{2}+c_{1}e^{-i\varphi}&0\end{array} \right) \tag{2}\]  
where \(\varphi\) denotes the Bloch momentum. Then, the eigenvalues are calculated to be  
\[\lambda_{\pm}=\pm\sqrt{c_{1}^{2}-c_{2}^{2}+2 ic_{1}c_{2}\cos\varphi}, \tag{3}\]  
The system has chiral symmetry, which is defined by \(\sigma_{z}\mathbf{H}(\varphi)\sigma_{z}=-\mathbf{H}(\varphi)\), with \(\sigma_{z}\) representing the Pauli matrix. The eigenvalues of the system come in \(\pm\)pairs due to chiral symmetry. In Equation (3), the eigenvalues appear in the form of \(\pm\) pair of square roots. Exceptional points emerge as two eigenvalues are of equal values, corresponding to the zero value of the square root. The condition takes place as \(c_{1}=c_{2}\) at \(\varphi=\pm\pi/2\).