Fig. 3: Three-band SSH model.
a Non-Hermitian three-band model illustrating non-reciprocal inter-unit-cell hopping with intra-unit-cell hoppings d1 and d2, and inter-unit-cell hoppings teθ and te−θ. The dashed box highlights the unit cell. b Phase diagrams of the Hamiltonian (Eq. (6)) on the (t, θ) plane for d1 = 1.0 and d2 = 0.2. Black dots signify Dirac points. Red and blue lines indicate two pairs of EPs associated with the second and third bands, and with the first and second bands, respectively. In the Hermitian limit (θ = 0), two phases, 13 and \({1}^{1}{\bar{1}}^{2}\), are present; for the non-Hermitian case (θ ≠ 0), five phases, 13, \({1}^{1}{\bar{1}}^{2}\), \({1}^{1}{\bar{2}}^{1}\), \({\bar{1}}^{1}{2}^{1}\), and 31, are distinguished by the boundaries formed by pairs of EPs. The strips' colors (Yellow, Blue, and Red) correspond to the energy band index (1st, 2nd, and 3rd). c Real energy surfaces at θ = 0.1 (left) and θ = 0.4 (right) in the parameter space (t, k), with paths along the Brillouin zone (k ∈ [0, 2π]) and t = 0, 1. EPs are marked by red and blue dots, related to the EPs shown in b. The \({1}^{1}{\bar{1}}^{2}\) phase includes three separate paths; blue and yellow paths encompass two EPs each, resulting in a π phase change. In the 31 phase, a single 3-cycle band path encircles two EPs, and three loops result in zero phase change.
