Solvable semi-infinite Fock-state-lattice Su-Schrieffer-Heeger model: Stable topological zero modes and the non-Hermitian bound effect

Fock-state lattice (FSL) offers a powerful quantum simulator for topological phenomena due to the unbounded scalability and ease of implementation. Nevertheless, the unique topological properties induced by its site-dependent coupling have remained elusive, mainly due to the challenge of handling an infinite state space without translational symmetry. Here, we rigorously analyze the topological features of a semi-infinite FSL-based Su-Schrieffer-Heeger (SSH) model, in both Hermitian and non-Hermitian realms, by mapping it to the solvable Jaynes-Cummings model via a unitary displacement transformation. We find a topological zero mode persisting across all parameter regimes, which is more stable than the conventional SSH model. It originates from the bound state at the inherent domain wall under anisotropic conditions. With gain and loss introduced, we predict a non-Hermitian bound effect, i.e., any state overlapping with the bound state will quickly stabilize to the domain wall, with the minimal stabilization time occurring in the vicinity of exceptional point. The parity-time phase transition can be observed by the oscillating-to-steady crossover of dynamics in the subspace orthogonal to the bound state. Furthermore, a concrete experimental proposal based on the trapped-ion setup is provided.