A Solvable Semi-infinite Fock-state-lattice SSH Model: the Stable Topological Zero Mode and the Non-Hermitian Bound Effect

Abstract

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 (JC) model via a unitary displacement transformation. We find a more stable topological zero mode than the conventional SSH model, originating from the bound state at the inherent domain wall under anisotropic conditions. With gain and loss introduced, we predict a non-Hermitian bound effect (NHBE), 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 (EP). The paritytime (PT ) 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.

Figures (6)

• Figure 1: A semi-infinite anisotropic SSH model and its topological phase diagram. [(a), (b)] A spin-boson system with JC coupling and a resonant spin driving, can be mapped onto a SSH chain, with intracell coupling $J_1$, anisotropic intercell coupling $J_2\left( n \right) =J_2\sqrt{n+1}$, and the $n$-th unit cell $\{\ket{n,\uparrow},\ket{n,\downarrow }\}$. [(c), (d)] The anisotropic SSH model exhibits a topological phase for $J_1 < J_2$, hosting a single topological edge state at the left boundary, and a mixed phase for $J_1 > J_2$, characterized by a topological domain wall separating the trivial and topological region, with a bound state localized at the interface.
• Figure 2: Comparison of the eigenenergies between isotropic and anisotropic SSH models. (a) The eigenenergy spectrum of the conventional isotropic SSH model is numerically obtained under open boundary conditions with unit-cell number truncated at $n=50$. It exhibits topological zero modes in the topological phase ($J_1<J_2$). (b) The eigenenergy spectrum of the anisotropic SSH model is plotted using the analytical eigenvalues in Eq. (\ref{['eq:eigenenergy']}) with $n \leqslant 50$, showing a stable topological zero mode regardless of $J_1/J_2$. (c) Left and right edge states corresponding to the two-fold zero modes at $J_1/J_2=0.25$ (orange circle) in (a). [(d), (e)] The left-edge-localized state (d) and the domain-wall-bounded state (e), are plotted by using the analytical expression of zero modes $\ket{\alpha,\downarrow}$ at $J_1/J_2=0.5$ (blue square) and $J_1/J_2=4$ (blue pentagram) in (b), respectively.
• Figure 3: Non-Hermitian bound effect (NHBE) in the NH anisotropic SSH model. [(a)-(c)] In the $J_2 > \gamma$ parameter regime, the complex eigenspectrum hosts a single gain mode corresponding to the bound state $\ket{\varphi_0}$ (a). Both the boson number distribution (b) and projection probabilities onto representative eigenmodes (c) demonstrate that the renormalized time-evolved state is rapidly dominated by the exponentially amplified $\ket{\varphi_0}$, ultimately stabilizing at the domain-wall position. [(d)-(f)] In the $J_2 < \gamma$ parameter regime, additional gain modes besides the bound state emerge with $|\mathrm{Im}(E)| < \gamma$ (d). The boson number distribution (e) and eigenmode projections (f) show that $\ket{\varphi_0}$ sustains its dominance in mode competition dynamics, eventually giving rise to NHBE. (g) Stabilization time $\tau$ versus $\gamma$ for representative initial states $\ket{n, \downarrow}$ ($n=10, 20, 30, 40, 50$) reveals that the fastest stabilization is near $\gamma = J_2$. Parameters: $J_1=1$, $J_2=0.2$; $\gamma=0.15$ [(a)-(c)], $\gamma=0.5$ [(d)-(f)]; the initial state is chosen as $\ket{ 50,\downarrow}$ [(b), (c), (e), (f)].
• Figure 4: $\mathcal{P}\mathcal{T}$ quantum phase transition observed in the restricted Hilbert space governed by $\hat{H}_1$. [(a), (b)] Real (a) and imaginary (b) components of eigenvalues of $\hat{H}_1$. [(c), (d)] In the PTS phase, both boson-number dynamics (c) and spin-boson entanglement (d) exhibit oscillatory behaviors, with entanglement quantified by the von Neumann entropy $S(\hat{\rho}_\mathrm{boson})$. [(e), (f)] In the PTB phase, boson number dynamics (e) and entanglement entropy (f) rapidly converge to the values determined by the largest-gain mode $\ket{\tilde{\psi}_{0,+}^R}$ in $\hat{H}_1$, as marked by the green dot in Fig. \ref{['fig3']}(d), after transient mode competition dynamics. Parameters: $J_1=1$, $J_2=0.2$; $\gamma=0.15$ [(c), (d)], $\gamma=0.25$[(e), (f)]; the initial state is chosen as $\ket{10,\uparrow}$ [(c)-(f)].
• Figure A1: The Von Neumann entropy $S(\hat{\rho}_{\mathrm{boson}})/\ln 2$ of the eigenstates $\ket{\psi_{n,\pm}}$ in Eq. (\ref{['eigenstates']}) as a function of $\gamma$. For simplicity, $n$ ranges from 0 to 5, with each value corresponding to a distinct color. The parameters are set to $J_1 = 1$ and $J_2 = 0.2$. When $\gamma \leq J_2 \sqrt{n+1}$, $\ket{\psi_{n,\pm}}$ are maximally entangled, with $S(\hat{\rho}_{\mathrm{boson}})/\ln 2 = 1$. When $\gamma > J_2 \sqrt{n+1}$, $S(\hat{\rho}_{\mathrm{boson}})/\ln 2$ decreases monotonically with $\gamma/J_2$. The red dashed lines indicate EP ($\gamma = J_2 \sqrt{n+1}$) of each subspace $h_n$ in $\hat{H}_1$. As $n$ increases, the distance between them decreases.