Physical relevance of time-independent scattering calculations in non-Hermitian systems: The role of time-growing bound states

Abstract

Time-independent scattering methods are widely employed to analyze transport in non-Hermitian systems. Their application, however, rests on a critical yet often overlooked assumption: that an incident wave is a pure superposition of scattering states. In practice, any physically realistic, spatially localized wave packet will generally have a nonzero overlap with the system's bound states, thereby violating this premise. While this violation is inconsequential in Hermitian systems, it can invalidate the conventional scattering picture in their non-Hermitian counterparts. The underlying cause is the emergence of time-growing bound states, which manifest as poles of the scattering matrix ($S$ matrix) in the first quadrant of the complex wave-number plane. Any initial overlap with these states becomes exponentially amplified, eventually dominating the long-time dynamics. Consequently, the actual evolution of a wave packet diverges dramatically from the conventional scattering picture, rendering the transmission and reflection coefficients derived from time-independent scattering methods unphysical. Using tight-binding models with non-Hermiticity introduced via imaginary on-site potentials or asymmetric hopping, we demonstrate that parameter regimes supporting such growing states are common. We therefore conclude that an analysis of $S$-matrix poles is an indispensable step to confirm the physical relevance of time-independent scattering calculations in non-Hermitian systems.

Figures (6)

• Figure 1: Schematic illustration of a non-Hermitian scattering system with gain and loss. The system consists of a scattering center coupled to two semi-infinite tight-binding chains.
• Figure 2: Reflection and transmission coefficients ($R_L$ and $T_L$) for left-incident waves as a function of $\gamma_1$, with fixed parameters $\gamma_0 = 1$ and $k = \pi/3$. Solid lines represent results from the time-independent scattering method, while circles and squares depict results from wave-packet evolution simulations. A significant divergence between the two approaches is observed when $\gamma_1$ exceeds the critical value $\gamma_c = 1.5$, as highlighted by the shaded region. The wave-packet simulations used a system size $L=800$, with wave-packet parameters $\sigma=40$ and $j_0 = -200$. $R_L$ and $T_L$ were calculated at time $t=240$.
• Figure 3: Time evolution of the wave packet for (a) $\gamma_1 = 1.2$, below the critical value $\gamma_c$, and (b) $\gamma_1 = 1.8$, above $\gamma_c$. (c) Semilogarithmic plot of the intensity profile $|\Psi_j|^2$ at $t=70$ for $\gamma_1 = 1.8$. The intensity in the central region exhibits an exponential decay, following $C_1 e^{-2 \alpha |j|}$ with $\alpha=0.322 \pm 0.001$. (d) Semilogarithmic plot of the total intensity $\sum_{j} |\Psi_j|^2$ versus time for $\gamma_1 = 1.8$. The total intensity grows exponentially over time, following $C_2 e^{2 \Gamma t}$ with $\Gamma = 0.655 \pm 0.001$. These simulations were performed using a system with $\gamma_0=1$, $L=800$, and an initial Gaussian wave packet characterized by $k =\pi/3$, $\sigma=40$, and $j_0 = -200$.
• Figure 4: Eigenvalues and eigenstates of the finite system. (a) Complex energy spectrum for $\gamma_1 = 1.2$ (below the critical value $\gamma_c$), showing eigenvalues clustered near the real axis. (b) A representative eigenstate from this spectrum [marked by the blue cross in (a)] is extended throughout the system. (c) Complex energy spectrum for $\gamma_1 = 1.8$ (above $\gamma_c$), where an isolated eigenvalue with a large positive imaginary part ($E \approx 0.655i$) emerges. (d) The corresponding eigenstate [marked by the red cross in (c)] is a bound state, strongly localized at the scattering center. Its intensity profile follows an exponential decay $e^{-2 \alpha |j|}$, with a fitted decay constant $\alpha=0.322 \pm 0.001$. The calculations were performed for a system with $\gamma_0=1$ and $L=800$.
• Figure 5: Locations of $S$-matrix poles in the complex $k$ plane for (a) $\gamma_1 = 1.2$, below the critical value $\gamma_c$, and (b) $\gamma_1 = 1.8$, above $\gamma_c$. (c) Trajectories of the $S$-matrix poles as a function of $\gamma_1$. The color bar shows the value of $\gamma_1$. A pole crosses the real axis and enters the first quadrant (shaded region) at the critical value $\gamma_c=1.5$, indicating the emergence of a time-growing bound state in the system. For these calculations, $\gamma_0$ was fixed at 1.