From Chiral Topological Dynamics to Chiral Topological Amplification: Real vs Imaginary Parameters in a Hermitian Bosonic Chain

TL;DR

This work demonstrates that a Hermitian quadratic bosonic chain can realize non-Hermitian topological dynamics by engineering its dynamical matrix. In the real-parameter regime, the system is unitarily equivalent to four copies of the non-Hermitian SSH model (nSSH2), hosting a Möbius phase with fractional winding and a chiral dynamical order parameter under quenches. In the purely imaginary regime, the dynamical matrix maps to a different nSSH1-like model, removing the Möbius phase but inducing sublattice-dependent directional amplification with a direct topological origin. The study provides analytic expressions for Loschmidt amplitudes, DTOP, and amplification matrices, and discusses feasible realizations in optomechanical and superconducting platforms, offering a fully Hermitian route to explore non-Hermitian topology and dynamics. Overall, the work links topological phase structure to dynamical responses and amplification in a unified bosonic framework with experimental relevance.

Abstract

We propose a Hermitian quadratic bosonic model (QBH) whose dynamical matrix exhibits distinct topological and dynamical phenomena depending on whether the hopping and pairing amplitudes are real or purely imaginary. In the real-parameter regime, the dynamical matrix is unitarily equivalent to four decoupled copies of the sublattice-symmetric non-Hermitian Su-Schrieffer-Heeger (nSSH2) model, thereby inheriting its topological phases and energy spectrum-including the Möbius phase, a gapless topological phase with fractional winding number, having no Hermitian counterpart. We show that the dynamics generated by the QBH Hamiltonian naturally reproduce non-Hermitian time evolution, without invoking nonlinear Schrödinger dynamics or ad hoc normalization. It is demonstrated by analytically calculating the Loschmidt amplitude and computing the dynamical topological order parameter under periodic boundary conditions, which displays a distinct chiral response in the Möbius phase. In contrast, when the hopping and pairing terms are taken to be purely imaginary, the dynamical matrix becomes unitarily equivalent to a different version of the sublattice-symmetric non-Hermitian Su-Schrieffer-Heeger (nSSH1) model that supports only two topological phases: trivial and non-trivial, and the Möbius phase disappears. The latter system exhibits sublattice-dependent chiral amplification under open boundary conditions. We show that this amplification arises from the non-trivial topology of the dynamical matrix, establishing a clear link between topological phase and amplification behavior in the imaginary-parameter regime.

Figures (11)

• Figure 1: Top: Cartoon illustration of the QBH model as two copies of Hermitian SSH chains with intra-chain hopping (solid lines) and inter-chain pairing (dotted lines). Bottom: Comparison of various phenomena in real and imaginary parameter regimes of the model.
• Figure 2: Schematic diagram of three different winding numbers and the corresponding parametric energy plots of $H_{nSSH2}(k)$. (a) $\delta=-0.9$, no exceptional point is enclosed by the closed curve, $\nu_1=\nu_2=\nu=0$. (b) The complex energy bands form two isolated closed loops with real energy gap. (c) $\delta=-0.1$, EP1 is enclosed while EP2 is not, $\nu_1=1$, $\nu_2=0$ which makes $\nu=1/2$. (d) The two energy loops come close together and merge to form a single bigger loop and there is no real energy gap. (e) $\delta=0.9$, both the exceptional points are enclosed, $\nu_1=\nu_2=1$ which makes $\nu=1$. (f) The two loops again separate from each other and there is a real gap in between them. We take $J=1$ and $\theta=0.4$ in (a)-(f).
• Figure 3: Real part (a) and imaginary part (b) of the spectrum of $G_{QB}(k)$ as function of $\delta$. We fix $J=1$ and $\theta=0.4$ in both the plots.
• Figure 4: Spectrum of the dynamical matrix $G_{QB}$ for OBC. It is purely real and there are two gapped phases: trivial phase and non-trivial phase, with zero modes present in the non-trivial phase. The parameters are $J=1$ and $\theta=0.4$.
• Figure 5: Dynamical response for the real parameter regime. The Pancharatnam geometric phase, $\phi_{\rm pgp}(k,t)$ in (a), the return rate ($RR(t)$) in (b), the $DTOP_-$ and $DTOP_+$ in (c) and (d), respectively, for the quench from an initial Hamiltonian with parameters $J^i=1$, $\delta^i=-0.9$, $\theta^i=0$ to a final Hamiltonian with parameters $J^f=1$, $\delta^f=0.9$, $\theta^f=0.4$.