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

TL;DR

This paper introduces a Hermitian bosonic model with real or imaginary parameters that exhibits distinct topological phases and dynamical phenomena, including a novel M"obius phase and topological amplification, linking topology to amplification behavior.

Contribution

It reveals how real and imaginary parameters in a Hermitian bosonic chain lead to different topological phases and dynamical effects, including a new M"obius phase and topological amplification.

Findings

Real parameters produce a M"obius topological phase with fractional winding number.

Imaginary parameters support trivial and non-trivial topological phases without the M"obius phase.

Topological properties in the imaginary regime cause sublattice-dependent chiral amplification.

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 M\"obius 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 Schr\"odinger 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 M\"obius 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 M\"obius 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.