Topological aspects of periodically driven non-Hermitian Su-Schrieffer-Heeger model

TL;DR

This work investigates topological properties of a non-Hermitian Su-Schrieffer-Heeger model under periodic driving, introducing a bi-orthonormal Pancharatnam-Zak geometric phase $\gamma$ as a gauge-invariant topological index that captures zero-mode behavior. The authors analyze both undriven and Floquet-driven cases, identifying three phases—trivial insulator, non-trivial insulator, and Möbius metallic phase—where the driving amplitude acts as a control parameter for phase transitions. While zero-energy edge modes exist in insulator phases, they are non-robust in the Möbius metallic phase, and bulk-boundary correspondence is not straightforward in the driven non-Hermitian context. The work demonstrates how Floquet theory and bi-orthogonal geometry unify the understanding of topology in driven non-Hermitian lattices and shows that external driving can tune phase structure with potential implications for engineered quantum systems.

Abstract

A non-Hermitian generalization of the Su-Schrieffer-Heeger model driven by a periodic external potential is investigated, and its topological features are explored. We find that the bi-orthonormal geometric phase acts as a topological index, well capturing the presence/absence of the zero modes. The model is observed to display trivial and non-trivial insulator phases and a topologically non-trivial M${ö}$bius metallic phase. The driving field amplitude is shown to be a control parameter causing topological phase transitions in this model. While the system displays zero modes in the metallic phase apart from the non-trivial insulator phase, the metallic zero modes are not robust, as the ones found in the insulating phase. We further find that zero modes' energy converges slowly to zero as a function of the number of dimers in the M${ö}$bius metallic phase compared to the non-trivial insulating phase.

Figures (8)

• Figure 1: A schematic diagram of the driven non-Hermitian SSH model. The red and green circles denote $A$ and $B$ sublattice sites, respectively. The lattice constant is $a$, and two sites in an unit cell are separated by distance $b$. The intracell hopping amplitude is $v$, whereas the intercell amplitudes are $w e^{i\textbf{A}(t)}$ and $w e^{\theta-i\textbf{A}(t)}$ respectively for to and fro tunnelling in the presence of a time-periodic vector potential $\textbf{A}(t)$ and a non-Hermiticity measure $\theta$.
• Figure 2: Plot of real and imaginary parts of energy eigenvalues $E_{\pm}(k)$ as a function of $k$. Here, the red continuous and dot-dash curves respectively denote the real and imaginary part of the upper band's energy; whereas the blue dotted and dashed curves respectively indicate the real and imaginary part of the lower band's energy.
• Figure 3: Plots showing an agreement between analytically obtained energy spectrum with PBC (top) and numerically obtained energy spectrum for $N=300$ with OBC (bottom). The real part of energy is plotted as a function of $\kappa$ for $\theta = 0.5$. Apart from the zero modes, the two spectra closely match.
• Figure 4: Geometric phase $\gamma$ in a driven non-Hermitian SSH model as a function of vector potential's amplitude $(A_0)$ and ratio $\kappa~(\equiv w/v)$ for non-Hermiticity parameter $\theta=0.75$ (a), and the same as a function of $A_0$ and $\theta$ for $\kappa=1.5$ (b). Here the blue region represents the trivial insulating phase ($\gamma(2\pi) = 0$), while the yellow region indicates the non-trivial insulating phase ($\gamma(2\pi) = \pi$). The green region denotes the Möbius metallic phase ($\gamma(4\pi) = \pi$).
• Figure 5: Quasienergy spectrum versus amplitude of vector potential $(A_0)$ of a driven Hermitian SSH model (top) and a driven non-Hermitian SSH model (bottom) at high-frequency driving and with OBC. The Möbius strip phase appears and reappears only in the driven non-Hermitian model. All parameters are shown at the headings.