Floquet Non-Bloch Formalism for a Non-Hermitian Ladder: From Theoretical Framework to Topolectrical Circuits

TL;DR

This work addresses the challenge of defining topological invariants in periodically driven non-Hermitian systems by developing a Floquet non-Bloch formalism that constructs a generalized Brillouin zone (GBZ) via a high-frequency Magnus expansion and a rotating frame. It analyzes a driven non-Hermitian Creutz ladder, deriving an effective Floquet Hamiltonian and two chiral-symmetric partner frames whose invariants, when combined, reproduce the edge-state structure and restore bulk-boundary correspondence under drive. The authors introduce Floquet non-Bloch invariants based on $R'_{\pm}(\beta)$ along the GBZ, yielding a phase diagram in the $(\omega,t_V)$ plane with zero and $\pi$ edge modes, and they illustrate a concrete topolectrical-circuit (TEC) implementation that emulates the driven NH Creutz ladder and directly visualizes skin modes and Floquet edge states via impedance measurements. Overall, the work bridges Floquet engineering and non-Hermitian topology and provides an experimentally viable TEC platform to realize and study driven NH topological phases. The results have potential impact for controllable exploration of NH-driven topological phenomena in laboratory settings.

Abstract

Periodically driven systems intertwined with non-Hermiticity opens a rich arena for topological phases that transcend conventional Hermitian limits. The physical significance of these phases hinges on obtaining the topological invariants that restore the bulk-boundary correspondence, a task well explored for static non-Hermitian (NH) systems, while it remains elusive for the driven scenario. Here, we address this problem by constructing a generalized Floquet non-Bloch framework that analytically captures the spectral and topological properties of time-periodic NH systems. Employing a high-frequency Magnus expansion, we analytically derive an effective Floquet Hamiltonian and formulate the generalized Brillouin zone for a periodically driven quasi-one-dimensional system, namely, the Creutz ladder with a staggered complex potential. Our study demonstrates that the skin effect remains robust (despite the absence of non-reciprocal hopping) across a broad range of driving parameters, and is notably amplified in the low-frequency regime due to emergent longer-range couplings. We further employ a symmetric time frame approach that generates chiral-partner Hamiltonians, whose invariants, when appropriately combined, account for the full edge-state structure. To substantiate the theoretical framework, we propose a topolectrical circuit (TEC) that serves as a viable experimental setting. Apart from capturing the skin modes, the proposed TEC design faithfully reproduces the presence of distinct Floquet edge states, as revealed through the voltage and impedance profiles, respectively. Thus, our work not only offers a theoretical framework for exploring NH-driven systems, but also provides an experimentally feasible TEC architecture for realizing these phenomena stated above in a laboratory.

Figures (10)

• Figure 1: Schematic representation of the quasi-1D Creutz ladder, where $a_n$ and $b_n$ denote the two distinct sublattices. The different hopping amplitudes, $t_H$, $t_V$, $t_D$, denote the horizontal, vertical, and diagonal hoppings, respectively.
• Figure 2: Panel (a) shows the Floquet quasi-energy spectrum corresponding to the harmonic drive, plotted as a function of the vertical hopping, $t_V$. Panel (b) demonstrates the bulk invariants $\nu_{0,\pi}$ evaluated from the coordinates of GBZ, which correctly coincides with panel (a). The rest of the parameters are chosen as, $t_H=0.6$, $t_D=0.6$, $V_0=0.3$, $\gamma=0.4$, and $\omega=3$.
• Figure 3: Panels (a) and (b) display the probability distributions of the eigenstates for two specific values of $t_V$, namely, $t_V = 0.3$ and $t_V = -1.5$, as indicated in Fig. \ref{['fig:2']}(a) by the $\triangle$ and $\lozenge$ symbols, respectively. The corresponding Floquet GBZs, denoted as $C_\beta$, are shown in panels (c) and (d). These GBZ coordinates are then used to compute $\nu_{0,\pi}$, presented in Fig. \ref{['fig:2']}(d), which match accurately with the open boundary quasi-energy spectrum. The rest of the parameters are chosen as, $t_H=0.6$, $t_D=0.6$, $V_0=0.3$, $\gamma=0.4$, and $\omega=3$.
• Figure 4: The associated expansion coefficients corresponding to the Hamiltonian for $t_V=1$ and $t_V=-1$ in the coordinate basis have been shown in panels (a) and (b), respectively. Here, $l$ and $m$ denote the base indices, namely, the row and column indices of the Hamiltonian matrix in the coordinate basis. While panels (c) and (d) denote the distribution of the NH skin modes at the edges corresponding to positive ($t_V=1$) and negative ($t_V=-1$) values of $t_V$, respectively. The other parameters have been chosen as $t_H=t_D=0.6$, $V_0=0.3$, $\omega=1$, and $\gamma=0.4$.
• Figure 5: Panels (a) and (b) display the trajectories of $R^{\prime}_{+}$ and $R^{\prime}_{-}$ on the complex plane along the GBZ for $t_V = 0.3$ ($\triangle$) and $t_V = -1.5$ ($\lozenge$), respectively. Notably, both the loops encircle the origin, indicating the presence of a topological phase. Furthermore, $R^{\prime}_{+}$ rotates in the clockwise direction, while $R^{\prime}_{-}$ rotates counterclockwise, resulting in a net finite winding of the $R$ vectors. All other parameters are the same as those used in Fig. \ref{['fig:2']}.