Boundary Floquet Control of Bulk non-Hermitian Systems

Abstract

Non-Hermitian systems provide a powerful platform for engineering and controlling nonequilibrium phenomena beyond Hermitian settings, with the presence of non-Hermitian skin effect broadening the scope of dynamical control. Here, we develop a general theory of non-Hermitian systems driven exclusively at their boundaries, providing a unified description of the driving-frequency dependence of bulk spectra and dynamics in the thermodynamic limit. Our framework extends non-Bloch band theory to time-periodic systems at arbitrary boundary driving frequencies. Applying it to representative models, we demonstrate boundary-driving-induced parity-time symmetry breaking, with the driving frequency serving as a control knob and the driving amplitude providing an additional handle in finite-size systems. These results establish boundary Floquet driving as a versatile mechanism for controlling bulk properties of non-Hermitian systems and open new routes for dynamical engineering in driven open systems.

Figures (3)

• Figure 1: Key mechanism underlying the boundary Floquet control of bulk systems with the non-Hermitian skin effect. (a) Schematic illustration of the boundary driving protocol for the time-periodic Hamiltonian $H(t)=H_0+V(t)$, with the static bulk Hamiltonian $H_0$ in Eq. \ref{['eq:H0']} and the boundary drive $V(t)$ with period duration $T$ in Eq. \ref{['eq:boundary_step']}. (b) An example of two skin modes of the static Hamiltonian $H_0$ whose energies differ by $2\pi/T$, labeled in the upper panel of (c). They exhibit distinct localization behaviors and are resonantly coupled via $V(t)$. (c) Upper panel: OBC spectrum of the static Hamiltonian $H_0$ (i.e., $V(t)=0$). Lower panel: OBC quasienergy spectrum of $H(t)$ with $V(t)\neq 0$. We have set $V=0.01$ in the lower panel; other parameters are $t_1=2$, $t_2=0.15$, $\gamma=0.16$, $T=0.9$, and $L=80$. (d) Lyapunov exponent $\lambda=\lim_{t\to\infty}[\ln\braket{\psi(t)|\psi(t)}]/(2t)$ as a dynamical signature during the evolution for an initial state $\ket{\psi(0)}=\ket{L/2}$, corresponding to the largest imaginary part of the OBC quasienergy spectrum. Solid colored lines denote finite-size results for different system sizes, while the dashed purple line shows the thermodynamic-limit prediction from our Floquet non-Bloch band theory. Our theory also predicts the threshold value $T_c$ for the Floquet-induced parity-time symmetry breaking, shown as the dashed gray line in (d).
• Figure 2: Phase diagrams of the ratio $\eta$ as the fraction of OBC quasienergies that are complex, showing the Floquet-induced PT-symmetry breaking and its pronounced finite-size dependence. (a) The phase diagram of the system size $L$ and driving period $T$, with $\gamma=0.16$ and $V=0.01$. (b) The phase diagram of the non-Hermitian strength $\gamma$ and driving period $T$, with $L=200$ and $V=0.01$. (c) The phase diagram of the system size $L$ and driving strength $V$, with $T=2$ and $\gamma=0.16$. (d) The phase diagram of the system size $L$ and non-Hermitian strength $\gamma$, with $T=2$ and $V=0.01$. In all plots, we set $t_1=2$ and $t_2=0.15$. The solid lines in (a) and (b) are theoretical predictions from our Floquet GBZ theory. The red dashed lines indicate the scaling relation in Eq. \ref{['eq:scaling']}: $L_c\propto -\log V$ in (c) and $L_c\propto 1/\gamma$ in (d), revealing that even weak boundary driving can trigger a bulk spectral transition, only beyond a parametrically large system size.
• Figure 3: The Floquet GBZ and aGBZs (top row), and the corresponding OBC quasienergy spectra (bottom row), for representative driving periods $T$ with $\gamma=0.16$, $t_1=2$, $t_2=0.15$, and $V=0.01$. Here ${\rm aGBZ}_{\ell}$ denotes the auxiliary GBZ associated with the $\ell$th Floquet replica $E\pm2\pi \ell/T$. As $T$ increases, neighboring Floquet replicas approach and hybridize, which is captured geometrically by the evolution of these curves in the top row. In the bottom row, colored points are obtained by exactly diagonalizing the OBC Floquet operator $U_{\text{F}}$ for different system sizes, while solid lines are the thermodynamic-limit predictions from our Floquet GBZ theory. This shows that Floquet-zone hybridization in the large-$T$ regime gives rise to cusps in the Floquet GBZ from which complex spectral branches emerge.