Boundary-sensitive non-Hermiticity of Floquet Hamiltonian: spectral transition and scale-free localization

Abstract

We report a novel mechanism of boundary-sensitive PT symmetry breaking in one-dimensional Floquet systems. By designing a time-periodic driving protocol, we realize a Floquet Hamiltonian that is Hermitian under periodic boundary conditions yet acquires non-Hermitian boundary terms under open boundary conditions due to the non-commutativity of driving Hamiltonians. We establish that a PT symmetry breaking transition occurs when the quasienergy bandwidth expands to cover the entire frequency Brillouin zone. This condition highlights a crucial difference from static non-Hermitian systems, where such transitions typically require band touching. Furthermore, we demonstrate that in the PT-broken phase, the eigenstates exhibit scale-free localization, a phenomenon arising from the specific system-size scaling of non-Hermitian terms. Finally, we provide a general framework for constructing multi-band models that exhibit this boundary-induced phase transition.

Figures (5)

• Figure 1: (a) Real part of the quasienergy spectrum under open boundary conditions, as a function of $\lambda$. Inset show the zoomed-in spectrum under open and periodic boundary conditions for model \ref{['eq:Hamiltonian']}. (b) Imaginary part of the quasienergy spectrum under open boundary conditions as a function of $\lambda$. The proportion of complex eigenvalues $P_{\text{com}}$ is superimposed (scale on the right axis). (c) Magnified view of the spectrum in the vicinity of an exceptional point. Solid black lines represent a square-root fit characterizing the bifurcation of the imaginary part. (d) Phase diagram created using $P_{\text{com}}$ as the order parameter, where white dots represent the threshold value of $\lambda$ for $\mathcal{PT}$ symmetry breaking, as a function of system size $N$. We set $T=1$.
• Figure 2: Illustration of $\mathcal{PT}$ symmetry transition by tracking trajectories of a pair of eigenvalues $\xi_{1,2}$ of the Floquet operator $U_F$ under open boundary conditions.
• Figure 3: Scale-free localization in the $\mathcal{PT}$-broken phase for model \ref{['eq:Hamiltonian']}. (a) Finite-size scaling of the averaged imaginary part of quasienergies. $\lambda=3$ and $4$. (b) The eigenvalue-resolved mean position ratio $\langle x\rangle_n/N$, plotted for various system sizes $N$. The eigenstate index $n$ is sorted in ascending order of the imaginary part of the corresponding quasienergy. The inset displays wavefunction profiles of selected eigenstates.
• Figure 4: Quasienergy spectra of the two-band models as functions of certain parameters. (a) Type-I model [Eq. \ref{['eq:twoband1']}], varying $t$. (b) Type-II model [Eq. \ref{['eq:twoband2']}], varying $t_1$. The vertical black dashed line marks the critical threshold for $\mathcal{PT}$ symmetry breaking. The real parts of quasienergies are represented by blue and black curves for two bands (scale on the left axis); the imaginary parts are shown in red (scale on the right axis). We set $T=1$.
• Figure 5: (a) Magnitude of matrix elements of the non-Hermitian part of $H_{F,\text{OBC}}$, system size $N=1000$. (b) Diagonal and secondary diagonal matrix elements, as marked by solid lines in (a). The inset shows a logarithmically scaled plot. (c) Diagonal matrix elements for different system sizes $N$. (d) Averaged bulk perturbation $\Gamma_p$ as a function of $1/N$. $\lambda=3$ for all the plots.