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As one indispensable type of nonreciprocal mechanism, a system with temporal modulations is intrinsically open in the physical sense and inevitably non-Hermitian, but the space and time degrees of freedom are nonseparable in a large variety of circumstances, which restrains the application of the non-Bloch band theory. Here, we investigate the spatially photonic crystals (PhCs) composed of spatiotemporal modulation materials (STMs) and homogeneous media, dubbed as the STMPhC, wherein the spatial and temporal modulations are deliberately designed to be correlated. To bypass the difficulty of the spatiotemporal correlation, we first employ the effective medium theory to account for the dispersion of fundamental bands under the influence of Floquet sidebands. Based on the continuum generalized Brillouin zone condition, we then analytically give the criteria for the existence of the non-Hermitian skin effect in the STM. Assisted by developing a numerical method that embeds the plane wave expansion in the transfer matrix, we establish the non-Bloch band theory for the low-frequency Floquet bands in the STMPhCs, in which the identification of the generalized Brillouin zone is central. We finally delve into the topological properties, including non-Bloch Zak phases and non-Bloch bulk-boundary correspondence. Our work validates the idea that the effective medium assists the non-Bloch band theory applied to the STMPhCs, which delivers a prescription to broaden the horizons of non-Bloch theory. Published by the American Physical Society 2024
Ding et al. (Tue,) studied this question.