We propose a method to generate a real-energy flat band in a two-dimensional (2D) non-Hermitian Lieb lattice. The coincidence of the flat band eigenstate in both real and momentum spaces is essential for the proposed flat band, which is flexible at the appropriate match between the synthetic magnetic flux and non-Hermiticity. The proposed method is not limited to the 2D non-Hermitian Lieb lattice, and is applied to the 2D non-Hermitian Tasaki's decorated square lattice, dice lattice, and kagome lattice. Our findings make a step forward for the flat band engineering in 2D non-Hermitian optical lattices.

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The dice lattice supports Aharonov-Bohm caging when all the energy bands are flat for the half-quantum magnetic flux enclosed in each plaquette of the lattice. We analytically investigate the eigenstates and discuss the localization dynamics. We find that arbitrary excitation is compactly confined within the excited-site-related snowflake structures of the dice lattice; as a consequence that the nonzero-energy flatband localizes in the single snowflake, whereas the zero-energy flatband localizes in three nearest snowflakes that are connected in the form of a trident star. The localization dynamics of an arbitrary excitation is grasped from two dynamical behaviors of single-site excitation. For the single-site excitation at the center of a snowflake, the excitation is localized in that snowflake; whereas for the single-site excitation at the branch site of a snowflake, the excitation is localized in the three snowflakes that the branch site belongs to. Our findings deepen the understanding of destructive interference and the dynamics of Aharonov-Bohm caging in the dice lattice.