Non-Hermitian butterfly spectra in a family of quasiperiodic lattices

We propose a family of exactly solvable quasiperiodic lattice models with analytical complex mobility edges, which can incorporate mosaic modulations as a straightforward generalization. By sweeping a potential tuning parameter $\ensuremath{\delta}$, we demonstrate a kind of interesting butterflylike spectra in a complex energy plane, which depicts energy-dependent extended-localized transitions sharing common exact non-Hermitian mobility edges. Applying Avila's global theory, we are able to analytically calculate the Lyapunov exponents and determine the mobility edges exactly. For the minimal model without mosaic modulation, we obtain a compactly analytic formula for the complex mobility edges, which indicates clearly mobility edges having a loop structure in the complex energy plane. Together with an analytical estimation of the range of the complex energy spectrum, we can obtain the true mobility edge. The non-Hermitian mobility edges are further verified by numerical calculations of the fractal dimension and spatial distribution of the wave functions. Tuning the parameters of non-Hermitian potentials, we also investigate the variations of the non-Hermitian mobility edges and the corresponding butterfly spectra, which exhibit a richness of spectrum structures.