Moiré superlattices, a twisted functional structure crossing the periodic and nonperiodic potentials, have recently attracted great interest in multidisciplinary fields, including optics and ultracold atoms, because of their unique band structures, physical properties, and potential implications. Driven by recent experiments on quantum phenomena of bosonic gases, the atomic Bose–Einstein condensates in moiré optical lattices, by which other quantum gases such as ultracold fermionic atoms are trapped, could be readily achieved in ultracold atom laboratories, whereas the associated nonlinear localization mechanism remains unexploited. Here, we report the nonlinear localization theory of ultracold atomic Fermi gases in two-dimensional moiré optical lattices. The linear Bloch-wave spectrum of such a twisted structure exhibits rich nontrivial flat bands, which are separated by different finite bandgaps wherein the existence, properties, and dynamics of localized superfluid Fermi gas structures of two types, gap solitons and gap vortices (topological modes) with vortex charge S = 1, are studied numerically. Our results demonstrate the wide stability regions and robustness of these localized structures, opening up a new avenue for studying soliton physics and moiré physics in ultracold atoms beyond bosonic gases.

Bose–Einstein condensate (BEC) exhibits a variety of fascinating and unexpected macroscopic phenomena, and has attracted sustained attention in recent years—particularly in the field of solitons and associated nonlinear phenomena. Meanwhile, optical lattices have emerged as a versatile toolbox for understanding the properties and controlling the dynamics of BEC, among which the realization of bright gap solitons is an iconic result. However, the dark gap solitons are still experimentally unproven, and their properties in more than one dimension remain unknown. In light of this, we describe, numerically and theoretically, the formation and stability properties of gap-type dark localized modes in the context of ultracold atoms trapped in optical lattices. Two kinds of stable dark localized modes—gap solitons and soliton clusters—are predicted in both the one- and two-dimensional geometries. The vortical counterparts of both modes are also constructed in two dimensions. A unique feature is the existence of a nonlinear Bloch-wave background on which all above gap modes are situated. By employing linear-stability analysis and direct simulations, stability regions of the predicted modes are obtained. Our results offer the possibility of observing dark gap localized structures with cutting-edge techniques in ultracold atoms experiments and beyond, including in optics with photonic crystals and lattices.