Gap solitons of the Wannier and Bloch types in spin-orbit-coupled Bose-Einstein condensates with a moiré lattice

Gap solitons (GSs) bifurcating from flat bands, which may be represented in terms of Wannier functions, have garnered significant interest due to their strong localization with extremely small norms. Moiré lattices (MLs), with multiple flat bands, offer an appropriate platform for creating such solitons. We explore the formation mechanism and stability of GSs in spin-1 Bose-Einstein condensates under the combined action of the Rashba spin-orbit coupling (SOC) and an ML potential. We identify five Wannier-type GS families bifurcating from the lowest five energy bands in the spectrum induced by the ML with sufficiently large period and depth. These fundamental GSs serve as basic elements for constructing more complex Wannier-type GS states. Reducing the lattice period and depth triggers a transition from the Wannier-type GSs to ones of the Bloch type, the latter exhibiting higher norm thresholds and pronounced spatial broadening near edges of the energy bands. In addition to tuning the lattice-potential parameters, adjusting the SOC strength can also modulate the flatness of energy bands and enhance the localization of gap solitons, enabling reversible transitions between the GSs of the Wannier and Bloch types. Distinctive properties of GSs in the quasiperiodic ML are uncovered too. Thus, we propose the theoretical foundation for the creation of and manipulations with strongly localized GSs.

💡 Research Summary

The paper investigates gap solitons (GSs) in a spin‑1 Bose‑Einstein condensate (BEC) subjected simultaneously to Rashba spin‑orbit coupling (SOC) and a two‑dimensional moiré lattice (ML) formed by superimposing two square optical lattices with a relative twist angle. Using the three‑component Gross‑Pitaevskii equations, the authors incorporate density‑density (c₀) and spin‑exchange (c₂) interaction coefficients together with the SOC strength γ. The ML potential is parameterized by its depth V₀, period a, and twist angle θ; when θ satisfies a Pythagorean relation the lattice is strictly periodic, otherwise it is quasiperiodic.

First, the linear band‑gap structure is computed via Bloch theory for two representative twist angles, θ = arctan(3/4) and θ = arctan(5/12). The authors show that large lattice periods and deep potentials generate multiple flat bands separated by wide gaps. Both increasing the period a and strengthening SOC (γ) lower the band energies and flatten the bands, thereby widening the gaps. This flat‑band landscape is the prerequisite for the emergence of Wannier‑type (WT) gap solitons, which can be excited with arbitrarily small norms because the nonlinearity acts as a perturbation on an essentially dispersionless background.

In the nonlinear regime, the authors focus on the antiferromagnetic interaction case (c₀ < 0, c₂ > 0) and fix θ = arctan(3/4), a = 2π, V₀ = 4, γ = 0.5. Numerical solutions obtained by a squared‑operator iteration reveal five distinct families of WT‑GSs, each bifurcating from one of the lowest five flat bands (μ₁…μ₅). The density profiles exhibit ring‑shaped or multi‑hump structures; the ±1 components share identical amplitudes but opposite phases, while the m = 0 component carries a distinct phase pattern. The norm‑versus‑chemical‑potential curves start at the flat‑band edges and descend to N ≈ 10⁻⁸ as μ approaches the band edge, confirming the ultra‑low norm threshold. Linear stability analysis shows that most of the WT families are dynamically stable across their existence intervals, with only narrow regions of weak instability near the middle of the bands.

When the lattice period or depth is reduced, the flat bands disappear and the system supports Bloch‑type (BT) gap solitons instead. BT‑GSs are characterized by phase structures that match the corresponding Bloch eigenstates near the band edges, higher norm thresholds (N ~ 10⁻³–10⁻²), and pronounced spatial broadening. The transition from WT to BT solitons is reversible: increasing a or γ restores flatness and re‑establishes WT‑GSs, while decreasing them drives the system back to BT‑GSs. Thus SOC serves as a tunable knob for band flattening and soliton type control.

The study also explores quasiperiodic MLs (θ not satisfying the Pythagorean condition). Even in this case, flat bands and sizable gaps persist, allowing WT‑GSs to form, but their spatial localization becomes non‑uniform due to the lack of translational symmetry. BT‑GSs in quasiperiodic lattices spread over larger regions, reflecting the more complex spectral structure.

Overall, the work provides a comprehensive theoretical framework for creating and manipulating strongly localized gap solitons in spin‑orbit‑coupled BECs with moiré lattices. It demonstrates that (i) multiple flat bands in MLs enable ultra‑low‑norm Wannier‑type solitons, (ii) lattice geometry (period, depth) and SOC strength can reversibly switch solitons between Wannier and Bloch types, and (iii) quasiperiodicity introduces novel localization features. These findings open pathways for experimental realization of highly localized nonlinear excitations, with potential applications in quantum information processing, topological matter, and nonlinear atom‑optics.