Moiré and frustration physics of dipolar supersolids under periodic confinement

Abstract

We study the ground-state phases of a two-dimensional dipolar supersolid subjected to external periodic confinement by numerically solving the extended Gross--Pitaevskii equation. Focusing on a regime in which the unconfined system forms an intrinsic triangular droplet crystal, we consider triangular, honeycomb, and square optical lattices and classify them into isostructural and heterostructural settings relative to the spontaneous supersolid order. We map out the stationary states as functions of the lattice depth $V_0$ and the commensurability ratio between the intrinsic droplet spacing and the external lattice period. For triangular and honeycomb confinements, the competition between the soft self-organized supersolid lattice and the rigid external potential can generate long-wavelength moiré superstructures in the weak- to intermediate-lattice regime, together with a sequence of reconstructed states including ring-like clusters and stripe-segment configurations. By contrast, the square lattice introduces strong symmetry mismatch between the intrinsic $C_6$ order and the imposed $C_4$ geometry, leading to frustration-induced anisotropic states and symmetry-reduced cluster arrangements. Our results establish dipolar supersolids under periodic confinement as an unconventional route to exploring moiré physics, where moiré superstructures arise from the competition between a self-organized soft lattice and an externally imposed rigid one.

Figures (11)

• Figure 1: (a)-(c) Schematic illustrations of the system under study. A 2D dipolar supersolid (orange spheres forming an intrinsic triangular array) is subjected to an external periodic optical potential $V_{\mathrm{ext}}$ with (a) triangular, (b) honeycomb, and (c) square geometry. The competition between the intrinsic droplet spacing $a_{\mathrm{int}}$ and the imposed lattice period $a_{\mathrm{latt}}$ generates commensurability effects and geometric frustration.
• Figure 2: Energies of converged stationary branches obtained from different initial states for the system subjected to a triangular potential, with a fixed periodicity ratio $R=1/2$. Each curve corresponds to a different initial configuration including uniform (uni), triangular (tri), hexagonal (hex), and stripe (str). Energies are measured relative to the relaxed uniform state, so $\Delta_{\rm uni}=0$.
• Figure 3: Ground-state density evolution of the dipolar Bose gas in a triangular optical lattice for $R < 1$. (a), (e) Superposition of the intrinsic triangular density (red dots) and the external potential (color map) for $R = 1/2$ and $R = 2/3$, respectively. The arrows mark the three competing length scales: $a_{\text{int}}$ (red), $a_{\text{pot}}$ (black), and the emergent moiré period $a_m$ (orange). (b)--(d) Density profiles for $R = 1/2$ at increasing lattice depths $V_0 = 0.0175$, $0.025$, and $0.05$, respectively. (f)--(h) Density profiles for $R = 2/3$ at increasing lattice depths $V_0 = 0.01$, $0.015$, and $0.02$, respectively. In each density panel, a geometric marker indicates the initial state from which the displayed ground-state density is obtained, using the same symbol convention as in Fig. \ref{['fig2']}. The star in panels (b)--(d) corresponds to the hexagonal initial state, the triangle in (f), the asterisk in (g), and the square in (h) indicate the respective initial states that yield the lowest energy at each $V_0$.
• Figure 4: Ground-state density evolution of the dipolar Bose gas in a triangular optical lattice for $R > 1$. (a), (e) Superposition of the intrinsic triangular density (red dots) and the external potential (color map) for $R = 3/2$ and $R = 5/2$, respectively. The arrows indicate the three relevant length scales: $a_{\rm int}$ (red), $a_{\rm pot}$ (dark), and the moiré period $a_m$ (orange). (b)--(d) Density profiles for $R = 3/2$ at increasing lattice depths $V_0 = 0.005$, $0.0075$, and $0.025$, respectively. (f)--(h) Density profiles for $R = 5/2$ at increasing lattice depths $V_0 = 0.025$, $0.05$, and $0.125$, respectively. Intermediate lattice depths induce symmetry-broken stripe-like states: a clearly developed stripe-like state for $R = 3/2$ and a paired-droplet or multi-scale stripe texture for $R = 5/2$. In each density panel, the geometric marker identifies the initial state yielding the ground-state density, using the same symbol convention as in Fig. \ref{['fig2']}.
• Figure 5: Ground-state density evolution of the dipolar Bose gas in a honeycomb optical lattice for $R < 1$. (a), (e) Superposition of the intrinsic triangular density (red dots) and the external potential (color map) for $R = 1/2$ and $R = 2/3$, respectively. The arrows indicate the three length scales $a_{\rm int}$ (red), $a_{\rm pot}$ (dark), and $a_m$ (orange). Unlike the triangular lattice, the potential maxima coincide with the intrinsic droplet positions. (b)--(d) Density profiles for $R = 1/2$ at $V_0 = 0.02$, $0.025$, and $0.027$, respectively. (f)--(h) Density profiles for $R = 2/3$ at $V_0 = 0.0075$, $0.01$, and $0.025$, respectively. The intermediate panels (c) and (g) display a symmetry-broken stripe phase and a ring-droplet lattice, respectively. In each density panel, the geometric marker identifies the initial state yielding the ground-state density, using the same symbol convention as in Fig. \ref{['fig2']}.