Excitations and anisotropic sound in planar dipolar supersolids with tilted dipoles

TL;DR

The paper develops an anisotropic hydrodynamic framework for planar dipolar supersolids with dipoles tilted into the plane, introducing orientational elastic terms that capture broken rotational symmetry. By combining a quadratic supersolid Lagrangian with ground-state-derived elastic parameters, it yields analytic expressions for sound speeds along principal axes and for the stripe phase in arbitrary directions, and validates these against full BdG calculations. The key contribution is a unified description of long-wavelength excitations that links tilt-induced anisotropy to measurable quantities like sound speeds and superfluid fractions, with connections to recent stripe experiments and smectic-like interpretations. This framework enables quantitative interpretation of experiments on tilted dipolar systems and sets the stage for exploring anisotropic supersolid dynamics in engineered 2D and stripe configurations.

Abstract

We investigate the collective excitations of anisotropic dipolar supersolids in planar confinement, focusing on triangular and stripe phases in situations where the dipoles are titled to have a component in the plane. Using Bogoliubov-de Gennes calculations and hydrodynamic theory, we identify the elastic parameters that govern the long-wavelength dynamics, including two orientational coefficients that capture the broken rotational symmetry induced by dipole tilt. Analytical expressions for the speeds of sound are obtained along the principal axes for triangular supersolids and along any propagation direction for the stripe supersolid. Our results provide a unified framework for understanding sound propagation in anisotropic dipolar supersolids and establish connections to recent experiments on sound propagation in striped Bose-Einstein condensates.

Figures (5)

• Figure 1: Ground state and band structure of an anisotropic 2D dipolar supersolid supersolid. Ground state (a) density isosurface and (b) $z=0$ density slice. (c) The band structure along the special lines, with the first Brillouin zone and special points shown for reference. In (a) the green line indicates the direction of the dipole polarization. The red (blue) isosurface are at $0.5\times10^{20}/$m$^{3}$ ($4\times10^{20}/$m$^{3}$). The dashed line indicates the conventional cell. In (b) the primitive cell (solid line), conventional cell (dashed line) and the lattice vectors are shown. Parameters: $^{164}$Dy gas with $a_{dd}=130.8\,a_0$, $a_s=100.7\,a_0$, $\rho_0=0.075/a_{dd}^2$ and axial confinement of $\omega_z/2\pi=72.4\,$Hz and the dipoles tilted by an angle of $\alpha=38^\circ$. The lattice vector angle is $\theta=55.1^\circ$ with length $a=5.38\,\mu$m.
• Figure 2: Primitive cells and distortions. (a) Primitive 2D unit cell used in subplots (b)-(e) to give examples of: (b) Normal strain in the $x$-direction; (c) Normal strain in the $y$-direction; (d) Shear strain; (e) Rotation. The 1D stripe cell is illustrated subject to: (f) Normal strain in the $y$-direction; (g) Rotation. In (b)-(g) the blue cells and grey vectors illustrate the unperturbed cell and lattice vectors, while the magenta cells and vectors represent the perturbed cell and lattice vectors. The horizontal red vectors illustrate the component of the tilted dipoles in the $xy$ plane.
• Figure 3: Speeds of sound and elastic parameters for an anisotropic triangular supersolid. (a) First sound speeds ($c_+$) and (b) second ($c_-$) and transverse ($c_t$) sound speeds along principal axes as a function of dipole tilt. The markers indicate the speeds of sound obtained from BdG calculations whereas the lines are from the hydrodynamic theory. Inset to subplot (a) reveals the difference in speeds near $\alpha=35^\circ$. (c,d) Corresponding elastic parameters obtained from ground state calculations. Subplot (d) focuses on parameters relevant to the transverse speeds of sound. (e) Corresponding superfluid fractions obtained from ground state calculations. The vertical dashed line indicates the first-order transition of the ground state between triangular and stripe phases. Other parameters as in Fig. \ref{['fig:gsbs']}.
• Figure 4: Stripe phase density profile and excitation properties. (a) Density of the stripe state on the $y$-axis for the tilt angles $\alpha=0^\circ,30^\circ,$ and $45^\circ$, as labeled in the subplot. Speeds of sound versus propagation angle $\theta$ for the (b) lower $c_-$ and (c) upper $c_+$ branch. Solid lines are hydrodynamic results and markers indicate speeds of sound obtained from BdG calculations. Hydrodynamic $c_-$ results are shown in (c) for context. The lowest three excitations bands ($\nu=1,2,3$) for propagation along (d) ${q}_x$ and (e) ${q}_y$ obtained from BdG calculations. Results shown for $\alpha=0^\circ$ (solid) and $\alpha=45^\circ$ (dashed). Other parameters for this calculation are given in Table \ref{['tab:stripeparams']}.
• Figure 5: Superfluid smectic interpretation of the stripe state. Schematic of smectic stripe state with (a) untilted and (b) tilted dipoles. For tilted dipoles the smectic stripe layers prefer to align parallel to the $x$ axis. (c), (d) The elastic parameters for the stripe state as a function of tilt angle. Other system parameters in Table \ref{['tab:stripeparams']}.