Excitations and anisotropic sound in planar dipolar supersolids with tilted dipoles

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.

💡 Research Summary

This paper presents a comprehensive study of the collective excitations and anisotropic sound propagation in planar dipolar supersolids when the dipoles are tilted so that they possess an in‑plane component. The authors focus on two representative crystalline orders that emerge in a pancake‑shaped Bose‑Einstein condensate of magnetic atoms: a two‑dimensional (2D) triangular‑like lattice (which becomes orthorhombic under tilt) and a one‑dimensional (1D) stripe lattice. Using an extended mean‑field framework that includes beyond‑mean‑field Lee‑Huang‑Yang corrections, they first compute ground‑state density patterns and the corresponding Bogoliubov–de Gennes (BdG) excitation spectra for a range of tilt angles α, scattering lengths a_s, and densities ρ_0.

The BdG spectra reveal the expected Goldstone branches associated with broken U(1) gauge symmetry (phase) and broken translational symmetry (phonons), together with a transverse shear mode that appears only for the 2D lattice. Importantly, the slopes of the gapless branches—i.e., the sound speeds—are different along the principal crystallographic axes (Γ–X versus Γ–Y) when the dipoles are tilted, signalling clear anisotropy. For the stripe phase, the single acoustic branch exhibits a continuous angular dependence: the sound speed varies smoothly with the angle between the propagation direction and the stripe orientation.

To rationalise these findings, the authors develop a hydrodynamic theory that extends the isotropic supersolid formalism to include anisotropic elasticity. Three slow fields are introduced: the areal density ρ, the superfluid phase ϕ, and the displacement vector u = (u_x,u_y). The displacement gradient u_{ij}=∂j u_i is split into a symmetric strain ε{ij} and an antisymmetric rotation w=−½(u_{xy}−u_{yx}). The energy density is expanded to quadratic order in these variables, yielding:

• An elastic tensor C_{ijkl} with four independent components for the orthorhombic 2D lattice (C_{xxxx}, C_{yyyy}, C_{xxyy}, C_{xyxy}).
• Two orientational coefficients A and B that describe, respectively, the pure rotational stiffness and the coupling between rotation and shear strain—both of which vanish when the dipoles are aligned perpendicular to the plane.
• A bulk modulus α_{ρρ}=∂²E/∂ρ² and mixed density‑strain couplings γ_{ij}=∂²E/(∂ε_{ij}∂ρ), which are small but essential for quantitative agreement with BdG.
• A superfluid density tensor ρ_{s,ij}= (1/m)∂²E/∂v_i∂v_j that captures the reduction of the superfluid fraction caused by density modulation.

From the quadratic Lagrangian built with these parameters, the authors derive the equations of motion for the hydrodynamic fields and solve for plane‑wave solutions e^{i(q·r−ωt)}. The resulting dispersion relations give analytic expressions for the sound speeds. For the 2D lattice, the speeds along the principal axes are

c_x = √