$\mathbb{Z}_n$ symmetry broken supersolid in spin-orbit-coupled Bose-Einstein condensates

Supersolid is an exotic state of matter characterized by both superfluid properties and periodic particle density modulation, due to spontaneous breaking of U(1) gauge symmetry and spatial translation symmetry, respectively. For conventional supersolids,continuous translation symmetry breaking is accompanied by one gapless Goldstone mode in the excitation spectra. An interesting question naturally arises: what is the consequence of breaking discrete translation symmetry for supersolids? In this work, we propose the concept of $\mathbb{Z}_n$ supersolid resulting from spontaneous breaking of a discrete $\mathbb{Z}_n$ symmetry, or equivalently, a discrete translation symmetry. This $\mathbb{Z}_n$ supersolid is realized in the stripe phase of spin-orbit-coupled Bose-Einstein condensate under an external periodic potential with period $1/n$ of intrinsic stripe period. For $n\geq2$, there are $n$ degenerate ground states with spontaneously broken lattice translation symmetry. The low-energy excitations of $\mathbb{Z}_n$ supersolid include a pseudo-Goldstone mode, whose excitation gap at long wavelength limit is found to decrease fast with $n$. We further numerically show that, when confined in a harmonic trap, a spin-dependent perturbation can result in the transition between degenerate ground states of $\mathbb{Z}_n$ supersolid. With the integer $n$ tunable using the experimental technique of generating subwavelength optical lattice, the $\mathbb{Z}_n$ supersolid proposed here offers a cold atom platform to simulate physics related with generic $\mathbb{Z}_n$ symmetry breaking, which is interesting not only in the field of cold atoms, but also in particle physics and cosmology.