Abstract
We determine the hydrodynamic modes of the superfluid analog of a smectic-A liquid crystal phase, i.e., a state in which both gauge invariance and translational invariance along a single direction are spontaneously broken. Such a superfluid smectic provides an idealized description of the incommensurate supersolid state realized in Bose–Einstein condensates with strong dipolar interactions as well as of the stripe phase in Bose gases with spin–orbit coupling. We show that the presence of a finite normal fluid density in the ground state of these systems gives rise to a well-defined second-sound type mode even at zero temperature. It replaces the diffusive permeation mode of a normal smectic phase and is directly connected with the classic description of supersolids by Andreev and Lifshitz in terms of a propagating defect mode. An analytic expression is derived for the two sound velocities that appear in the longitudinal excitation spectrum. It only depends on the low-energy parameters associated with the two independent broken symmetries, which are the effective layer compression modulus and the superfluid fraction.
Highlights
The question whether superfluidity might persist even in a solid state has a long history
An upper bound on the associated superfluid fraction fs < 1 that only involves the inhomogeneous density profile was given by Leggett [3]
Supersolid states in a commensurate situation, i.e., with an integer number of atoms per unit cell, require a fine-tuning to a vanishing value of the defect density and are not generic
Summary
The question whether superfluidity might persist even in a solid state has a long history. We analyze the spectrum of hydrodynamic and Goldstone modes for a general class of supersolids that exhibit a mass-density wave along a single direction They may be thought of as a superfluid version of a classical smectic-A liquid crystal [17]. There are two appendices, one on the Leggett bound for the superfluid fraction in superfluids with an inhomogeneous density profile and a second one on the hydrodynamic modes transverse to the direction of spatial order This allows to connect our results to earlier work by Radzihovsky and Vishwanath on superfluid liquid crystal states in imbalanced Fermi superfluids with either Larkin-Ovchinnikov or Fulde-Ferrell type order along a single direction [23, 24]
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