$O(N)$ smectic $\sigma$-model

A unidirectional "density" wave order in an otherwise isotropic environment is guaranteed to display a smecticlike Goldstone mode. Examples of such "soft" states include conventional smectic liquid crystals, putative Fulde-Ferrell-Larkin-Ovchinnikov superfluids, and helical states of frustrated bosons and spins. Here we develop generalized spin-smectic $\sigma$-models that break $O(N)$ internal symmetry in addition to the $d$-dimensional rotational and uniaxial translational symmetries. We explore long-wavelength properties of such strongly fluctuating states, show that they are characterized by a "double-power-law" static structure peak, and analyze their asymptotic symmetry-reduced crossover to conventional low-energy modes. We also present the associated Ginzburg-Landau theory, describing phase transition into such spin-smectic states, and discuss experimental realization of such models.