Quantum smectic and supersolid order in helium films and vortex arrays

A flux liquid can condense into a smectic crystal in a pure-layered superconductor with the magnetic field oriented nearly parallel to the layers. Similar order can arise in low-temperature ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ films with a highly anisotropic periodic substrate potential. If the smectic order is commensurate with the layering, this periodic array is stable to quenched random pointlike disorder. By tilting and adjusting the magnitude of the applied field, both incommensurate and tilted smectic and crystalline phases are found for vortex arrays. Related variations are possible by changing the chemical potential in the helium system. We discuss transport near the second-order smectic freezing transition, and show that permeation modes in superconductors lead to a small nonzero resistivity and a large but finite tilt modulus in the smectic crystal. In helium films, the theory predicts a nonzero superfluid density and propagating third sound modes, showing that the quantum smectic is always simultaneously crystalline and superfluid.