Theory of superconducting quantum liquid crystals*

A novel superconductor state, whose order parameter oscillates producing a {\em pair density wave} (PDW), in which charge, spin and superconducting orders are intertwined, was recently proposed\footnote{E. Berg, E. Fradkin, S.A. Kivelson, Phys.Rev. B79, 064515 (2009)}. The PDW necessarily has charge density wave(CDW) modulations with double period and a novel homogeneous superconducting order with $4e$ charged quasi-particles(4eSC)\footnote{E. Berg, E. Fradkin, S.A. Kivelson, Nature Phys. 5, 830 (2009)}. The PDW is an anisotropic quantum liquid with nematic symmetry. In this work, we present a theory of smectic and superconductor fluctuations, given by PDW, CDW and 4eSC order parameters, in a nematic background. Our theory closely resembles the description of the smectic-nematic phase transition in classical liquid crystals. We compute the effect of the director fluctuations on the correlations of each ordered phase. We also make contact with the theory of thermal melting.