Pair-Density-Wave Superconducting States and Electronic Liquid Crystal Phases

In conventional superconductors the Cooper pairs have a zero center of mass momentum. In this paper we present a theory of superconducting states where the Cooper pairs have a nonzero center of mass momentum, inhomogeneous superconducting states known as a pair-density-waves (PDW) states. We show that in a system of spin-1/2 fermions in 2 dimensions in an electronic nematic spin triplet phase where rotational symmetry is broken both in real and in spin space PDW phases arise naturally in a theory that can be analyzed using controlled approximations. We show that several superfluid phases that may arise in this phase can be treated within a controlled BCS mean field theory, with the strength of the spin-triplet nematic order parameter playing the role of the small parameter of this theory. We find that in a spin-triplet nematic phase, in addition of a triplet $p$-wave and spin-singlet $d$-wave (or $s$ depending on the nematic phase) uniform superconducting states, it is also possible to have a $d$-wave (or $s$) PDW superconductor. The PDW phases found here can be either unidirectional, bidirectional or tridirectional depending on the spin-triplet nematic phase and which superconducting channel is dominant. In addition, a triple-helix state is found in a particular channel. We show that these PDW phases are present in the weak coupling limit, in contrast to the usual Fulde-Ferrell-Larkin-Ovchinnikov phases which require strong coupling physics in addition to a large magnetic field (and often both).