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 (PDWs) states. We show that in a system of spin-1/2 fermions in two dimensions in an electronic nematic spin-triplet phase where rotational symmetry is broken in both real-and spin-space PDWphases 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 to 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) PDWsuperconductor. The PDWphases 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).