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-$\frac{1}{2}$ fermions in two dimensions in an electronic nematic spin-triplet phase where rotational symmetry is broken in both real- and 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 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 $s$) 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).

Figure 1

(a) Distortion of the FS for the fermions with spin-up (black) and spin-down (red) for a triplet $\alpha$ nematic phase with $l=2$ (Wu et al. [63]). (b),(c) Distortion of the FS for the fermions with spin plus (black) and minus (red) with respect to the spin quantization axis (blue) for a $\beta$ nematic phase with $l=2$ and $l=-1$, respectively. (d) Putative nematic spin order in a copper oxide plane in YBCO [21].

Figure 2

SC susceptibility for the $\alpha$ phase in the $s$-wave and $d$-wave channels in the direction $\varphi =n\pi /2$.

Figure 3

SC susceptibility for the $\beta$ phase in the $s$-wave and $d$-wave channels in the direction $\varphi =n\pi /2$.

Figure 4

(a) Phase diagram as a function of the parameters $T/{\Delta }_{\mathrm{BCS}}$ and $\delta /{\Delta }_{\mathrm{BCS}}$. The light blue region corresponds to the uniform $d$-wave SC state (with $\mathit{Q}=0$). The yellow and green regions correspond to the PDW and the bidirectional SC states, respectively. $P$ labels the tricritical point discussed in the text. (b) Order parameter of the uniform $d$-wave SC state. (c)–(e) Direction of the inhomogeneous superconducting wave vector $\mathit{Q}$ for (c) $0.23\gtrsim T/{\Delta }_{\mathrm{BCS}}\gtrsim 0.055$, (d) $0.055\gtrsim T/{\Delta }_{\mathrm{BCS}}>0$, and (e) $T=0$. Smooth curves show continuous phase transitions and first-order transitions are shown as dashed curves.

Figure 5

Phase diagram in the nematic $\beta$ phase for $s$-wave pairing. Here $n$ is the magnitude of the order parameter of the spin-triplet nematic $\beta$ phase. All transitions from the non-SC $\beta$ phase are continuous. Notice the complex sequence of phases with different types of homogeneous and inhomogeneous orders and their sequence of multicritical points. Here $L$ labels the Lifshitz point discussed in the text. Smooth curves show continuous phase transitions and first-order transitions are shown as dashed curves. The same caveats we pointed out in the case of the $\alpha$ phase apply here too. See text for details.

Figure 6

Phase diagram in the nematic $\beta$ phase for $d$-wave pairing. Here $n$ is the magnitude of the order parameter of the spin-triplet nematic $\beta$ phase. $L$ is the Lifshitz point separating the uniform $d$-wave SC from the unidirectional PDW phase. $P_2$ is a tricritical point separating a continuous non-SC high-temperature phase from the bidirectional PDW phase, and $P_1$ is a bicritical point separating the low-temperature bidirectional phase oriented along the antinodal direction of the $d$ wave from a bidirectional PDW oriented along the nodal direction. Smooth curves show continuous phase transitions and first-order transitions are shown as dashed curves. See text for details.

Figure 7

Superconducting susceptibility for the FF state.

Figure 8

Quartic effective interaction for the FF state.

Figure 9

Superconducting susceptibility for the LO state.

Figure 10

Diagrams that contribute to the coefficient $C_4^{\mathrm{PDW}}$.

Figure 11

Susceptibility for bidirectional superconducting order.

Figure 12

Feynman diagrams for the coefficients $c_{4B1}$ and $c_{4B2}$ (see text).