Optical conductivity from pair density waves

We present a theory of optical conductivity in systems with finite-momentum Cooper pairs. In contrast to the BCS pairing where ac conductivity is purely imaginary in the clean limit, there is nonzero ac absorption across the superconducting gap for finite-momentum pairing if we break the Galilean symmetry explicitly in the electronic Hamiltonian. Vertex correction is crucial for maintaining the gauge invariance in the mean-field formalism and dramatically changes the optical conductivity in the direction of the pairing momentum. We carried out a self-consistent calculation and gave an explicit formula for optical conductivity in a simple case. This result applies to the Fulde-Ferrell-Larkin-Ovchinnikov state and candidates with pair density waves proposed for high-$T_{\text{c}}$ cuprates. It may help detect pair density waves and determine the pairing gap as well as the direction of the pairing momentum in experiments.

Figure 1

Two examples of finite-momentum pairing. (a) FFLO pairing. The dark orange region is occupied by both spins, while the light orange region is occupied by only up spin. The blue shaded regions on the Fermi surface are gapped out by pairing. (b) Amperean pairing. A different pairing mechanism without spin splitting, where the vicinity of a hot spot on the Fermi surface is gapped out, and the pairing momentum is close to $2k_F$.

Figure 2

The self-consistent equation of the mean-field Green's function and the diagrams included in this approximation. The solid line represents the two-component Nambu spinor, and the dashed line represents the electron-electron interaction mediated by a boson, e.g., phonon. We have ignored the correction of the interaction since it is not important for our purpose. All diagrams without the crossing of the interaction line are included.

Figure 3

The self-consistent vertex correction and the diagrams included in the corrected electromagnetic response function $K_{\mu \nu }$ (defined as $j_{\mu }=K_{\mu \nu }{A}_{\nu }$). The solid line represents the Nambu spinor, the dashed line represents the electron-electron interaction, and the wavy line represents the electromagnetic field. The second diagram on the first line of $K_{\mu \nu }$ is the paramagnetic response $P_{\mu \nu }$.

Figure 4

Optical conductivity of the FFLO state calculated for tight-binding bands on a two-dimensional square lattice, $t_2/t_1=0.35$. The spin splitting is set to be $0.4t_1$, which is about 4% of the bandwidth, and the pairing momentum is $(0.1/a,0.1/a)$. (a) Conductivity in the direction of the pairing momentum. The dashed orange line is the bare result, and the blue line is the corrected result. (b) Conductivity in the perpendicular direction. The vertex correction is identically zero in this direction by symmetry.