Leggett Modes and the Anderson-Higgs Mechanism in Superconductors without Inversion Symmetry

We develop a microscopic and gauge-invariant theory for collective modes resulting from the phase of the superconducting order parameter in noncentrosymmetric superconductors. Considering various crystal symmetries, we derive the corresponding gauge mode ${\omega }_G(\mathbf{q})$ and find, in particular, new Leggett modes ${\omega }_L(\mathbf{q})$ with characteristic properties that are unique to noncentrosymmetric superconductors. We calculate their mass and dispersion that reflect the underlying spin-orbit coupling and thus the balance between triplet and singlet superconductivity occurring simultaneously. Finally, we demonstrate the role of the Anderson-Higgs mechanism: while the long-range Coulomb interaction shifts ${\omega }_G(\mathbf{q})$ to the condensate plasma mode ${\omega }_P(\mathbf{q})$, it leaves the mass ${\mathrm{\Lambda }}_0$ of the new Leggett mode unaffected and only slightly modifies its dispersion.

Figure 1

Illustration of various calculated collective modes ($T=0$) common to all NCS. The Anderson-Higgs mechanism shifts the gauge mode ${\omega }_G$ (dashed line) to the plasma mode ${\omega }_P$ usually lying in the pair-breaking continuum. The new Leggett modes (solid green lines) unique to NCS are only slightly changed by this process (not visible) and the mass ${\mathrm{\Lambda }}_0$ remains unchanged. Importantly, in some cases ${\mathrm{\Lambda }}_0\to 0$ is possible, see discussion of Fig. 3; thus, the Leggett modes might be easily observable. Note that the slope of the Leggett modes depend on the ratio $t={\mathrm{\Delta }}_{\text{tr}}/{\mathrm{\Delta }}_s$ as discussed in connection with Fig. 2.

Figure 2

Slope of the dispersion of the Leggett mode for NCS systems with $C_{4v}$ point group symmetry as a function of the unit vectors ${\widehat{\mathbf{q}}}_x$, ${\widehat{\mathbf{q}}}_y$, ${\widehat{\mathbf{q}}}_z$. (a) Comparison of the slope for $t=0.5$ (upward parabola) and $t=1.5$ (downward parabola), (b) slope along the ${\widehat{\mathbf{q}}}_z$ direction for various $t={\mathrm{\Delta }}_{\text{tr}}/{\mathrm{\Delta }}_s$.

Figure 3

Normalized mass of the Leggett mode for NCS systems with $C_{4v}$ point group symmetry for fixed ${\lambda }_s=0.1$ as a function of the mixing term and various ${\lambda }_{\text{tr}}$: ${\lambda }_{\text{tr}}=0.025$ (upper solid line), ${\lambda }_{\text{tr}}=0.05$ (middle solid line) and ${\lambda }_{\text{tr}}=0.075$ (lower solid line). The dashed lines correspond to Eq. (18), which is an analytical solution in the limit of small $t$.