Electromagnetic Responses of Vortex Lattices in Unconventional Superconductors

TL;DR

The work shows that the rigid flow of a vortex lattice in high magnetic fields is not guaranteed for all lattice configurations; it occurs only for structures that minimize the appropriate Abrikosov-like free energy factors ($β_A^{(1)}$ for single-component and $β_A^{(2)}$ for two-component order parameters). Extending to $d$-wave pairing introduces field-induced lattice anisotropy (rhombic) that, however, does not translate into diagonal in-plane conductivities, while elasticity exhibits anisotropic tilt moduli. The analysis highlights that the vortex-lattice response is governed by lattice energetics and collective fluctuations rather than a simple superposition of single-vortex dynamics, with potential implications for pinning and the elasticity framework of vortex matter in unconventional superconductors. The results provide a unified view of how lattice structure, multi-component order parameters, and pairing symmetry shape electromagnetic responses and elastic properties of vortex matter in high-field regimes.

Abstract

The electro-magnetic responses of ordered vortex lattices in unconventional superconductors are studied in a high field approximation. In the cases with a vortex lattice formed within the lowest Landau level of the superconducting order parameter (OP) such as a conventional s-wave paired system with a single OP and a nonchiral spin triplet paired one with multiple components of OPs, the vanishing of the superfluid stiffness for a gauge field disturbance perpendicular to the applied uniform magnetic field is found to be ensured only for the vortex lattice structures minimizing the free energy. The notion of the vanishing superfluid stiff ness ensured by minimization of the free energy is found to be satisfied in a more complex d-wave pairing case where the vortex lattice in lower fields has an anisotropic structure de viated from the six-fold hexagonal symmetry. Interestingly, such an anisotropy in the vortex lattice structure of a d-wave paired superconductor is reflected not in the resulting vortex flow conductivities obtained after minimizing the free energy but in the elastic energy describing the harmonic fluctuation around the vortex lattice state. Relevance of the obtained results to the vortex pinning effects are discussed.

Figures (4)

• Figure 1: Parallelogram expressing the unit cell of the conventional vortex lattice.
• Figure 2: Four types of lattice structures of vortices occurring in the GL model (\ref{['GLtriplet']}) with $\rho=0$ (see Fig.3) : (a) rectangular (Rec), (b) square (SQ), (c) rhombic (Rh), and triangular (T) lattices. In the figures, the dark green dots are zero points (vortices) of $\Delta_1$, while those of $\Delta_2$ are expressed by light green dots.
• Figure 3: Phase diagram of structural transitions among the vortex lattices expressed in Fig.2 of the model (\ref{['GLtriplet']}) with $\rho=0$. In the state denoted by A, one of the two OPs vanishes, and the familiar triangular vortex lattice of the nonvanishing OP occurs.
• Figure 4: Squashing deformation (black arrows) of a nearly triangular lattice (blue dashed lines) to a square lattice (red solid lines) occurring with increasing the field in the $d$-wave superconductor.