Elliptical Vortex and Oblique Vortex Lattice in the FeSe Superconductor Based on the Nematicity and Mixed Superconducting Orders

The electronic nematic phase is characterized as an ordered state of matter with rotational symmetry breaking and has been well studied in the quantum Hall system and the under-doped high-$T_c$ superconductors, regardless of cuprate or pnictide family. The nematic state in high-$T_c$ systems is often related to the structural transition or electronic instability in the normal phase. Nevertheless, the electronic states below the superconducting transition is still an open question. With high-resolution scanning tunneling microscope measurements, direct observation of vortex core in FeSe thin films reveals the nematic superconducting state [Song \emph{et al}., Science 332, 1410 (2011)]. Here, motivated by the experiment, we construct the extended Ginzburg-Landau free energy to describe the elliptical vortex, where a mixed isotropic \emph{s}-wave and anisotropic \emph{d}-wave superconducting order is coupled to the nematic order. The nematic order induces the mixture of two superconducting orders and enhance the anisotropic interaction between the two superconducting orders, resulting in a symmetry breaking from $C_4$ to $C_2$. Consequently, the vortex cores are stretched into an elliptical shape. In the equilibrium state, the elliptical vortices assemble an oblique vortex lattice with $r>\sqrt{3}$, being well consistent with experimental results.