We present resistivity and thermal-conductivity measurements of superconducting FeSe in intense magnetic fields up to 35 T applied parallel to the ab plane. At low temperatures, the upper critical field µ0H ab c2 shows an anomalous upturn, while thermal conductivity exhibits a discontinuous jump at µ0H * ≈ 24 T well below µ0H ab c2 , indicating a first-order phase transition in the superconducting state. This demonstrates the emergence of a distinct field-induced superconducting phase. Moreover, the broad resistive transition at high temperatures abruptly becomes sharp upon entering the highfield phase, indicating a dramatic change of the magnetic-flux properties. We attribute the high-field phase to the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state, where the formation of planar nodes gives rise to a segmentation of the flux-line lattice. We point out that strongly orbital-dependent pairing as well as spin-orbit interactions, the multiband nature, and the extremely small Fermi energy are important for the formation of the FFLO state in FeSe.Exotic superconductivity with a nontrivial Cooperpairing state has been a longstanding issue of interest in condensed-matter physics. Among possible exotic states, a spatially nonuniform superconducting state in the presence of strong magnetic fields caused by the paramagnetism of conduction electrons has been the subject of great interest after the pioneering work by Fulde and Ferrell as well as Larkin and Ovchinnikov (FFLO) [1,2]. In the FFLO state, pair breaking due to the Pauli paramagnetic effect is reduced by forming a new pairing state (k↑, −k + q↓) with |q| ∼ gµ B H/ υ F (υ F is the Fermi velocity, g the g-factor, and µ B the Bohr magneton) between Zeeman split parts of the Fermi surface, instead of (k↑, −k↓) pairing in BCS superconductors [Figs. 1(a) and 1(b)]. The fascinating aspect of the FFLO state is that the superconducting order parameter, in its simplest form, is modulated as ∆ ∝ sin q · r, and periodic planar nodes appear perpendicular to the magnetic field near the upper critical field H c2 , leading to a segmentation of the vortices into pieces of length Λ = π/|q| [ Fig. 1(c)].Despite tremendous efforts in the search for the FFLO states in the past half century, indications of its experimental realization have been reported in only a few candidate materials, including quasi-two-dimensional (2D) organic superconductors and the heavy-fermion superconductor CeCoIn 5 [3-5]. In both systems, a thermodynamic phase transition occurs below H c2 and a high-field superconducting phase emerges at low temperatures [6][7][8][9]. In the former, each superconducting layer is very weakly coupled via the Josephson effect. The FFLO state is observed in a magnetic field H applied parallel to the layers, where the magnetic flux is concentrated in the regions between the layers forming coreless Josephson vortices. Therefore, the segmentation of the vortices by FFLO nodes, which is one of the most fascinating properties of the FFLO state, is not expected. The presence of the FFLO...

FeSe is argued as a superconductor in the Bardeen-Cooper-Schrieffer Bose-Einstein-condensation crossover regime where the superconducting-gap size and the superconducting transition temperature Tc are comparable to the Fermi energy. In this regime, vortex bound states should be well quantized and the preformed pairs above Tc may yield a pseudogap in the quasiparticle-excitation spectrum. We performed spectroscopic-imaging scanning tunneling microscopy to search for these features. We found Friedel-like oscillations near the vortex, which manifest the quantized levels, whereas the pseudogap was not detected. These apparently conflicting observations may be related to the multi-band nature of FeSe.