image: (a) Schematic diagram of the distribution of Cooper-pair pairing momentum across different layers in the orbital-FFLO state. (b) Schematic of the Josephson vortex soild. (c) Schematic of the Josephson vortex liquid . (d) Phase diagram of the system.
Credit: ©Science China Press
Ising superconductors, represented by transition‑metal dichalcogenides (TMDs), form a special class of superconducting materials. In 2015, Ising superconductivity was discovered in TMDs, where the in‑plane upper critical field exceeds the classical theoretical limit, revealing that spin–orbit coupling can also protect superconductivity. Building on this, in 2017 Prof. Chaoxing Liu at Pennsylvania State University theoretically predicted that, under an in‑plane magnetic field, bilayer 2H‑stacked TMD Ising superconductors would undergo a first‑order phase transition into an unusual state in which Cooper pairs carry opposite momenta of equal magnitude—the “orbital‑FFLO state.” This prediction was ultimately confirmed in 2023, when the team led by Prof. Jurgen Smet at the Max Planck Institute for Solid State Research successfully observed the first-order transition in experiments on bilayer MoS₂. Around the same time, Prof. Jianting Ye’s group at the University of Groningen reported an important finding in multilayer NbSe₂: the upper critical field shows a pronounced upturn near the superconducting transition temperature, likewise indicating the presence of an orbital‑FFLO state. Unlike the bilayer case, however, how the finite momentum of Cooper pairs is distributed among layers in multilayer systems (e.g., alternating between adjacent layers or aligned in the same direction) remains theoretically unresolved, leaving open physical questions crucial for understanding the high‑field phase diagram of Ising superconductors.
Recently, Hongyi Yan and Prof. Haiwen Liu at the School of Physics and Astronomy, Beijing Normal University, together with Prof. Xincheng Xie (Center for Interdisciplinary Studies of Theoretical Physics and Information Sciences, Fudan University / Center for Quantum Materials Science, Peking University), Prof. Ding Zhang (Tsinghua University), Prof. Yi Liu (Renmin University of China), and collaborators, carried out a systematic study of exotic superconducting phases in multilayer/bulk Ising superconductors. Based on the Lawrence–Doniach model, their work demonstrates that bulk layered Ising superconductors can form a finite‑momentum pairing configuration (the orbital‑FFLO state) under in‑plane magnetic fields, and establishes a quantitative method to calculate the melting line of the Josephson vortex lattice. The study reveals how orbital effects influence the high‑field phase diagram of Ising superconductors and provides a theoretical basis for understanding vortex phase transitions at high magnetic fields. The results have been published in National Science Review (2026, Issue X) under the title “Orbital effect induced finite-momentum pairing and Josephson vortex lattice melting in layered Ising superconductors”.
The team proposed a new finite‑momentum configuration: under an in‑plane magnetic field, the difference in Cooper‑pair momentum between adjacent layers becomes a field‑dependent constant. This special “momentum‑staircase” arrangement can effectively compensate the kinetic‑energy cost induced by the magnetic vector potential, thereby lowering the free energy and making it more stable than other configurations at high fields. In addition, for bulk superconductors, the study finds that Josephson vortices introduced by an in‑plane field can drive another first‑order phase transition. As the field increases, vortices first form an ordered lattice and become progressively denser; at even higher fields, the dense Josephson vortex lattice “melts” into a disordered vortex liquid due to thermal fluctuations. By deriving the elastic moduli of the vortex lattice and combining them with the Lindemann criterion, the team quantitatively computed the melting line and successfully explained recent experimental results. Overall, the work highlights the decisive role of orbital effects in shaping the high‑field phase diagram of Ising superconductors and provides a theoretical foundation for understanding vortex phase transitions in the high‑field regime.
Meanwhile, the Josephson vortex state induced by an in-plane magnetic field between superconducting layers leads to another type of first-order phase transition. As the field increases, Josephson vortices first form a lattice and gradually evolve into a densely packed arrangement; at even higher fields, the vortex lattice melts into a Josephson vortex liquid, resulting in a first-order transition. By deriving the elastic moduli of the dense Josephson vortex lattice and combining them with the Lindemann criterion, the team quantitatively determined the melting line , developed a theoretical framework for Josephson vortex-lattice melting in layered Ising superconductors, and successfully explained the relevent experimental results reported recently.
This work reveals how orbital effects shape the high-field phase diagram of Ising superconductors, and provides a theoretical basis for understanding vortex phase transitions at high magnetic fields.
Journal
National Science Review