Signatures of bosonic Landau levels in a finite-momentum superconductor.

Charged particles subjected to magnetic fields form Landau levels (LLs). Originally studied in the context of electrons in metals, fermionic LLs continue to attract interest as hosts of exotic electronic phenomena. Bosonic LLs are also expected to realize novel quantum phenomena, but, apart from recent advances in synthetic systems, they remain relatively unexplored. Cooper pairs in superconductors-composite bosons formed by electrons-represent a potential condensed-matter platform for bosonic LLs. Under certain conditions, an applied magnetic field is expected to stabilize an unusual superconductor with finite-momentum Cooper pairs and exert control over bosonic LLs. Here we report thermodynamic signatures, observed by torque magnetometry, of bosonic LL transitions in the layered superconductor BaNbS. By applying an in-plane magnetic field, we observe an abrupt, partial suppression of diamagnetism below the upper critical magnetic field, which is suggestive of an emergent phase within the superconducting state. With increasing out-of-plane magnetic field, we observe a series of sharp modulations in the upper critical magnetic field that are indicative of distinct vortex states and with a structure that agrees with predictions for Cooper pair LL transitions in a finite-momentum superconductor. By applying Onsager's quantization rule, we extract the momentum. Furthermore, study of the fermionic LLs shows evidence for a non-zero Berry phase. This suggests opportunities to study bosonic LLs, topological superconductivity, and their interplay via transport, scattering, scanning probe and exfoliation techniques.