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

## Abstract

Charged particles subjected to magnetic fields form Landau levels (LLs). Originally studied in the context of electrons in metals1, fermionic LLs continue to attract interest as hosts of exotic electronic phenomena2,3. Bosonic LLs are also expected to realize novel quantum phenomena4,5, but, apart from recent advances in synthetic systems6,7, 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 pairs8,9 and exert control over bosonic LLs10,11,12,13. Here we report thermodynamic signatures, observed by torque magnetometry, of bosonic LL transitions in the layered superconductor Ba6Nb11S28. 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 superconductor10,11,12,13,14. By applying Onsager’s quantization rule15, 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 transport16, scattering17, scanning probe18 and exfoliation techniques19.

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## Data availability

The data presented in this article are available from the Harvard Dataverse at https://doi.org/10.7910/DVN/PLWWKA.

## Code availability

The codes used for the density functional theory and analytical calculations in this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We are grateful to S. K. Yip, M. Shayegan and M. T. Randeria for discussions. This research is funded in part by the Gordon and Betty Moore Foundation through grant GBMF9070 to J.G.C. (instrumentation development), the Office of Naval Research (ONR) under award N00014-21-1-2591 (advanced characterization), the US Department of Energy (DOE) Office of Science, Basic Energy Sciences, under award DE-SC0019300 (material synthesis) and award DE-SC0022028 (structure analysis), the STC Center for Integrated Quantum Materials, NSF grant DMR-1231319 (E.K.) and the DOE Office of Basic Energy Sciences under award DE-SC0018945 (L.F.). S.F. acknowledges support from the Rutgers Center for Materials Theory Distinguished Postdoctoral Fellowship. Computations were performed on the Cannon cluster supported by the FAS Division of Science Research Computing Group (FASRC) at Harvard University. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement DMR-1157490, the State of Florida, and DOE.

## Author information

Authors

### Contributions

A.D. and J.G.C. conceived the project. A.D. grew the single crystals, characterized the materials, and performed the measurements with T.S., J.Z., D.G. and M.K. supporting. A.D. performed the analytical calculations and S.F. and A.D. the electronic structure calculations with L.F. and E.K. supporting. A.D. and J.G.C. wrote the manuscript with contributions and discussions from all authors. J.G.C. supervised the project.

### Corresponding author

Correspondence to J. G. Checkelsky.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks the anonymous reviewers for their contribution to the peer review of this work.

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## Extended data figures and tables

### Extended Data Fig. 1 Torque magnetometry.

a Non-collinear response of Μ to Η results in τ which can be measured using torque magnetometry techniques. b Capacitive torque magnetometer (sample S1). c Resistive torque magnetometer (sample S2). Scale bars are 1 mm. d δC(Η) at θ = 89.7º and Τ = 0.3 K for sample S1. (inset) The experimental geometry for S1. δC < 0 is consistent with diamagnetic response of $$M$$. e Comparison of ρxx(T) and Mτ(T) measured with μ0H = 0.4 mT for sample S1. f Comparison of ρxx(H) and Mτ(H) at θ = 89.7º and T = 0.3 K for sample S1. g $$M(H)$$ loop for θ = 0º at T = 0.4 K measured by SQUID. h Mτ(H) loop for θ = 46º at T = 0.3 K measured by torque magnetometry (S1). (inset) τ(H) from which we obtain Mτ(H) in the main panel. i δ quantum oscillation frequency F(θH) (orange points) with fit to equation (2) (dashed line). (inset) θ where Mτ(H) oscillates around zero corresponds to θ = 90º.

### Extended Data Fig. 2 Survey of H-T-θ phase space (sample S1).

a 𝜕Μτ(H)/𝜕H of up-sweep Mτ(H) (see Fig. 2a) at θ = 89.7º and various fixed temperatures, vertically offset for clarity. The peak-like structure marking H is traced by a dashed guide to the eye. b Up- (solid) and down-sweep (dashed) Mτ(H) measured at T = 0.31 K at various angles between 80º and 90º, vertically offset for clarity. c Up- (solid) and down-sweep (dashed) Mτ(H) measured at θ = 89.2º at various fixed temperatures, vertically offset for clarity. d 𝜕Μτ(H)/𝜕H of up-sweep Mτ(H) at θ = 89.2º (shown in c) for various fixed temperatures, vertically offset for clarity. The peak-like structure marking H is absent here. e T-H phase diagram at θ = 89.2º showing Hc2 and Hc1.

### Extended Data Fig. 3 H-T phase diagram (sample S2).

a Up- (solid) and down-sweep (dashed) Mτ(H) measured at θ = 88.9º for various temperatures, vertically offset for clarity. b 𝜕Μτ(H)/𝜕H of up-sweep Mτ(H) at θ = 88.9º (shown in a) for fixed temperatures down to T = 33 mK, vertically offset for clarity. The peak-like structure at H is traced by a dashed guide to the eye. c T-H phase diagram at θ = 88.9º showing Hc2, H, and Hc1. Black circle marks Tc.

### Extended Data Fig. 4 Mτ(H) at fixed T and θ (sample S1).

a Up- (solid) and down-sweeps (dashed) of Mτ(H) for T = 395 mK at equally spaced, fixed θ between 89.30º and 89.95º, vertically offset for clarity. b 𝜕Μτ(H)/𝜕H of up-sweep in (a), vertically offset for clarity. Guide to the eye (grey dashed line) traces corrugation of H(θ). c Up- (solid) and down-sweeps (dashed) of Mτ(H) for T = 460 mK at equally spaced, fixed θ between 89.75º and 89.95º, vertically offset for clarity. d 𝜕Μτ(H)/𝜕H of up-sweep in (c), vertically offset for clarity. Guide to the eye (grey dashed line) traces corrugation of H(θ).

### Extended Data Fig. 5 Low temperature Mτ(H) at fixed θ.

a Up- and b down-sweeps of Mτ(H) for T = 20 mK at equally spaced, fixed θ between 88.45º and 89.95º, vertically offset for clarity. Guide to the eye (grey dashed line) in (b) traces angular evolution of Hʹ(θ) c 𝜕Μτ(H)/𝜕H of up-sweep Mτ(H) in (a), vertically offset for clarity. Guide to the eye (grey dashed line) traces corrugation of H(θ) (inset) Up- and down-sweep Mτ(H) (solid and dashed, respectively) at fixed angles near 90º for a third sample, S3.

### Extended Data Fig. 6 Ginzburg-Landau modeling.

a GL free energy δF(qx, qy) in the finite-q pairing state of an isotropic superconductor. Finite-q pairing with |q| = q0 minimizes δF (red contour). b (black line) Radial cut of δF(q) for H = 0. (orange points) For H ≠ 0, discrete LLs are formed. The LL with (2n+1)½/𝓁 closest to q0 is the optimal solution. c Superconductivity opens a gap ΔSC around the Fermi energy EF (orange). d Magnetic field causes Zeeman splitting of a spin-degenerate band (dashed) into spin-up (blue) and spin-down (red) bands. e Zeeman split bands create two Fermi surface contours in k-space separated by q.

### Extended Data Fig. 7 Electronic structure modeling.

a The Fermi surface contours for monolayer H-NbS2. The supercell Brillouin zones and corresponding M points are shown as white lines and open blue circles, respectively. b Electronic structure resulting from 3 × 3 zone-folding. This is formed by overlapping the nine cells marked by white lines in (a). c Fermi surface contours from first-principles calculations capturing monolayer H-NbS2 and the spacer layer 3 × 3 perturbation. This resembles (b) but with additional band gaps and renormalization due to the spacer layer perturbation. d Electronic structure around K, Kʹ from first-principles with the δ pocket corresponding to fδ identified. e Electronic structure around M from first-principles showing the inner α1 and outer α2 pockets. f Bloch sphere showing spin-texture along the δ pocket. (inset) Expanded view near the north pole of the Bloch sphere. Blue arrow shows sense of spin evolution for clockwise motion along the δ pocket, see white arrow in (d). g Bloch sphere showing spin-texture along the α1 (blue) and α2 (red) pockets. Blue (red) arrow shows sense of spin evolution for clockwise motion along the α12) pocket, see white arrows in (e). h Coordinate system used for M point kp model. i Bloch sphere showing the segments $${{\mathscr{C}}}_{a}$$ (red) and $${{\mathscr{C}}}_{b}$$ (blue) traced by the spin-1/2 eigenvector $${\rm{|}}\varPsi {\rm{\rangle }}$$ which together form a closed  contour $${\mathscr{C}}$$. j Berry curvature ΩB(kx,ky). For ΔM ≠ 0, ΩB integrated over the area Ak defined by the FS contour yields φB ≠ π.

### Extended Data Fig. 8 de Haas-van Alphen oscillations.

a $${M}_{\tau }^{{osc}}$$(H) at T = 0.3 K for various angles. The oscillations are aligned across 75º when plotted versus H indicating the two-dimensionality of the FSs. b $${M}_{\tau }^{{osc}}(H)$$ at θ = 18º for various temperatures. c $${\widetilde{A}}_{{FFT}}(T)$$ for the α and δ oscillations (points) with fits to $${R}_{T}^{i}$$ (Methods) to extract the effective mass. d $${M}_{\tau }^{{osc}}(1/H)$$ measured at T = 20 mK for θ = 8.2º (blue). The result of applying a sliding window centered at various equally spaced 1/H0 are shown as gray and orange traces, vertically offset for clarity. e FFT of un-windowed (blue) and windowed (gray) $${M}_{\tau }^{\mathrm{osc}}(1/H)$$ in (d), vertically offset for clarity. The peak at $${f}_{{{\rm{\alpha }}}^{\star }}$$ (grey dashed guide) appears for 1/ μ0H0 < 0.21 T−1, orange trace in (d) and (e).

### Extended Data Fig. 9 Phase analysis.

a Mτ(H) measured at T = 0.3 K and θ = 80º showing both the Meissner effect (left) and dHvA quantum oscillation (right). b $${M}_{\tau }^{\mathrm{osc}}$$ (1/H) at T = 0.3 K (T = 20 mK) for sample(s) S1 (S4-S5) at angles near θ = 16º. c-f Phase-shift function K(f, φB)around the δ (left) and α (right) oscillations for samples S1, S3, S4, and S5 using $${M}_{\tau }^{\mathrm{osc}}$$ shown in (b). The red contours are drawn at 98% of the local maxima. K(f, φB) for S2 is shown in Fig. 4b.

### Extended Data Fig. 10 Calculated Landau level spectrum.

a $$M$$-point pockets resulting from the kp model with the experimentally determined SOC parameters. The inner (outer) contour in red (blue) corresponds to the measured $${f}_{{{\rm{\alpha }}}_{1}}$$ ($${f}_{{{\rm{\alpha }}}_{2}}$$) frequency. The intersecting dashed green circles are the contours in the presence of mirror symmetry. b Calculated density of states in the presence of H showing oscillations due to Landau quantization of the electronic structure in (a). (inset) FFT power spectrum shows three modes corresponding to $${f}_{{{\rm{\alpha }}}_{1}}$$ (red), $${f}_{{\alpha }_{2}}$$ (blue), and the breakdown frequency $${f}_{{{\rm{\alpha }}}^{\star }}$$ (green). c The breakdown contribution at $${f}_{{{\rm{\alpha }}}^{\star }}$$ in the FFT power spectrum vanishes at μ0Hmb ≈ 6 T. d FFT power spectrum at various values of the chemical potential EF showing a Landau fan. By comparing with the observed pocket sizes, we can estimate EF.

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Devarakonda, A., Suzuki, T., Fang, S. et al. Signatures of bosonic Landau levels in a finite-momentum superconductor. Nature 599, 51–56 (2021). https://doi.org/10.1038/s41586-021-03915-3

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• DOI: https://doi.org/10.1038/s41586-021-03915-3