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Published February 13, 2024 | Published
Journal Article Open

Quantum interference in superposed lattices

  • 1. ROR icon California Institute of Technology


Charge transport in solids at low temperature reveals a material’s mesoscopic properties and structure. Under a magnetic field, Shubnikov–de Haas (SdH) oscillations inform complex quantum transport phenomena that are not limited by the ground state characteristics and have facilitated extensive explorations of quantum and topological interest in two- and three-dimensional materials. Here, in elemental metal Cr with two incommensurately superposed lattices of ions and a spin-density-wave ground state, we reveal that the phases of several low-frequency SdH oscillations in σₓₓₓₓ) and σᵧᵧ (ρᵧᵧ) are no longer identical but opposite. These relationships contrast with the SdH oscillations from normal cyclotron orbits that maintain identical phases between σₓₓₓₓ) and σᵧᵧ (ρᵧᵧ) . We trace the origin of the low-frequency SdH oscillations to quantum interference effects arising from the incommensurate orbits of Cr’s superposed reciprocal lattices and explain the observed π-phase shift by the reconnection of anisotropic joint open and closed orbits.

Copyright and License

© 2024 the Author(s). Published by PNAS. This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY).


We thank M.G. Dronova and D.M. Silevitch for technical assistance during dHvA measurements. Y.F. acknowledges the support from Okinawa Institute of Science and Technology Graduate University with subsidy funding from the Cabinet Office, Government of Japan. Y.W. acknowledges the startup support from the University of Tennessee, Knoxville. T.F.R. acknowledges support from the US Department of Energy Basic Energy Sciences Award No. DE-SC0014866 for the data analysis and manuscript preparation performed at Caltech. H.C. acknowledges support from US NSF CAREER grant DMR-1945023.


Y.F. designed research; Y.F., Y.W., T.F.R., P.B.L., and H.C. performed research; Y.F., Y.W., T.F.R., P.B.L., and H.C. analyzed data; and Y.F. wrote the paper.

Data Availability

All study data are included in the article and/or SI Appendix.

Conflict of Interest

The authors declare no competing interest.


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February 6, 2024
February 20, 2024