## Abstract

Following nearly a century of research, it remains a puzzle that the low-lying excitations of metals are remarkably well explained by effective single-particle theories of non-interacting bands^{1,2,3,4}. The abundance of interactions in real materials raises the question of direct spectroscopic signatures of phenomena beyond effective single-particle, single-band behaviour. Here we report the identification of quantum oscillations (QOs) in the three-dimensional topological semimetal CoSi, which defy the standard description in two fundamental aspects. First, the oscillation frequency corresponds to the difference of semiclassical quasiparticle (QP) orbits of two bands, which are forbidden as half of the trajectory would oppose the Lorentz force. Second, the oscillations exist up to above 50 K, in strong contrast to all other oscillatory components, which vanish below a few kelvin. Our findings are in excellent agreement with generic model calculations of QOs of the QP lifetime (QPL). Because the only precondition for their existence is a nonlinear coupling of at least two electronic orbits, for example, owing to QP scattering on defects or collective excitations, such QOs of the QPL are generic for any metal featuring Landau quantization with several orbits. They are consistent with certain frequencies in topological semimetals^{5,6,7,8,9}, unconventional superconductors^{10,11}, rare-earth compounds^{12,13,14} and Rashba systems^{15}, and permit to identify and gauge correlation phenomena, for example, in two-dimensional materials^{16,17} and multiband metals^{18}.

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## Data availability

Data reported in this paper are available at https://doi.org/10.5281/zenodo.7957067.

## References

Luttinger, J. M. Theory of the de Haas-van Alphen effect for a system of interacting fermions.

*Phys. Rev.***121**, 1251–1258 (1961).Engelsberg, S. & Simpson, G. Influence of electron-phonon interactions on the de Haas-van Alphen effect.

*Phys. Rev. B***2**, 1657–1665 (1970).Wasserman, A., Springford, M. & Hewson, A. Theory of the de Haas-van Alphen effect for heavy-fermion alloys.

*J. Phys. Condens. Matter***1**, 2669–2676 (1989).Taillefer, L., Newbury, R., Lonzarich, G., Fisk, Z. & Smith, J. Direct observation of heavy quasiparticles in UPt

_{3}via the dHvA effect.*J. Mag. Mag. Mater.***63–64**, 372–376 (1987).van Delft, M. R. et al. Electron-hole tunneling revealed by quantum oscillations in the nodal-line semimetal HfSiS.

*Phys. Rev. Lett.***121**, 256602 (2018).Müller, C. S. A. et al. Determination of the Fermi surface and field-induced quasiparticle tunneling around the Dirac nodal loop in ZrSiS.

*Phys. Rev. Res.***2**, 023217 (2020).Ding, L. et al. Quantum oscillations, magnetic breakdown and thermal Hall effect in Co

_{3}Sn_{2}S_{2}.*J. Phys. D Appl. Phys.***54**, 454003 (2021).Pavlosiuk, O., Swatek, P. W., Wang, J.-P., Wiśniewski, P. & Kaczorowski, D. Giant magnetoresistance, Fermi-surface topology, Shoenberg effect, and vanishing quantum oscillations in the type-II Dirac semimetal candidates MoSi

_{2}and WSi_{2}.*Phys. Rev. B***105**, 075141 (2022).Broyles, C. et al. Revealing a 3D Fermi surface and electron-hole tunneling in UTe

_{2}with quantum oscillations. Preprint at https://arxiv.org/abs/2303.09050 (2023).Reiss, P. et al. Quenched nematic criticality and two superconducting domes in an iron-based superconductor.

*Nat. Phys.***16**, 89–94 (2020).Sebastian, S. E. & Proust, C. Quantum oscillations in hole-doped cuprates.

*Annu. Rev. Condens. Matter Phys.***6**, 411–430 (2015).McMullan, G. J. et al. The Fermi surface and f-valence electron count of UPt

_{3}.*New J. Phys.***10**, 053029 (2008).Shishido, H. et al. Anomalous change in the de Haas–van Alphen oscillations of CeCoIn

_{5}at ultralow temperatures.*Phys. Rev. Lett.***120**, 177201 (2018).Dalgaard, K. J., Lei, S., Wiedmann, S., Bremholm, M. & Schoop, L. M. Anomalous Shubnikov-de Haas quantum oscillations in rare-earth tritelluride NdTe

_{3}.*Phys. Rev. B***102**, 245109 (2020).Sunko, V. et al. Maximal Rashba-like spin splitting via kinetic-energy-coupled inversion-symmetry breaking.

*Nature***549**, 492–496 (2017).Phinney, I. et al. Strong interminivalley scattering in twisted bilayer graphene revealed by high-temperature magneto-oscillations.

*Phys. Rev. Lett.***127**, 056802 (2021).Broyles, C. et al. Effect of the interlayer ordering on the Fermi surface of Kagome superconductor CsV

_{3}Sb_{5}revealed by quantum oscillations.*Phys. Rev. Lett.***129**, 157001 (2022).Sanchez, D. S. et al. Topological chiral crystals with helicoid-arc quantum states.

*Nature***567**, 500–505 (2019).Landau, L. Diamagnetismus der Metalle.

*Z. Phys.***64**, 629–637 (1930).de Haas, W. J. & van Alphen, P. M. The dependence of the susceptibility of diamagnetic metals upon the field.

*Proc. Netherlands Roy. Acad. Sci.***33**, 1106–1118 (1930).Shoenberg, D.

*Magnetic Oscillations in Metals*(Cambridge Univ. Press, 1984).Peierls, R. Zur Theorie des Diamagnetismus von Leitungselektronen.

*Z. Phys.***80**, 763–791 (1933).Onsager, L. Interpretation of the de Haas-van Alphen effect.

*Lond. Edinb. Dublin Philos. Mag. J. Sci.***43**, 1006–1008 (1952).Lifshitz, I. M. & Kosevich, A. Theory of magnetic susceptibility in metals at low temperatures.

*Sov. Phys. JETP***2**, 636–645 (1956).Kartsovnik, M. V. High magnetic fields: a tool for studying electronic properties of layered organic metals.

*Chem. Rev.***104**, 5737–5781 (2004).Julian, S. R. in

*Strongly Correlated Systems*(eds Avella, A. & Mancini, F.) 137–172 (Springer, 2015).Tan, B. S. et al. Unconventional Fermi surface in an insulating state.

*Science***349**, 287–290 (2015).Han, Z., Li, T., Zhang, L., Sullivan, G. & Du, R.-R. Anomalous conductance oscillations in the hybridization gap of InAs/GaSb quantum wells.

*Phys. Rev. Lett.***123**, 126803 (2019).Knolle, J. & Cooper, N. R. Quantum oscillations without a Fermi surface and the anomalous de Haas–van Alphen effect.

*Phys. Rev. Lett.***115**, 146401 (2015).Sodemann, I., Chowdhury, D. & Senthil, T. Quantum oscillations in insulators with neutral Fermi surfaces.

*Phys. Rev. B***97**, 045152 (2018).Dingle, R. B. Some magnetic properties of metals II. The influence of collisions on the magnetic behaviour of large systems.

*Proc. R. Soc. Lond. A Math. Phys. Sci.***211**, 517–525 (1952).Polyanovsky, V. Magnetointersubband oscillations of conductivity in a two-dimensional electronic system.

*Sov. Phys. Semicond.***22**, 1408–1409 (1988).Polyanovsky, V. High-temperature quantum oscillations of the magnetoresistance in layered systems.

*Phys. Rev. B***47**, 1985–1990 (1993).Huber, N. et al. Network of topological nodal planes, multifold degeneracies, and Weyl points in CoSi.

*Phys. Rev. Lett.***129**, 026401 (2022).Rao, Z. et al. Observation of unconventional chiral fermions with long Fermi arcs in CoSi.

*Nature***567**, 496–499 (2019).Guo, C. et al. Quasi-symmetry-protected topology in a semi-metal.

*Nat. Phys.***18**, 813–818 (2022).Alexandrov, A. S. & Kabanov, V. V. Combination quantum oscillations in canonical single-band Fermi liquids.

*Phys. Rev. B***76**, 233101 (2007).Allocca, A. A. & Cooper, N. R. Low-frequency quantum oscillations from interactions in layered metals.

*Phys. Rev. Res.***3**, L042009 (2021).Cohen, M. H. & Falicov, L. M. Magnetic breakdown in crystals.

*Phys. Rev. Lett.***7**, 231–233 (1961).Blount, E. I. Bloch electrons in a magnetic field.

*Phys. Rev.***126**, 1636–1653 (1962).Chambers, R. G. Magnetic breakdown in real metals.

*Proc. Phys. Soc.***88**, 701–715 (1966).Bergmann, G. Weak localization in thin films: a time-of-flight experiment with conduction electrons.

*Phys. Rep.***107**, 1–58 (1984).Lee, P. A. & Stone, A. D. Universal conductance fluctuations in metals.

*Phys. Rev. Lett.***55**, 1622–1625 (1985).Fu, Y. et al. Quantum transport evidence of topological band structures of Kagome superconductor CsV

_{3}Sb_{5}.*Phys. Rev. Lett.***127**, 207002 (2021).Allocca, A. A. & Cooper, N. R. Fluctuation-dominated quantum oscillations in excitonic insulators. Preprint at https://arxiv.org/abs/2302.06633 (2023).

Yuan, Q.-Q. et al. Quasiparticle interference evidence of the topological Fermi arc states in chiral fermionic semimetal CoSi.

*Sci. Adv.***5**, eaaw9485 (2019).Wu, D. S. et al. Single crystal growth and magnetoresistivity of topological semimetal CoSi.

*Chin. Phys. Lett.***36**, 077102 (2019).Xu, X. et al. Crystal growth and quantum oscillations in the topological chiral semimetal CoSi.

*Phys. Rev. B***100**, 045104 (2019).Wang, H. et al. de Haas–van Alphen quantum oscillations and electronic structure in the large-Chern-number topological chiral semimetal CoSi.

*Phys. Rev. B***102**, 115129 (2020).Sasmal, S. et al. Shubnikov-de Haas and de Haas-van Alphen oscillations in Czochralski grown CoSi single crystal.

*J. Phys. Condens. Matter***34**, 425702 (2022).Neubauer, A. et al. Ultra-high vacuum compatible image furnace.

*Rev. Sci. Instrum.***82**, 013902 (2011).Bauer, A., Benka, G., Regnat, A., Franz, C. & Pfleiderer, C. Ultra-high vacuum compatible preparation chain for intermetallic compounds.

*Rev. Sci. Instrum.***87**, 113902 (2016).Springford, M. The anisotropy of conduction electron scattering in the noble metals.

*Adv. Phys.***20**, 493–550 (1971).Paul, D. M. & Springford, M. Accurate measurement of changes in electron scattering in the de Haas-van Alphen effect.

*J. Low Temp. Phys.***27**, 561–569 (1977).Blaha, P. et al. WIEN2k: an APW+lo program for calculating the properties of solids.

*J. Chem. Phys.***152**, 074101 (2020).Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple.

*Phys. Rev. Lett.***77**, 3865–3868 (1996).Tang, P., Zhou, Q. & Zhang, S.-C. Multiple types of topological fermions in transition metal silicides.

*Phys. Rev. Lett.***119**, 206402 (2017).Pshenay-Severin, D. A., Ivanov, Y. V., Burkov, A. A. & Burkov, A. T. Band structure and unconventional electronic topology of CoSi.

*J. Phys. Condens. Matter***30**, 135501 (2018).Wilde, M. & Pfleiderer, C. Large curvature near a small gap.

*Nat. Phys.***18**, 731–732 (2022).O’Brien, T. E., Diez, M. & Beenakker, C. W. J. Magnetic breakdown and Klein tunneling in a type-II Weyl semimetal.

*Phys. Rev. Lett.***116**, 236401 (2016).Alexandrov, A. S. & Bratkovsky, A. M. New fundamental dHvA frequency in canonical low-dimensional Fermi liquids.

*Phys. Lett. A***234**, 53–58 (1997).Stark, R. W. & Friedberg, C. B. Quantum interference of electron waves in a normal metal.

*Phys. Rev. Lett.***26**, 556–559 (1971).Kaganov, M. I. & Slutskin, A. A. Coherent magnetic breakdown.

*Phys. Rep.***98**, 189–271 (1983).Leadley, D. R. et al. Intersubband resonant scattering in GaAs-Ga

_{1−x}Al_{x}As heterojunctions.*Phys. Rev. B***46**, 12439–12447 (1992).Coleridge, P. T. Inter-subband scattering in a 2D electron gas.

*Semicond. Sci. Technol.***5**, 961–966 (1990).Raikh, M. E. & Shahbazyan, T. V. Magnetointersubband oscillations of conductivity in a two-dimensional electronic system.

*Phys. Rev. B***49**, 5531–5540 (1994).Goran, A. V., Bykov, A. A., Toropov, A. I. & Vitkalov, S. A. Effect of electron-electron scattering on magnetointersubband resistance oscillations of two-dimensional electrons in GaAs quantum wells.

*Phys. Rev. B***80**, 193305 (2009).Grigoriev, P. D. Theory of the Shubnikov–de Haas effect in quasi-two-dimensional metals.

*Phys. Rev. B***67**, 144401 (2003).Thomas, I. O., Kabanov, V. V. & Alexandrov, A. S. Shubnikov–de Haas effect in multiband quasi-two-dimensional metals.

*Phys. Rev. B***77**, 075434 (2008).Leeb, V., & Knolle, J. On the theory of difference frequency quantum oscillations. Preprint at https://arxiv.org/abs/2306.10760 (2023).

Bastin, A., Lewiner, C., Betbeder-matibet, O. & Nozieres, P. Quantum oscillations of the Hall effect of a Fermion gas with random impurity scattering.

*J. Phys. Chem. Solids***32**, 1811–1824 (1971).Mañes, J. L. Existence of bulk chiral fermions and crystal symmetry.

*Phys. Rev. B***85**, 155118 (2012).Bradlyn, B. et al. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals.

*Science***353**, aaf5037 (2016).Yasuoka, H., Sherwood, R., Wernick, J. & Wertheim, G. Local moment formation in substituted and cobalt-rich CoSi.

*Mater. Res. Bull.***9**, 223–231 (1974).Wernick, J., Wertheim, G. & Sherwood, R. Magnetic behavior of the monosilicides of the 3d-transition elements.

*Mater. Res. Bull.***7**, 1431–1441 (1972).Wertheim, G. K., Wernick, J. H. & Buchanan, D. N. E. Mössbauer effect in Co

_{1−}_{x}Fe_{x}Si.*J. Appl. Phys.***37**, 3333–3337 (1966).Kawarazaki, S., Yasuoka, H. & Nakamura, Y. Moment formation on Co atom in FeSi–CoSi mixed system -Co

^{59}NMR in the paramagnetic state.*Solid State Commun.***10**, 919–921 (1972).

## Acknowledgements

We wish to thank A. Schnyder for discussions. J.K. acknowledges helpful discussions with N. R. Cooper. V.L. acknowledges support from the Studienstiftung des deutschen Volkes. N.H. and V.L. acknowledge support from the TUM Graduate School. M.A.W. and C.P. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - TRR 80 - 107745057; TRR 360 - 492547816 and DFG-GACR project WI3320/3 - 323760292. C.P. acknowledges support through DFG SPP 2137 (Skyrmionics) under grant no. PF393/19 (project ID 403191981), ERC Advanced Grant no. 788031 (ExQuiSid) and Germany’s excellence strategy under EXC-2111 390814868. J.K. acknowledges support from the Imperial-TUM flagship partnership. This research is part of the Munich Quantum Valley, which is supported by the Bavarian State Government with funds from the Hightech Agenda Bayern Plus.

## Author information

### Authors and Affiliations

### Contributions

M.A.W. and C.P. conceived and started this study. M.A.W., C.P. and J.K. proposed the interpretation. G.B. and A.B. prepared and characterized the samples. N.H. conducted the measurements and analysed the data. M.A.W. and V.L. performed band-structure calculations. M.A.W. and N.H. connected the experimental data with the calculated band structure. V.L. and J.K. developed the theoretical analysis. C.P., M.A.W. and J.K. wrote the manuscript, with contributions from N.H. and V.L. All authors discussed the data and commented on the manuscript.

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## Extended data figures and tables

### Extended Data Fig. 1 Equivalence of predicted QO orbits when invoking MB (ref. ^{}34) or hidden quasi-symmetries (ref. ^{36}).

Panels are organized in the spirit of a flow chart. **a**, Approach 1 comprises nodal planes and MB^{34}. Three mutually perpendicular nodal planes at the R point as depicted at the top of the panel enforce pairwise band degeneracies of the FS sheets labelled A, B and C, D. Numerically calculated MB with probabilities close to 1 occur between sheet pairs (C,B) and (A,D) along the lines as marked. **b**, Extremal cross-section for the three selected planes. In the (001) plane, the cross-sections are doubly degenerate. In the (111) and (110) planes, four different cross-sections exist, with MB junctions denoted by coloured circles. The cross-sectional areas are always pairwise identical. **c**, Effective extremal orbits corresponding to panels **b** and e. For all field directions, two dominant QO frequencies exist. **d**, Approach 2 comprises nodal planes and hidden quasi-symmetries^{36}. Three mutually perpendicular nodal planes at the R point are enclosed by regimes of hidden quasi-symmetry up to the enclosing surfaces (green), as depicted at the top of the panel. Within the green regimes, near degeneracies are approximated by exact degeneracies in first-order perturbation theory. The FS sheets of the perturbation Hamiltonian are labelled as orbital (1/2) and spin (+/−) degrees of freedom, for which sheet pairs (1−/2+) and (1+/2−) intersect at the surfaces of the quasi-symmetry. **e**, Extremal cross-sections for the same three selected planes shown in **b**. In the (001) plane, the cross-sections are doubly degenerate, identical to **b**. In the (111) and (110) planes, the cross-sections correspond to the extremal orbits as depicted in **b** with breakdown gaps that approach zero. Taken together, approaches 1 and 2 yield the same orbits as depicted in **c** and thus the same two QO frequencies.

### Extended Data Fig. 2 Temperature dependence of Shubnikov–de Haas oscillations in the Hall resistivity of CoSi.

Oscillation amplitudes shown in panels **b**–**d** were analysed in the magnetic field range between 9 and 18 T with **B** applied along the [001] direction and normalized to the amplitude of *f*_{α} at *T* = 20 mK. **a**, Oscillatory component of *ρ*_{xy} as a function of inverse magnetic field at different temperatures. Curves are shifted by a constant. **b**, Oscillation amplitudes of the *α* and *β* frequencies. Lines represent a fit with the temperature reduction factor *R*_{T} in the LK formalism. Effective masses inferred from the fit are \({m}_{\alpha }^{* }=(0.92\pm 0.01){m}_{{\rm{e}}}\) and \({m}_{\beta }^{* }=(0.95\pm 0.01){m}_{{\rm{e}}}\). Error bars are smaller than the data points. **c**, Oscillation amplitudes of the difference frequency *f*_{β−α}. The red line represents a two-component fit of the temperature reduction factor, yielding effective masses of (0.07 ± 0.01)*m*_{e} and (1.6 ± 0.3)*m*_{e}, corresponding to the difference and the sum of the individual masses within the error bars and consistent with the QPL oscillations reported in this work. The black lines represent the individual components of the fit, in which HT and LT denote the high-temperature and low-temperature behaviour, respectively. **d**, Amplitude of the sum frequency *f*_{α+β}. The line represents a LK fit yielding a cyclotron mass of (2.6 ± 0.4)*m*_{e}.

### Extended Data Fig. 3 Calculated FS orbits near the Γ point of CoSi under various conditions.

The observation of essentially identical values of *f*_{α} and *f*_{β} in different samples studied by different groups^{34,36,47,48,49,50} establishes the same energy of the electronic bands at the R point and thus a lack of sensitivity of the band structure to defect-induced charge doping. In turn, charge conservation enforces a position of the bands with respect to the Fermi level at the Γ point within 1 meV. No QO frequencies of around 100 T are expected under these conditions. **a**, Calculated band structure in the vicinity of the Γ point of CoSi. For the FS pockets at the R point observed experimentally, a value of *E*_{F} = (+7 ± 1) meV at the Γ point, marked by the black horizontal line, corresponds to charge neutrality. The grey shading depicts a range of ±20 meV beyond which hole carriers vanish altogether and charge conservation would be violated substantially (see Methods for details). Coloured horizontal lines at *E*_{F} = +17 meV and *E*_{F} = +1.5 meV denote locations in which FS cross-sections with QO frequencies of around 100 T are expected, but for **B** ∥ ⟨001⟩ only. **b**, FS sheets centred at Γ for *E*_{F} = +17 meV, in which the sheets depicted in blue and green shading correspond to the blue and green bands shown in panel **a**. **c**, Upper panel, calculated angular dispersion *f*(*θ*) of the blue band for *E*_{F} = +17 meV as compared with the experimental data (red symbols) shown in Fig. 2i. Lower panel, calculated dispersion of the cyclotron mass (blue) as compared with the experimental value (red), at which strong disagreement would be expected. QOs with a frequency of around 100 T are expected for **B** ∥ ⟨001⟩ only, in which the predicted mass differs strongly from experiment. **d**, Green FS sheet centred at Γ for *E*_{F} = +1.5 meV. **e**, Upper panel, calculated angular dispersion *f*(*θ*) of the green band for *E*_{F} = +1.5 meV as compared with the experimental data (red symbols) shown in Fig. 2i. Lower panel, calculated dispersion of the cyclotron mass (green) as compared with the experimental value (red), in which strong disagreement would be expected. QOs with a frequency of around 100 T are expected for **B** ∥ ⟨001⟩ only, in which the predicted mass differs strongly from experiment.

### Extended Data Fig. 4 Analysis of the phase relation between the Shubnikov–de Haas oscillations at different temperatures.

**a**, Oscillatory component of the transverse magnetoresistivity, \({\widetilde{\rho }}_{xx}\), and Hall resistivity, \({\widetilde{\rho }}_{xy}\), as a function of inverse magnetic field at *T* = 2.5 K. A pronounced beating pattern originates from oscillations at *f*_{α} and *f*_{β}. Nodes of the beating pattern are indicated by vertical lines. The envelope of the beating pattern may be expressed as cos(2π(*f*_{β} − *f*_{α})/(2*B*) + (*φ*_{β} − *φ*_{α})/2, in which *φ*_{α} and *φ*_{β} are the phases of *f*_{α} and *f*_{β}, respectively. An analysis using the frequencies determined from the FFT peaks, notably *f*_{α} = 565 T and *f*_{β} = 663 T, yields a phase difference of *φ*_{β} − *φ*_{α} = 0.16π. We note that this value sensitively depends on the precise value of the frequency difference *f*_{β} − *f*_{α}. **b**, Oscillatory component of the resistivities as a function of inverse applied magnetic field at *T* = 20 K. The oscillations at *f*_{α} and *f*_{β} are strongly suppressed at this temperature. Only the slow oscillations at *f*_{β−α} are visible. Here minima of the oscillations coincide with the nodes of the beating pattern shown in panel **a**. Neglecting the amplitude, the oscillation at *f*_{β−α} may be described by a term reading cos(2π*f*_{β}_{−α}/*B* + *φ*_{β−α}), which oscillates with the same frequency as the nodes in the beating pattern. The phase *φ*_{β−α} matches the phase difference *φ*_{β} − *φ*_{α}.

### Extended Data Fig. 5 Quantitative assessment of the effects of magnetic interactions in CoSi.

Changes of the magnetization owing to de Haas–van Alphen oscillations may generate variations of the internal field that result in oscillatory signal components observed in the physical properties. This feedback is known as magnetic interaction. Shown is the derivative of the oscillatory part of the measured magnetization in SI units as a function of applied magnetic field. For contributions on the order \({\rm{d}}\mathop{M}\limits^{ \sim }\,/\,{\rm{d}}H\approx 1\), oscillatory signal contributions owing to magnetic interactions are commonly expected^{21}. The de Haas–van Alphen contribution observed experimentally is well below this limit. Numerical simulations of the effect of magnetic interactions expected in our study establish that the amplitudes of the de Haas–van Alphen oscillations at *f*_{α} and *f*_{β} are several orders of magnitude below the value required to account for the oscillatory signal components at the difference of the frequencies we observed experimentally.

### Extended Data Fig. 6 Characteristic frequencies of QOs of the QPL for generic types of band pair.

Precondition for QOs of the QPL is a general form of interaction, such as scattering from defects or collective excitations, that generate a nonlinear intraorbit or interorbit coupling. **a**, Left, two linear bands with equal Fermi velocities, *v*_{F,α} = *v*_{F,β}, and a small offset. This configuration may be expected near multifold band crossings as in CoSi. The crossings need not be topological. Right, the small variation of *f*_{β} − *f*_{α} owing to the band offset leads to a strongly reduced suppression of the dephasing of the oscillation amplitude as a function of temperature. **b**, Left, parabolic bands with equal masses, *m*_{α} = *m*_{β}, and a small offset. Right, the difference frequency *f*_{β} − *f*_{α} is independent of the Fermi level, as the cyclotron masses are equal. The dephasing as a function of temperature and hence the decay of the oscillation amplitude with increasing temperature is suppressed completely. **c**, Left, Rashba-type spin–orbit splitting, in which the Rashba parameter *α*_{R} ≠ 0. Right, the temperature dephasing depends on *E*_{F}, as the slope of *f*_{β} − *f*_{α} is not constant.

### Extended Data Fig. 7 Predicted QO spectra and possible mechanism of interband scattering.

The dependence of the QO frequencies on energy is key to understanding their temperature dependence. **a**, FFT amplitude of equation (11) over an extended *B*-field range at *T* = 0. As well as equation (11), leading contributions up to the fourth order in the Dingle factor are included, which can be evaluated systematically. Experimental values for \({m}_{\lambda }^{* }\) and *T*_{D,λ} ≈ 0.5 K were used consistent with CoSi, as well as *f*_{α}(*E*), *f*_{β}(*E*) from the DFT calculations. **b**, Cut along *E* = *E*_{F} for CoSi. This may be compared with the spectrum recorded experimentally. Peaks shown in panels **a** and **b** seem broad because of the log scale. **c**, Band structure of the effective **k** ⋅ **p** Hamiltonian around the R point equation (12) used to model a possible mechanism for nonlinear interband scattering. **d**, Schematic FSs in the effective **k** ⋅ **p** Hamiltonian (projection on the *k*_{z} = 0 plane with respect to R) in which the spin is polarized perpendicular to the FS and orientated in opposing directions on the two bands. This is consistent with results obtained in DFT calculations^{36}.

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Huber, N., Leeb, V., Bauer, A. *et al.* Quantum oscillations of the quasiparticle lifetime in a metal.
*Nature* **621**, 276–281 (2023). https://doi.org/10.1038/s41586-023-06330-y

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DOI: https://doi.org/10.1038/s41586-023-06330-y

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