Abstract
Stable composite objects, such as hadrons, nuclei, atoms, molecules and superconducting pairs, formed by attractive forces are ubiquitous in nature. By contrast, composite objects stabilized by means of repulsive forces were long thought to be theoretical constructions owing to their fragility in naturally occurring systems. Surprisingly, the formation of bound atom pairs by strong repulsive interactions has been demonstrated experimentally in optical lattices1. Despite this success, repulsively bound particle pairs were believed to have no analogue in condensed matter owing to strong decay channels. Here we present spectroscopic signatures of repulsively bound three-magnon states and bound magnon pairs in the Ising-like chain antiferromagnet BaCo2V2O8. In large transverse fields, below the quantum critical point, we identify repulsively bound magnon states by comparing terahertz spectroscopy measurements to theoretical results for the Heisenberg–Ising chain antiferromagnet, a paradigmatic quantum many-body model2,3,4,5. Our experimental results show that these high-energy, repulsively bound magnon states are well separated from continua, exhibit notable dynamical responses and, despite dissipation, are sufficiently long-lived to be identified. As the transport properties in spin chains can be altered by magnon bound states, we envision that such states could serve as resources for magnonics-based quantum information processing technologies6,7,8.
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Data availability
All data needed to evaluate the conclusions in the paper are included in this paper. Further data that support the plots and other analysis in this work are available from the corresponding author on request. The theoretical simulations data are available at https://doi.org/10.5281/zenodo.11521387 (ref. 44).
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Acknowledgements
We thank M. Garst, T. Giamarchi, S. Wolff, J. Wu and H. Zou for helpful discussions. We acknowledge support by the European Research Council (ERC) under the Horizon 2020 research and innovation programme, grant agreement no. 950560 (DynaQuanta), by the Natural Sciences and Engineering Research Council of Canada (NSERC) (funding reference nos. RGPIN-2021-04338 and DGECR-2021-00359) and by the Swiss National Science Foundation under Division II grants 200020-188687 and 200020-219400. This research was supported in part by the National Science Foundation under grant nos. NSF PHY-1748958 and PHY-2309135. We acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 107745057 - TRR 180 (F5), project number 277146847 - CRC 1238 (A02, B01, B05, C05), project number 277625399 - TRR 185 (B4), project number 247310070 - SFB 1143, project number 511713970 - CRC 1639, project number 390858490 - EXC 2147 Cluster of Excellence Complexity and Topology in Quantum Matter (CT.QMAT) and project number 390534769 - EXC 2004/1 Cluster of Excellence Matter and Light for Quantum Computing (ML4Q). We also acknowledge the support of the HFML-RU/FOM and the HLD at Helmholtz-Zentrum Dresden-Rossendorf (HZDR), members of the European Magnetic Field Laboratory (EMFL). Parts of this research were carried out at ELBE at the HZDR, a member of the Helmholtz Association.
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Z.W. conceived the experiment and coordinated the project. C.-M.H., J.-S.B. and C.K. performed the theoretical analysis. Z.W., A.P., J.M.K. and S.Z. carried out the spectroscopic measurements. D.I.G., T.L. and Z.W. performed the magnetization measurements. S.N., O.B. and T.L. prepared and characterized the single crystals. Z.W., C.-M.H., J.-S.B. and C.K. analysed the data and interpreted the results. Z.W., C.-M.H., J.-S.B. and C.K. wrote the manuscript, with input from T.L. and A.L. All authors commented on the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Magnetization of BaCo2V2O8 as a function of the applied magnetic field along the crystallographic [110] direction15, that is, B∥[110].
The solid line shows the theoretical result, as a sum of a paramagnetic (PM) Van Vleck contribution (dotted line) and the contribution of the one-dimensional Heisenberg–Ising (HI) model in equation (1) (dashed line).
Extended Data Fig. 2 Dynamical spin structure factor \(\boldsymbol{\mathcal{S}}({\boldsymbol{q}},{\boldsymbol{\omega }})\) as a function of momentum q and frequency ω (see equation (3)) at an applied field of B = 45 T > Bc for various values of \({{\boldsymbol{g}}}_{{\bf{s}}}^{{\boldsymbol{x}}}\) corresponding to different effective staggering of the magnetic field.
a, Without a staggered field (that is, \({g}_{{\rm{s}}}^{x}=0\)), the spin dynamics is characterized by a single cosine-shaped band of unbound-magnon excitations, labelled as M. b–d, With a finite staggered field (that is, \({g}_{{\rm{s}}}^{x}=0.31\), 0.66 and 0.94, respectively), this band is split into two bands, separated by a gap. The gap increases with increasing staggered field. The data in c correspond to the experimental value of the staggering in BaCo2V2O8 (see Fig. 4c). The blue lines are analytical results for the one-magnon excitations30.
Extended Data Fig. 3 Dynamical spin structure factor \(\boldsymbol{\mathcal{S}}({\boldsymbol{q}},{\boldsymbol{\omega }})\) as a function of momentum q and frequency ω (see equation (3)) at an applied field of B = 40.3 T ≈ Bc for various values of \({{\boldsymbol{g}}}_{{\bf{s}}}^{{\boldsymbol{x}}}\) corresponding to different effective staggering of the magnetic field.
Above the dashed line, the spectral weight of the repulsively bound two-magnon states (labelled as D) is multiplied by a factor of 10. The data in c correspond to the experimental value of the staggering for BaCo2V2O8 (see Fig. 4b).
Extended Data Fig. 4 Dynamical spin structure factor \(\boldsymbol{\mathcal{S}}({\boldsymbol{q}},{\boldsymbol{\omega }})\) as a function of momentum q and frequency ω (see equation (3)) at an applied field of B = 30 T (that is, 0 ≪ B < Bc) for various values of \({{\boldsymbol{g}}}_{{\bf{s}}}^{{\boldsymbol{x}}}\) corresponding to different effective staggering of the magnetic field.
a, Without a staggered field (that is, \({g}_{{\rm{s}}}^{x}=0\)), the high-energy features above the unbound-magnon excitation band are hard to identify. However, with increasing staggered field for \({g}_{{\rm{s}}}^{x}=0.31\) (b), for \({g}_{{\rm{s}}}^{x}=0.66\) (c) and for \({g}_{{\rm{s}}}^{x}=0.94\) (d), the spectral weight of the features of repulsively bound two-magnon and three-magnon excitations (labelled as D and T, respectively) can be clearly identified and is continuously enhanced. Above the dashed line, the spectral weight of the repulsively bound three-magnon states is multiplied by a factor of 10. The data in c correspond to the experimental value of the staggering for BaCo2V2O8 (see Fig. 4a).
Extended Data Fig. 5 Absorption coefficient measured in static magnetic fields.
a, The spectrum at 28 T with the single-magnon \({M}_{0}^{{\rm{u}}}\) and the repulsively bound two-magnon Dπ and Dπ/2 modes (see also Fig. 3). b, Zoom-in of the high-frequency spectral range corresponding to the Dπ/2 and the repulsively bound three-magnon Tπ/2 modes (as indicated by the arrows) measured at various fields.
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Wang, Z., Halati, CM., Bernier, JS. et al. Experimental observation of repulsively bound magnons. Nature (2024). https://doi.org/10.1038/s41586-024-07599-3
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DOI: https://doi.org/10.1038/s41586-024-07599-3
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