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  • Letter
  • Open Access

Anomalous thermal broadening in the Shastry-Sutherland model and SrCu2(BO3)2

Zhenjiu Wang1,2,3,*,†, Paul McClarty3,4,*, Dobromila Dankova3, Andreas Honecker5, and Alexander Wietek3,‡

  • *These authors contributed equally to this work.
  • †Contact author: wangzj@lzu.edu.cn
  • ‡Contact author: awietek@pks.mpg.de
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Phys. Rev. B 113, L041104 – Published 5 January, 2026

DOI: https://doi.org/10.1103/ldwx-2s7w

Abstract

In the quantum magnet SrCu2(BO3)2, an anomalous thermal broadening of the triplon modes has been measured at relatively low temperatures compared to the triplon gap Δ using both inelastic neutron scattering and Raman spectroscopy. Given how accurately a broad variety of physical phenomena in SrCu2(BO3)2 are captured by the spin S=1/2 Shastry-Sutherland model, it remains an open question whether the anomalous thermal broadening is also an intrinsic feature of this minimal model. However, few techniques are available for computing the finite-temperature dynamics of strongly interacting many-body systems. To address this problem, we have developed a broadly applicable numerical simulation method based on matrix-product states to simulate dynamical spectral functions at nonzero temperatures accurately, detailed in a companion paper [Phys. Rev. B 113, 024406 (2026)]. Using this technique, we demonstrate that the experimentally observed broadening is captured by the Shastry-Sutherland model. Perturbative calculations identify the origin of this phenomenon as singlet bound two-triplon states being thermally excited at an energy scale small compared to the gap Δ to the single triplon excitations at the experimentally relevant model parameters.

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Spectroscopy and complex-time correlations using minimally entangled typical thermal states

Zhenjiu Wang, Paul McClarty, Dobromila Dankova, Andreas Honecker, and Alexander Wietek
Phys. Rev. B 113, 024406 (2026)

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References (76)

  1. B. Sriram Shastry and B. Sutherland, Exact ground state of a quantum mechanical antiferromagnet, Phys. B+C 108, 1069 (1981).
  2. C. Lacroix, P. Mendels, and F. Mila, Introduction to Frustrated Magnetism: Materials, Experiments, Theory, Springer Series in Solid-State Sciences Vol. 164 (Springer Science & Business Media, 2011).
  3. H. Kageyama, K. Yoshimura, R. Stern, N. V. Mushnikov, K. Onizuka, M. Kato, K. Kosuge, C. P. Slichter, T. Goto, and Y. Ueda, Exact dimer ground state and quantized magnetization plateaus in the two-dimensional spin system SrCu2(BO3)2, Phys. Rev. Lett. 82, 3168 (1999).
  4. H. Kageyama, K. Onizuka, T. Yamauchi, Y. Ueda, S. Hane, H. Mitamura, T. Goto, K. Yoshimura, and K. Kosuge, Anomalous magnetizations in single crystalline SrCu2(BO3)2, J. Phys. Soc. Jpn. 68, 1821 (1999).
  5. K. Onizuka, H. Kageyama, Y. Narumi, K. Kindo, Y. Ueda, and T. Goto, 1/3 Magnetization plateau in SrCu2(BO3)2 - stripe order of excited triplets -, J. Phys. Soc. Jpn. 69, 1016 (2000).
  6. K. Kodama, M. Takigawa, M. Horvatić, C. Berthier, H. Kageyama, Y. Ueda, S. Miyahara, F. Becca, and F. Mila, Magnetic superstructure in the two-dimensional quantum antiferromagnet SrCu2(BO3)2, Science 298, 395 (2002).
  7. S. Miyahara and K. Ueda, Theory of the orthogonal dimer Heisenberg spin model for SrCu2(BO3)2, J. Phys.: Condens. Matter 15, R327 (2003).
  8. M. Takigawa, K. Kodama, M. Horvatić, C. Berthier, H. Kageyama, Y. Ueda, S. Miyahara, F. Becca, and F. Mila, The 18-magnetization plateau state in the 2D quantum antiferromagnet SrCu2(BO3)2: Spin superstructure, phase transition, and spin dynamics studied by high-field NMR, Phys. B: Condens. Matter 346-347, 27 (2004).
  9. K. Kodama, S. Miyahara, M. Takigawa, M. Horvatić, C. Berthier, F. Mila, H. Kageyama, and Y. Ueda, Field-induced effects of anisotropic magnetic interactions in SrCu2(BO3)2, J. Phys.: Condens. Matter 17, L61 (2005).
  10. F. Levy, I. Sheikin, C. Berthier, M. Horvatić, M. Takigawa, H. Kageyama, T. Waki, and Y. Ueda, Field dependence of the quantum ground state in the Shastry-Sutherland system SrCu2(BO3)2, Europhys. Lett. 81, 67004 (2008).
  11. S. E. Sebastian, N. Harrison, P. Sengupta, C. D. Batista, S. Francoual, E. Palm, T. Murphy, N. Marcano, H. A. Dabkowska, and B. D. Gaulin, Fractalization drives crystalline states in a frustrated spin system, Proc. Natl. Acad. Sci. USA 105, 20157 (2008).
  12. M. Jaime, R. Daou, S. A. Crooker, F. Weickert, A. Uchida, A. E. Feiguin, C. D. Batista, H. A. Dabkowska, and B. D. Gaulin, Magnetostriction and magnetic texture to 100.75 Tesla in frustrated SrCu2(BO3)2, Proc. Natl. Acad. Sci. USA 109, 12404 (2012).
  13. M. Takigawa, M. Horvatić, T. Waki, S. Krämer, C. Berthier, F. Lévy-Bertrand, I. Sheikin, H. Kageyama, Y. Ueda, and F. Mila, Incomplete devil's staircase in the magnetization curve of SrCu2(BO3)2, Phys. Rev. Lett. 110, 067210 (2013).
  14. Y. H. Matsuda, N. Abe, S. Takeyama, H. Kageyama, P. Corboz, A. Honecker, S. R. Manmana, G. R. Foltin, K. P. Schmidt, and F. Mila, Magnetization of SrCu2(BO3)2 in ultrahigh magnetic fields up to 118 T, Phys. Rev. Lett. 111, 137204 (2013).
  15. S. Haravifard, D. Graf, A. E. Feiguin, C. D. Batista, J. C. Lang, D. M. Silevitch, G. Srajer, B. D. Gaulin, H. A. Dabkowska, and T. F. Rosenbaum, Crystallization of spin superlattices with pressure and field in the layered magnet SrCu2(BO3)2, Nat. Commun. 7, 11956 (2016).
  16. T. Nomura, P. Corboz, A. Miyata, S. Zherlitsyn, Y. Ishii, Y. Kohama, Y. H. Matsuda, A. Ikeda, C. Zhong, H. Kageyama, and F. Mila, Unveiling new quantum phases in the Shastry-Sutherland compound SrCu2(BO3)2 up to the saturation magnetic field, Nat. Commun. 14, 3769 (2023).
  17. H. Nojiri, H. Kageyama, K. Onizuka, Y. Ueda, and M. Motokawa, Direct observation of the multiple spin gap excitations in two-dimensional dimer system SrCu2(BO3)2, J. Phys. Soc. Jpn. 68, 2906 (1999).
  18. P. Lemmens, M. Grove, M. Fischer, G. Güntherodt, V. N. Kotov, H. Kageyama, K. Onizuka, and Y. Ueda, Collective singlet excitations and evolution of Raman spectral weights in the 2D spin dimer compound SrCu2(BO3)2, Phys. Rev. Lett. 85, 2605 (2000).
  19. H. Kageyama, M. Nishi, N. Aso, K. Onizuka, T. Yosihama, K. Nukui, K. Kodama, K. Kakurai, and Y. Ueda, Direct evidence for the localized single-triplet excitations and the dispersive multitriplet excitations in SrCu2(BO3)2, Phys. Rev. Lett. 84, 5876 (2000).
  20. T. Rõõm, U. Nagel, E. Lippmaa, H. Kageyama, K. Onizuka, and Y. Ueda, Far-infrared study of the two-dimensional dimer spin system SrCu2(BO3)2, Phys. Rev. B 61, 14342 (2000).
  21. B. D. Gaulin, S. H. Lee, S. Haravifard, J. P. Castellan, A. J. Berlinsky, H. A. Dabkowska, Y. Qiu, and J. R. D. Copley, High-resolution study of spin excitations in the singlet ground state of SrCu2(BO3)2, Phys. Rev. Lett. 93, 267202 (2004).
  22. P. A. McClarty, F. Krüger, T. Guidi, S. F. Parker, K. Refson, A. W. Parker, D. Prabhakaran, and R. Coldea, Topological triplon modes and bound states in a Shastry–Sutherland magnet, Nat. Phys. 13, 736 (2017).
  23. D. Wulferding, Y. Choi, S. Lee, M. A. Prosnikov, Y. Gallais, P. Lemmens, C. Zhong, H. Kageyama, and K.-Y. Choi, Thermally populated versus field-induced triplon bound states in the Shastry-Sutherland lattice SrCu2(BO3)2, npj Quantum Mater. 6, 102 (2021).
  24. S. Miyahara and K. Ueda, Exact dimer ground state of the two dimensional Heisenberg spin system SrCu2(BO3)2, Phys. Rev. Lett. 82, 3701 (1999).
  25. S. Miyahara and K. Ueda, Thermodynamic properties of three-dimensional orthogonal dimer model for SrCu2(B03)2, J. Phys. Soc. Jpn. Suppl. B 69, 72 (2000).
  26. C. Knetter, A. Bühler, E. Müller-Hartmann, and G. S. Uhrig, Dispersion and symmetry of bound states in the Shastry-Sutherland model, Phys. Rev. Lett. 85, 3958 (2000).
  27. O. Cépas, K. Kakurai, L. P. Regnault, T. Ziman, J. P. Boucher, N. Aso, M. Nishi, H. Kageyama, and Y. Ueda, Dzyaloshinski-Moriya interaction in the 2D spin gap system SrCu2(BO3)2, Phys. Rev. Lett. 87, 167205 (2001).
  28. J. Romhányi, K. Totsuka, and K. Penc, Effect of Dzyaloshinskii-Moriya interactions on the phase diagram and magnetic excitations of SrCu2(BO3)2, Phys. Rev. B 83, 024413 (2011).
  29. J. Romhányi, K. Penc, and R. Ganesh, Hall effect of triplons in a dimerized quantum magnet, Nat. Commun. 6, 6805 (2015).
  30. K. Totsuka, S. Miyahara, and K. Ueda, Low-Lying magnetic excitation of the Shastry-Sutherland model, Phys. Rev. Lett. 86, 520 (2001).
  31. C. Knetter and G. S. Uhrig, Dynamic structure factor of the two-dimensional Shastry-Sutherland model, Phys. Rev. Lett. 92, 027204 (2004).
  32. T. Waki, K. Arai, M. Takigawa, Y. Saiga, Y. Uwatoko, H. Kageyama, and Y. Ueda, A novel ordered phase in SrCu2(BO3)2 under high pressure, J. Phys. Soc. Jpn. 76, 073710 (2007).
  33. P. Corboz and F. Mila, Crystals of bound states in the magnetization plateaus of the Shastry-Sutherland model, Phys. Rev. Lett. 112, 147203 (2014).
  34. M. E. Zayed, C. Rüegg, J. Larrea J., A. M. Läuchli, C. Panagopoulos, S. S. Saxena, M. Ellerby, D. F. McMorrow, T. Strässle, S. Klotz, 4-spin plaquette singlet state in the Shastry–Sutherland compound SrCu2(BO3)2, Nat. Phys. 13, 962 (2017).
  35. S. Haravifard, A. Banerjee, J. C. Lang, G. Srajer, D. M. Silevitch, B. D. Gaulin, H. A. Dabkowska, and T. F. Rosenbaum, Continuous and discontinuous quantum phase transitions in a model two-dimensional magnet, Proc. Natl. Acad. Sci. USA 109, 2286 (2012).
  36. T. Sakurai, Y. Hirao, K. Hijii, S. Okubo, H. Ohta, Y. Uwatoko, K. Kudo, and Y. Koike, Direct observation of the quantum phase transition of SrCu2(BO3)2 by high-pressure and terahertz electron spin resonance, J. Phys. Soc. Jpn. 87, 033701 (2018).
  37. C. Boos, S. P. G. Crone, I. A. Niesen, P. Corboz, K. P. Schmidt, and F. Mila, Competition between intermediate plaquette phases in SrCu2(BO3)2 under pressure, Phys. Rev. B 100, 140413(R) (2019).
  38. D. I. Badrtdinov, A. A. Tsirlin, V. V. Mazurenko, and F. Mila, SrCu2(BO3)2 under pressure: A first-principles study, Phys. Rev. B 101, 224424 (2020).
  39. J. L. Jiménez, S. P. G. Crone, E. Fogh, M. E. Zayed, R. Lortz, E. Pomjakushina, K. Conder, A. M. Läuchli, L. Weber, S. Wessel et al., A quantum magnetic analog to the critical point of water, Nature (London) 592, 370 (2021).
  40. P. C. G. Vlaar and P. Corboz, Tensor network study of the Shastry-Sutherland model with weak interlayer coupling, SciPost Phys. 15, 126 (2023).
  41. J. Yang, A. W. Sandvik, and L. Wang, Quantum criticality and spin liquid phase in the Shastry-Sutherland model, Phys. Rev. B 105, L060409 (2022).
  42. J. Y. Lee, Y.-Z. You, S. Sachdev, and A. Vishwanath, Signatures of a deconfined phase transition on the Shastry-Sutherland lattice: Applications to quantum critical SrCu2(BO3)2, Phys. Rev. X 9, 041037 (2019).
  43. A. Wietek, P. Corboz, S. Wessel, B. Normand, F. Mila, and A. Honecker, Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase, Phys. Rev. Res. 1, 033038 (2019).
  44. A. Honecker, F. Mila, and B. Normand, Multi-triplet bound states and finite-temperature dynamics in highly frustrated quantum spin ladders, Phys. Rev. B 94, 094402 (2016).
  45. E. Fogh, M. Nayak, O. Prokhnenko, M. Bartkowiak, K. Munakata, J.-R. Soh, A. A. Turrini, M. E. Zayed, E. Pomjakushina, H. Kageyama et al., Field-induced bound-state condensation and spin-nematic phase in SrCu2(BO3)2 revealed by neutron scattering up to 25.9 T, Nat. Commun. 15, 442 (2024).
  46. Z. Wang and C. D. Batista, Dynamics and instabilities of the Shastry-Sutherland model, Phys. Rev. Lett. 120, 247201 (2018).
  47. M. E. Zayed, C. Rüegg, T. Strässle, U. Stuhr, B. Roessli, M. Ay, J. Mesot, P. Link, E. Pomjakushina, M. Stingaciu, K. Conder, and H. M. Rønnow, Correlated decay of triplet excitations in the Shastry-Sutherland compound SrCu2(BO3)2, Phys. Rev. Lett. 113, 067201 (2014).
  48. S. El Shawish, J. Bonča, and I. Sega, Dynamic spin structure factor of SrCu2(BO3)2 at finite temperatures, Phys. Rev. B 72, 184409 (2005).
  49. P. Corboz and F. Mila, Tensor network study of the Shastry-Sutherland model in zero magnetic field, Phys. Rev. B 87, 115144 (2013).
  50. P. Czarnik, M. M. Rams, P. Corboz, and J. Dziarmaga, Tensor network study of the m=12 magnetization plateau in the Shastry-Sutherland model at finite temperature, Phys. Rev. B 103, 075113 (2021).
  51. A. Wietek, Y.-Y. He, S. R. White, A. Georges, and E. M. Stoudenmire, Stripes, antiferromagnetism, and the pseudogap in the doped Hubbard model at finite temperature, Phys. Rev. X 11, 031007 (2021).
  52. A. Wietek, R. Rossi, F. Šimkovic, M. Klett, P. Hansmann, M. Ferrero, E. M. Stoudenmire, T. Schäfer, and A. Georges, Mott insulating states with competing orders in the triangular lattice Hubbard model, Phys. Rev. X 11, 041013 (2021).
  53. C. Feng, A. Wietek, E. M. Stoudenmire, and R. R. P. Singh, Order, disorder, and monopole confinement in the spin-12 XXZ model on a pyrochlore tube, Phys. Rev. B 106, 075135 (2022).
  54. C. Feng, E. M. Stoudenmire, and A. Wietek, Bose-Einstein condensation in honeycomb dimer magnets and Yb2Si2O7, Phys. Rev. B 107, 205150 (2023).
  55. Z. Wang, P. McClarty, D. Dankova, A. Honecker, and A. Wietek, companion paper, Spectroscopy and complex-time correlations using minimally entangled typical thermal states, Phys. Rev. B 113, 024406 (2026).
  56. A. Koga and N. Kawakami, Quantum phase transitions in the Shastry-Sutherland model for SrCu2(BO3)2, Phys. Rev. Lett. 84, 4461 (2000).
  57. A. Läuchli, S. Wessel, and M. Sigrist, Phase diagram of the quadrumerized Shastry-Sutherland model, Phys. Rev. B 66, 014401 (2002).
  58. Z. Weihong, C. J. Hamer, and J. Oitmaa, Series expansions for a Heisenberg antiferromagnetic model for SrCu2(BO)3)2, Phys. Rev. B 60, 6608 (1999).
  59. C. Knetter, E. Müller-Hartmann, G. S. Uhrig, and E. Müller-Hartmann, Symmetries and triplet dispersion in a modified Shastry-Sutherland model for SrCu2(BO3)2, J. Phys.: Condens. Matter 12, 9069 (2000).
  60. S. R. White, Minimally entangled typical quantum states at finite temperature, Phys. Rev. Lett. 102, 190601 (2009).
  61. E. M. Stoudenmire and S. R. White, Minimally entangled typical thermal state algorithms, New J. Phys. 12, 055026 (2010).
  62. R. N. Silver, D. S. Sivia, and J. E. Gubernatis, Maximum-entropy method for analytic continuation of quantum Monte Carlo data, Phys. Rev. B 41, 2380 (1990).
  63. A. W. Sandvik, Stochastic method for analytic continuation of quantum Monte Carlo data, Phys. Rev. B 57, 10287 (1998).
  64. K. S. D. Beach, Identifying the maximum entropy method as a special limit of stochastic analytic continuation, arXiv:cond-mat/0403055.
  65. H. Shao and A. W. Sandvik, Progress on stochastic analytic continuation of quantum Monte Carlo data, Phys. Rep. 1003, 1 (2023).
  66. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ldwx-2s7w for details on the perturbation theory and spectroscopy using exact diagonalization, which includes Refs. [67, 68, 69, 70, 71, 72, 73, 74, 75, 76].
  67. F. H. L. Essler and R. M. Konik, Finite-temperature lineshapes in gapped quantum spin chains, Phys. Rev. B 78, 100403 (2008).
  68. A. J. A. James, F. H. L. Essler, and R. M. Konik, Finite-temperature dynamical structure factor of alternating Heisenberg chains, Phys. Rev. B 78, 094411 (2008).
  69. E. S. Klyushina, A. C. Tiegel, B. Fauseweh, A. T. M. N. Islam, J. T. Park, B. Klemke, A. Honecker, G. S. Uhrig, S. R. Manmana, and B. Lake, Magnetic excitations in the S=1/2 antiferromagnetic-ferromagnetic chain compound BaCu2V2O8 at zero and finite temperature, Phys. Rev. B 93, 241109 (2016).
  70. B. Fauseweh, F. Groitl, T. Keller, K. Rolfs, D. A. Tennant, K. Habicht, and G. S. Uhrig, Time-dependent correlations in quantum magnets at finite temperature, Phys. Rev. B 94, 180404 (2016).
  71. S. Sachdev and R. N. Bhatt, Bond-operator representation of quantum spins: Mean-field theory of frustrated quantum Heisenberg antiferromagnets, Phys. Rev. B 41, 9323 (1990).
  72. T. Momoi and K. Totsuka, Magnetization plateaus as insulator-superfluid transitions in quantum spin systems, Phys. Rev. B 61, 3231 (2000).
  73. T. Momoi and K. Totsuka, Magnetization plateaus of the Shastry-Sutherland model for SrCu2(BO3)2: Spin-density wave, supersolid, and bound states, Phys. Rev. B 62, 15067 (2000).
  74. C. Knetter, Perturbative Continuous Unitary Transformations: Spectral Properties of Low Dimensional Spin Systems, Ph.D. thesis, Universität zu Köln, 2003.
  75. Z. Weihong, C. J. Hamer, and J. Oitmaa, Series expansions for a Heisenberg antiferromagnetic model for SrCu2(BO3)2 (1998), arXiv:cond-mat/9811030 [cond-mat.str-el].
  76. S. Wessel, I. Niesen, J. Stapmanns, B. Normand, F. Mila, P. Corboz, and A. Honecker, Thermodynamic properties of the Shastry-Sutherland model from quantum Monte Carlo simulations, Phys. Rev. B 98, 174432 (2018).

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Phys. Rev. B 113, L041104– Published 5 January, 2026
  • Received 18 June 2024
  • Revised 7 March 2025
  • Accepted 18 November 2025
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