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Unconventional spin dynamics and supersolid excitations in the triangular-lattice XXZ model

Rafael Flores-Calderón1,*, Roderich Moessner1, and Frank Pollmann2

  • *Contact author: r.flores-calderon@tum.de
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Phys. Rev. B 112, 184423 – Published 20 November, 2025

DOI: https://doi.org/10.1103/xdwz-8k94

Abstract

Motivated by recent experiments, we investigate the spin-12 XXZ model on the triangular lattice with strong Ising anisotropy, combining large-scale numerical simulations and analytical methods to uncover unconventional spin dynamics at T=0. First, we compute the dynamical spin structure factor using density matrix renormalization group (DMRG) simulations and find excellent agreement with inelastic neutron scattering data on the layered compound K2Co(SeO3)2. The low-energy spectrum reveals a rotonlike minimum at the M point, absent in linear spin-wave theory, accompanied by peak intensity and a broad continuum above it. Near the Γ point, we observe an approximately linear dispersion with vanishing spectral weight. Second, we compare two analytical frameworks that reproduce the observed features. The first is a hard-core boson approach, which includes (i) an effective staggered boson model at zero magnetic field, (ii) perturbation theory applied to the one-third magnetization plateau, and (iii) a self-consistent mean-field Schwinger boson theory (SBT). The second framework is based on a variational supersolid quantum dimer model (QDM) ansatz, combined with a single-mode approximation. The SBT captures the broad continuum, the M-point minimum, and linear dispersion at Γ, whereas the QDM reproduces the roton minimum and linear dispersion at finite momentum near Γ. Remarkably, both the QDM wave function and the DMRG ground state exhibit nearly identical structure factors with pronounced transverse photonlike excitations. Together, our comprehensive theoretical and numerical analysis elucidates the microscopic origin of supersolid excitations in the XXZ triangular-lattice model and their proximity to a spin-liquid phase observed experimentally.

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References (75)

  1. A. J. Leggett, Can a solid be “superfluid”? Phys. Rev. Lett. 25, 1543 (1970).
  2. E. Kim and M. H.-W. Chan, Probable observation of a supersolid helium phase, Nature (London) 427, 225 (2004).
  3. S. Balibar, The enigma of supersolidity, Nature (London) 464, 176 (2010).
  4. D. Y. Kim and M. H. W. Chan, Absence of supersolidity in solid helium in porous vycor glass, Phys. Rev. Lett. 109, 155301 (2012).
  5. M. Boninsegni and N. V. Prokof'ev, Colloquium: Supersolids: What and where are they? Rev. Mod. Phys. 84, 759 (2012).
  6. J. Nyéki, A. Phillis, A. Ho, D. Lee, P. Coleman, J. Parpia, B. Cowan, and J. Saunders, Intertwined superfluid and density wave order in two-dimensional H4e, Nat. Phys. 13, 455 (2017).
  7. J.-R. Li, J. Lee, W. Huang, S. Burchesky, A. O. Jamison, and W. Ketterle, A stripe phase with supersolid properties in spin–orbit-coupled Bose–Einstein condensates, Nature (London) 543, 91 (2017).
  8. J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, Supersolid formation in a quantum gas breaking a continuous translational symmetry, Nature (London) 543, 87 (2017).
  9. L. Tanzi, S. A. Rößler, M. Fattori, L. Lepori, G. Semeghini, B. Capogrosso-Sansone, M. Lewenstein, G. Modugno, and F. Ferlaino, Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas, Nature (London) 574, 382 (2019).
  10. M. A. Norcia, T. Obata, A. Young, S. L. Campbell, and W. Ketterle, Two-dimensional supersolidity in a dipolar quantum gas, Nature (London) 596, 357 (2021).
  11. R. G. Melko, A. Paramekanti, A. A. Burkov, A. Vishwanath, D. N. Sheng, and L. Balents, Supersolid order from disorder: Hard-core bosons on the triangular lattice, Phys. Rev. Lett. 95, 127207 (2005).
  12. S. Wessel and M. Troyer, Supersolid hard-core bosons on the triangular lattice, Phys. Rev. Lett. 95, 127205 (2005).
  13. D. Heidarian and K. Damle, Persistent supersolid phase of hard-core bosons on the triangular lattice, Phys. Rev. Lett. 95, 127206 (2005).
  14. M. Boninsegni and N. V. Prokof'ev, Supersolid phase of hard-core bosons on a triangular lattice, Phys. Rev. Lett. 95, 237204 (2005).
  15. H. C. Jiang, M. Q. Weng, Z. Y. Weng, D. N. Sheng, and L. Balents, Supersolid order of frustrated hard-core bosons in a triangular lattice system, Phys. Rev. B 79, 020409(R) (2009).
  16. D. Heidarian and A. Paramekanti, Supersolidity in the triangular lattice spin-1/2 XXZ model: A variational perspective, Phys. Rev. Lett. 104, 015301 (2010).
  17. T. Chen, A. Ghasemi, J. Zhang, L. Shi, Z. Tagay, Y. Chen, L. Chen, E.-S. Choi, M. Jaime, M. Lee, Y. Hao, H. Cao, B. Winn, A. A. Podlesnyak, D. M. Pajerowski, R. Zhong, X. Xu, N. P. Armitage, R. Cava, and C. Broholm, Phase diagram and spectroscopic signatures of supersolids in quantum Ising magnet K2Co(SeO3)2, arXiv:2402.15869.
  18. M. Zhu, V. Romerio, N. Steiger, S. D. Nabi, N. Murai, S. Ohira-Kawamura, K. Y. Povarov, Y. Skourski, R. Sibille, L. Keller, Z. Yan, S. Gvasaliya, and A. Zheludev, Continuum excitations in a spin supersolid on a triangular lattice, Phys. Rev. Lett. 133, 186704 (2024).
  19. R. Moessner and S. L. Sondhi, Ising models of quantum frustration, Phys. Rev. B 63, 224401 (2001) .
  20. M. Drescher, L. Vanderstraeten, R. Moessner, and F. Pollmann, Dynamical signatures of symmetry-broken and liquid phases in an S=12 Heisenberg antiferromagnet on the triangular lattice, Phys. Rev. B 108, L220401 (2023).
  21. N. E. Sherman, M. Dupont, and J. E. Moore, Spectral function of the j1−j2 Heisenberg model on the triangular lattice, Phys. Rev. B 107, 165146 (2023).
  22. M. Zhu, L. M. Chinellato, V. Romerio, N. Murai, S. Ohira-Kawamura, C. Balz, Z. Yan, S. Gvasaliya, Y. Kato, C. D. Batista, and A. Zheludev, Wannier states and spin supersolid physics in the triangular antiferromagnet K2Co(SeO3)2), npj Quantum Mater. 10, 74 (2025).
  23. M. Ulaga, J. Kokalj, T. Tohyama, and P. Prelovšek, Easy-axis Heisenberg model on the triangular lattice: From supersolid to gapped solid, Phys. Rev. B 111, 174442 (2025).
  24. M. Ulaga, J. Kokalj, A. Wietek, A. Zorko, and P. Prelovšek, Finite-temperature properties of the easy-axis Heisenberg model on frustrated lattices, Phys. Rev. B 109, 035110 (2024).
  25. J. Xiang, C. Zhang, Y. Gao, W. Schmidt, K. Schmalzl, C.-W. Wang, B. Li, N. Xi, X.-Y. Liu, H. Jin, G. Li, J. Shen, Z. Chen, Y. Qi, Y. Wan, W. Jin, W. Li, P. Sun, and G. Su, Giant magnetocaloric effect in spin supersolid candidate Na2BaCo(PO4)2, Nature (London) 625, 270 (2024).
  26. H. Jia, B. Ma, Z. D. Wang, and G. Chen, Quantum spin supersolid as a precursory Dirac spin liquid in a triangular lattice antiferromagnet, Phys. Rev. Res. 6, 033031 (2024).
  27. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  28. J. Hauschild and F. Pollmann, Efficient numerical simulations with tensor networks: Tensor network python (tenpy), SciPost Phys. Lect. Notes 5 (2018).
  29. R. A. F. Calderon, R. Moessner, and F. Pollmann, Photon- and roton-like excitations of the triangular lattice XXZ supersolid, Zenodo (2025), https://doi.org/10.5281/zenodo.15673281.
  30. D. Sellmann, X.-F. Zhang, and S. Eggert, Phase diagram of the antiferromagnetic XXZ model on the triangular lattice, Phys. Rev. B 91, 081104(R) (2015).
  31. A. Sen, P. Dutt, K. Damle, and R. Moessner, Variational wave-function study of the triangular lattice supersolid, Phys. Rev. Lett. 100, 147204 (2008).
  32. O. Benton, O. Sikora, and N. Shannon, Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice, Phys. Rev. B 86, 075154 (2012).
  33. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  34. J. Hauschild, J. Unfried, S. Anand, B. Andrews, M. Bintz, U. Borla, S. Divic, M. Drescher, J. Geiger, M. Hefel, K. Hémery, W. Kadow, J. Kemp, N. Kirchner, V. S. Liu, G. Möller, D. Parker, M. Rader, A. Romen, S. Scalet et al., Tensor network python (tenpy) version 1, SciPost Phys. Codebases 41 (2024).
  35. J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Time-dependent variational principle for quantum lattices, Phys. Rev. Lett. 107, 070601 (2011).
  36. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B 94, 165116 (2016).
  37. L. Vanderstraeten, J. Haegeman, and F. Verstraete, Tangent-space methods for uniform matrix product states, SciPost Phys. Lect. Notes 7 (2019).
  38. M. Gohlke, R. Verresen, R. Moessner, and F. Pollmann, Dynamics of the Kitaev-Heisenberg model, Phys. Rev. Lett. 119, 157203 (2017).
  39. E. Stoudenmire and S. R. White, Studying two-dimensional systems with the density matrix renormalization group, Annu. Rev. Condens. Matter Phys. 3, 111 (2012).
  40. D. Yamamoto, G. Marmorini, and I. Danshita, Quantum phase diagram of the triangular-lattice XXZ model in a magnetic field, Phys. Rev. Lett. 112, 127203 (2014).
  41. B. Kleine, E. Müller-Hartmann, K. Frahm, and P. Fazekas, Spin-wave analysis of easy-axis quantum antiferromagnets on the triangular lattice, Z. Phys. B 87, 103 (1992).
  42. S. Streib and P. Kopietz, Hard-core boson approach to the spin-1/2 triangular-lattice antiferromagnet Cs2CuCl4 at finite temperatures in magnetic fields higher than the saturation field, Phys. Rev. B 92, 094442 (2015).
  43. F. Wang, F. Pollmann, and A. Vishwanath, Extended supersolid phase of frustrated hard-core bosons on a triangular lattice, Phys. Rev. Lett. 102, 017203 (2009).
  44. P. Chandra, P. Coleman, and A. I. Larkin, A quantum fluids approach to frustrated Heisenberg models, J. Phys.: Condens. Matter 2, 7933 (1990).
  45. V. I. Yukalov and H. Kleinert, Gapless Hartree-Fock-Bogoliubov approximation for Bose gases, Phys. Rev. A 73, 063612 (2006).
  46. C. J. Gazza and H. A. Ceccatto, A Schwinger boson approach to Heisenberg antiferromagnets on a triangular lattice, J. Phys.: Condens. Matter 5, L135 (1993).
  47. A. Mezio, C. N. Sposetti, L. O. Manuel, and A. E. Trumper, Broken discrete and continuous symmetries in two-dimensional spiral antiferromagnets, J. Phys.: Condens. Matter 25, 465602 (2013).
  48. E. A. Ghioldi, A. Mezio, L. O. Manuel, R. R. P. Singh, J. Oitmaa, and A. E. Trumper, Magnons and excitation continuum in XXZ triangular antiferromagnetic model: Application to Ba3CoSb2O9, Phys. Rev. B 91, 134423 (2015).
  49. G. H. Wannier, Antiferromagnetism. The triangular Ising net, Phys. Rev. 79, 357 (1950).
  50. C. Castelnovo, R. Moessner, and S. L. Sondhi, Spin ice, fractionalization, and topological order, Annu. Rev. Condens. Matter Phys. 3, 35 (2012).
  51. J. Knolle and R. Moessner, A field guide to spin liquids, Annu. Rev. Condens. Matter Phys. 10, 451 (2019).
  52. R. Moessner and J. T. Chalker, Low-temperature properties of classical geometrically frustrated antiferromagnets, Phys. Rev. B 58, 12049 (1998).
  53. R. Moessner and J. T. Chalker, Properties of a classical spin liquid: The Heisenberg pyrochlore antiferromagnet, Phys. Rev. Lett. 80, 2929 (1998).
  54. S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, Dipolar spin correlations in classical pyrochlore magnets, Phys. Rev. Lett. 93, 167204 (2004).
  55. C. L. Henley, Power-law spin correlations in pyrochlore antiferromagnets, Phys. Rev. B 71, 014424 (2005).
  56. H. Yan, O. Benton, A. H. Nevidomskyy, and R. Moessner, Classification of classical spin liquids: Detailed formalism and suite of examples, Phys. Rev. B 109, 174421 (2024).
  57. P. W. Anderson, Resonating valence bonds: A new kind of insulator? Mater. Res. Bull. 8, 153 (1973).
  58. P. W. Anderson, The resonating valence bond state in La2CuO4 and superconductivity, Science 235, 1196 (1987).
  59. D. S. Rokhsar and S. A. Kivelson, Superconductivity and the quantum hard-core dimer gas, Phys. Rev. Lett. 61, 2376 (1988).
  60. R. P. Feynman, Atomic theory of the two-fluid model of liquid helium, Phys. Rev. 94, 262 (1954).
  61. A. Griffin, T. Nikuni, and E. Zaremba, Bose-Condensed Gases at Finite Temperatures (Cambridge University Press, Cambridge, 2009).
  62. J. P. Toennies and A. F. Vilesov, Superfluid helium droplets: A uniquely cold nanomatrix for molecules and molecular complexes, Angew. Chem. Int. Ed. 43, 2622 (2004).
  63. J. Haegeman, S. Michalakis, B. Nachtergaele, T. J. Osborne, N. Schuch, and F. Verstraete, Elementary excitations in gapped quantum spin systems, Phys. Rev. Lett. 111, 080401 (2013).
  64. S. Yi, Ö. E. Müstecaplíoğlu, C.-P. Sun, and L. You, Single-mode approximation in a spinor-1 atomic condensate, Phys. Rev. A 66, 011601(R) (2002).
  65. D. E. Bruschi, J. Louko, E. Martín-Martínez, A. Dragan, and I. Fuentes, Unruh effect in quantum information beyond the single-mode approximation, Phys. Rev. A 82, 042332 (2010).
  66. A. M. Läuchli, S. Capponi, and F. F. Assaad, Dynamical dimer correlations at bipartite and non-bipartite Rokhsar–Kivelson points, J. Stat. Mech. (2008) P01010.
  67. E. Manousakis, The spin-1/2 Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides, Rev. Mod. Phys. 63, 1 (1991).
  68. Q. Chen, J. Stajic, S. Tan, and K. Levin, BCS–BEC crossover: From high temperature superconductors to ultracold superfluids, Phys. Rep. 412, 1 (2005).
  69. S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Magneto-roton theory of collective excitations in the fractional quantum Hall effect, Phys. Rev. B 33, 2481 (1986).
  70. R. Moessner and S. L. Sondhi, Three-dimensional resonating-valence-bond liquids and their excitations, Phys. Rev. B 68, 184512 (2003).
  71. A. Fancelli, R. Flores-Calderón, O. Benton, B. Lake, R. Moessner, and J. Reuther, Fragile spin liquid in three dimensions, Phys. Rev. B 111, 134413 (2025).
  72. R. Flores-Calderón, O. Benton, and R. Moessner, Irrational moments and signatures of higher-rank gauge theories in diluted classical spin liquids, Phys. Rev. Lett. 133, 106501 (2024).
  73. Y. D. Liao, H. Li, Z. Yan, H.-T. Wei, W. Li, Y. Qi, and Z. Y. Meng, Phase diagram of triangular lattice quantum Ising model under external field, Phys. Rev. B 103, 104416 (2021).
  74. S. Biswas and K. Damle, Singular ferromagnetic susceptibility of the transverse-field Ising antiferromagnet on the triangular lattice, Phys. Rev. B 97, 085114 (2018).
  75. M. Pretko and L. Radzihovsky, Symmetry-enriched fracton phases from supersolid duality, Phys. Rev. Lett. 121, 235301 (2018).

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Phys. Rev. B 112, 184423– Published 20 November, 2025
  • Received 5 July 2025
  • Revised 27 October 2025
  • Accepted 3 November 2025
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