- Open Access
Zeeman ladders in frustrated XYZ spin chains
Phys. Rev. Research 8, 033206 – Published 18 August, 2026
DOI: https://doi.org/10.1103/7pzn-3ygf
Abstract
We investigate the nature of the excitations captured by the dynamical response of XYZ triangular spin-1/2 ladders. We complement experimental inelastic neutron scattering results on the compound with numerically exact simulations based on time-dependent matrix product state methods. Our results show that bound states of spinon excitations can arise in XYZ beyond the requirement of strong Ising anisotropies. We analyze the role of the frustrated triangular couplings on the excitations giving rise to the spin dynamical structure factor and show how the features of the bound states manifest themselves in the different polarization channels.
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References (66)
- A. Auerbach, Interacting electrons and quantum magnetism, in Graduate Texts in Contemporary Physics, 1st ed. (Springer, New York, 1994).
- D. C. Mattis, The Theory of Magnetism Made Simple (World Scientific, Singapore, 2006).
- O. A. Starykh, Unusual ordered phases of highly frustrated magnets: A review, Rep. Prog. Phys. 78, 052502 (2015).
- A. Vasiliev, O. Volkova, E. Zvereva, and M. Markina, Milestones of low-D quantum magnetism, npj Quantum Mater. 3, 18 (2018).
- L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
- Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017).
- M. Klanjšek, H. Mayaffre, C. Berthier, M. Horvatić, B. Chiari, O. Piovesana, P. Bouillot, C. Kollath, E. Orignac, R. Citro, and T. Giamarchi, Controlling Luttinger liquid physics in spin ladders under a magnetic field, Phys. Rev. Lett. 101, 137207 (2008).
- P. Bouillot, C. Kollath, A. M. Läuchli, M. Zvonarev, B. Thielemann, C. Rüegg, E. Orignac, R. Citro, M. Klanjšek, C. Berthier, M. Horvatić, and T. Giamarchi, Statics and dynamics of weakly coupled antiferromagnetic spin-1/2 ladders in a magnetic field, Phys. Rev. B 83, 054407 (2011).
- D. Schmidiger, P. Bouillot, T. Guidi, R. Bewley, C. Kollath, T. Giamarchi, and A. Zheludev, Spectrum of a magnetized strong-leg quantum spin ladder, Phys. Rev. Lett. 111, 107202 (2013).
- Q. Faure, S. Takayoshi, V. Simonet, B. Grenier, M. Månsson, J. S. White, G. S. Tucker, C. Rüegg, P. Lejay, T. Giamarchi, and S. Petit, Tomonaga-Luttinger liquid spin dynamics in the quasi-one-dimensional Ising-like antiferromagnet , Phys. Rev. Lett. 123, 027204 (2019).
- Y. Cui, Y. Fan, Z. Hu, Z. He, W. Yu, and R. Yu, Field-induced antiferromagnetism and Tomonaga-Luttinger liquid behavior in the quasi-one-dimensional Ising antiferromagnet , Phys. Rev. B 105, 174428 (2022).
- Q. Faure, S. Takayoshi, S. Petit, V. Simonet, S. Raymond, L.-P. Regnault, M. Boehm, J. S. White, M. Månsson, C. Rüegg, P. Lejay, B. Canals, T. Lorenz, S. C. Furuya, T. Giamarchi, and B. Grenier, Topological quantum phase transition in the Ising-like antiferromagnetic spin chain , Nat. Phys. 14, 716 (2018).
- Y. Cui, H. Zou, N. Xi, Z. He, Y. X. Yang, L. Shu, G. H. Zhang, Z. Hu, T. Chen, R. Yu, J. Wu, and W. Yu, Quantum criticality of the Ising-like screw chain antiferromagnet in a transverse magnetic field, Phys. Rev. Lett. 123, 067203 (2019).
- R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Quantum criticality in an Ising chain: Experimental evidence for emergent symmetry, Science 327, 177 (2010).
- C. M. Morris, R. Valdés Aguilar, A. Ghosh, S. M. Koohpayeh, J. Krizan, R. J. Cava, O. Tchernyshyov, T. M. McQueen, and N. P. Armitage, Hierarchy of bound states in the one-dimensional ferromagnetic Ising chain investigated by high-resolution time-domain terahertz spectroscopy, Phys. Rev. Lett. 112, 137403 (2014).
- Z. Wang, J. Wu, W. Yang, A. K. Bera, D. Kamenskyi, A. T. M. N. Islam, S. Xu, J. M. Law, B. Lake, C. Wu, and A. Loidl, Experimental observation of Bethe strings, Nature (London) 554, 219 (2018).
- A. K. Bera, J. Wu, W. Yang, R. Bewley, M. Boehm, J. Xu, M. Bartkowiak, O. Prokhnenko, B. Klemke, A. T. M. N. Islam, J. M. Law, Z. Wang, and B. Lake, Dispersions of many-body Bethe strings, Nat. Phys. 16, 625 (2020).
- C.-M. Halati, Z. Wang, T. Lorenz, C. Kollath, and J.-S. Bernier, Repulsively bound magnon excitations of a spin-1/2 XXZ chain in a staggered transverse field, Phys. Rev. B 108, 224429 (2023).
- Z. Wang, C.-M. Halati, J.-S. Bernier, A. Ponomaryov, D. I. Gorbunov, S. Niesen, O. Breunig, J. M. Klopf, S. Zvyagin, T. Lorenz, A. Loidl, and C. Kollath, Experimental observation of repulsively bound magnons, Nature (London) 631, 760 (2024).
- B. Wehinger, F. T. Lisandrini, N. Kestin, P. Bouillot, S. Ward, B. Thielemann, R. Bewley, M. Boehm, D. Biner, K. W. Krämer, B. Normand, T. Giamarchi, C. Kollath, A. M. Läuchli, and C. Rüegg, Fingerprints of supersymmetric spin and charge dynamics observed by inelastic neutron scattering, Nat. Commun. 16, 3228 (2025).
- J. des Cloizeaux and J. J. Pearson, Spin-wave spectrum of the antiferromagnetic linear chain, Phys. Rev. 128, 2131 (1962).
- M. B. Stone, D. H. Reich, C. Broholm, K. Lefmann, C. Rischel, C. P. Landee, and M. M. Turnbull, Extended quantum critical phase in a magnetized spin-1/2 antiferromagnetic chain, Phys. Rev. Lett. 91, 037205 (2003).
- I. A. Zaliznyak, H. Woo, T. G. Perring, C. L. Broholm, C. D. Frost, and H. Takagi, Spinons in the strongly correlated copper oxide chains in , Phys. Rev. Lett. 93, 087202 (2004).
- B. Thielemann, C. Rüegg, H. M. Rønnow, A. M. Läuchli, J.-S. Caux, B. Normand, D. Biner, K. W. Krämer, H.-U. Güdel, J. Stahn, K. Habicht, K. Kiefer, M. Boehm, D. F. McMorrow, and J. Mesot, Direct observation of magnon fractionalization in the quantum spin ladder, Phys. Rev. Lett. 102, 107204 (2009).
- M. Mourigal, M. Enderle, A. Klöpperpieper, J.-S. Caux, A. Stunault, and H. M. Rønnow, Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain, Nat. Phys. 9, 435 (2013).
- B. Lake, D. A. Tennant, J.-S. Caux, T. Barthel, U. Schollwöck, S. E. Nagler, and C. D. Frost, Multispinon continua at zero and finite temperature in a near-ideal Heisenberg chain, Phys. Rev. Lett. 111, 137205 (2013).
- M. Greiter and S. Rachel, Valence bond solids for spin chains: Exact models, spinon confinement, and the Haldane gap, Phys. Rev. B 75, 184441 (2007).
- S. Rachel, R. Thomale, M. Führinger, P. Schmitteckert, and M. Greiter, Spinon confinement and the Haldane gap in spin chains, Phys. Rev. B 80, 180420(R) (2009).
- L. Vanderstraeten, E. Wybo, N. Chepiga, F. Verstraete, and F. Mila, Spinon confinement and deconfinement in spin-1 chains, Phys. Rev. B 101, 115138 (2020).
- B. M. McCoy and T. T. Wu, Two-dimensional Ising field theory in a magnetic field: Breakup of the cut in the two-point function, Phys. Rev. D 18, 1259 (1978).
- H. Shiba, Quantization of magnetic excitation continuum due to interchain coupling in nearly one-dimensional Ising-like antiferromagnets, Prog. Theor. Phys. 64, 466 (1980).
- S. B. Rutkevich, Energy spectrum of bound-spinons in the quantum Ising spin-chain ferromagnet, J. Stat. Phys. 131, 917 (2008).
- B. Grenier, S. Petit, V. Simonet, E. Canévet, L.-P. Regnault, S. Raymond, B. Canals, C. Berthier, and P. Lejay, Longitudinal and transverse Zeeman ladders in the Ising-like chain antiferromagnet , Phys. Rev. Lett. 114, 017201 (2015).
- Z. Wang, J. Wu, S. Xu, W. Yang, C. Wu, A. K. Bera, A. T. M. N. Islam, B. Lake, D. Kamenskyi, P. Gogoi, H. Engelkamp, N. Wang, J. Deisenhofer, and A. Loidl, From confined spinons to emergent fermions: Observation of elementary magnetic excitations in a transverse-field Ising chain, Phys. Rev. B 94, 125130 (2016).
- A. K. Bera, B. Lake, F. H. L. Essler, L. Vanderstraeten, C. Hubig, U. Schollwöck, A. T. M. N. Islam, A. Schneidewind, and D. L. Quintero-Castro, Spinon confinement in a quasi-one-dimensional anisotropic Heisenberg magnet, Phys. Rev. B 96, 054423 (2017).
- M. Mena, N. Hänni, S. Ward, E. Hirtenlechner, R. Bewley, C. Hubig, U. Schollwöck, B. Normand, K. W. Krämer, D. F. McMorrow, and C. Rüegg, Thermal control of spin excitations in the coupled Ising-chain material , Phys. Rev. Lett. 124, 257201 (2020).
- K. Matan, T. Ono, S. Ohira-Kawamura, K. Nakajima, Y. Nambu, and T. J. Sato, Breakdown of linear spin-wave theory and existence of spinon bound states in the frustrated kagome-lattice antiferromagnet, Phys. Rev. B 105, 134403 (2022).
- E. A. Ghioldi, S.-S. Zhang, Y. Kamiya, L. O. Manuel, A. E. Trumper, and C. D. Batista, Evidence of two-spinon bound states in the magnetic spectrum of , Phys. Rev. B 106, 064418 (2022).
- L. Facheris, S. D. Nabi, A. Glezer Moshe, U. Nagel, T. Rõõm, K. Y. Povarov, J. R. Stewart, Z. Yan, and A. Zheludev, Confinement of fractional excitations in a triangular lattice antiferromagnet, Phys. Rev. Lett. 130, 256702 (2023).
- K. Y. Povarov, L. Facheris, S. Velja, D. Blosser, Z. Yan, S. Gvasaliya, and A. Zheludev, Magnetization plateaux cascade in the frustrated quantum antiferromagnet , Phys. Rev. Res. 2, 043384 (2020).
- L. Facheris, K. Y. Povarov, S. D. Nabi, D. G. Mazzone, J. Lass, B. Roessli, E. Ressouche, Z. Yan, S. Gvasaliya, and A. Zheludev, Spin density wave versus fractional magnetization plateau in a triangular antiferromagnet, Phys. Rev. Lett. 129, 087201 (2022).
- L. Facheris, S. D. Nabi, K. Y. Povarov, Z. Yan, A. G. Moshe, U. Nagel, T. Rõõm, A. Podlesnyak, E. Ressouche, K. Beauvois, J. R. Stewart, P. Manuel, D. Khalyavin, F. Orlandi, and A. Zheludev, Magnetic field induced phases and spin Hamiltonian in , Phys. Rev. B 109, 104433 (2024).
- R. Coldea, D. A. Tennant, A. M. Tsvelik, and Z. Tylczynski, Experimental realization of a 2D fractional quantum spin liquid, Phys. Rev. Lett. 86, 1335 (2001).
- Y. Tokiwa, T. Radu, R. Coldea, H. Wilhelm, Z. Tylczynski, and F. Steglich, Magnetic phase transitions in the two-dimensional frustrated quantum antiferromagnet , Phys. Rev. B 73, 134414 (2006).
- O. A. Starykh, H. Katsura, and L. Balents, Extreme sensitivity of a frustrated quantum magnet: , Phys. Rev. B 82, 014421 (2010).
- A. I. Smirnov, K. Y. Povarov, S. V. Petrov, and A. Y. Shapiro, Magnetic resonance in the ordered phases of the two-dimensional frustrated quantum magnet , Magnetic resonance in the ordered phases of the two-dimensional frustrated quantum magnet , Phys. Rev. B 85, 184423 (2012).
- L. Facheris et al., Spin-density wave dynamics in a 2D distorted triangular lattice antiferromagnet, STFC ISIS Neutron and Muon Source, 2022, https://doi.org/10.5286/ISIS.E.RB2210048.
- O. Breunig, M. Garst, E. Sela, B. Buldmann, P. Becker, L. Bohatý, R. Müller, and T. Lorenz, Spin-1/2 chain system in a transverse magnetic field, Phys. Rev. Lett. 111, 187202 (2013).
- O. Breunig, M. Garst, A. Rosch, E. Sela, B. Buldmann, P. Becker, L. Bohatý, R. Müller, and T. Lorenz, Low-temperature ordered phases of the spin-1/2 XXZ chain system , Phys. Rev. B 91, 024423 (2015).
- S. R. White and A. E. Feiguin, Real-time evolution using the density matrix renormalization group, Phys. Rev. Lett. 93, 076401 (2004).
- A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces, J. Stat. Mech. (2004) P04005.
- U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
- S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
- S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter (Clarendon Press, Oxford, 1984).
- G. L. Squires, Introduction to the Theory of Thermal Neutron Scattering, 3rd ed. (Cambridge University Press, Cambridge, 2012).
- P. J. Brown, Magnetic form factors, in International Tables for Crystallography, edited by E. Prince (Wiley, 2006), Vol. C, pp. 454–461.
- G. Müller, H. Thomas, H. Beck, and J. C. Bonner, Quantum spin dynamics of the antiferromagnetic linear chain in zero and nonzero magnetic field, Phys. Rev. B 24, 1429 (1981).
- M. Karbach, G. Müller, A. H. Bougourzi, A. Fledderjohann, and K.-H. Mütter, Two-spinon dynamic structure factor of the one-dimensional s = Heisenberg antiferromagnet, Phys. Rev. B 55, 12510 (1997).
- T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, 2004).
- L. Facheris, S. Gvasaliya, V. Romerio, P. Steffens, and A. Zheludev, Longitudinal and transverse polarization of the 2D Zeeman ladder in , Institut Laue-Langevin (ILL), 2024, https://doi.org/10.5291/ILL-DATA.4-01-1818.
- M. Popovici, On the resolution of slow-neutron spectrometers. IV. The triple-axis spectrometer resolution function, spatial effects included, Acta Cryst. A 31, 507 (1975).
- A. Zheludev, ResLib: Resolution library for three-axis neutron spectroscopy (2009), https://neutron.phys.ethz.ch/Methods/reslib.html.
- C.-M. Halati, V. Romerio, A. Zheludev, and T. Giamarchi, Theoretical simulations data for article “Zeeman ladders in frustrated XYZ spin chains” [Dataset], Zenodo, 2025, https://zenodo.org/records/16940039.
- M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases 4 (2022).
- J.-I. Igarashi, Ground state and excitation spectrum of a spin-1/2 Ising-like ferromagnetic chain with competing interactions, J. Phys. Soc. Jpn. 58, 4600 (1989).
- S. Furukawa, M. Sato, S. Onoda, and A. Furusaki, Ground-state phase diagram of a spin-1/2 frustrated ferromagnetic XXZ chain: Haldane dimer phase and gapped/gapless chiral phases, Phys. Rev. B 86, 094417 (2012).