The Ising chain in a transverse field is a paradigmatic model for a host of physical phenomena, including spontaneous symmetry breaking, quantum criticality and duality. Although the quasi-one-dimensional ferromagnet CoNb2O6 has been regarded as the Ising chain’s best material realization, it exhibits substantial deviations from ideality. By combining terahertz spectroscopy and calculations, we show that CoNb2O6 is in fact described by a different model with bond-dependent interactions, which we call the ‘twisted Kitaev chain’, as these interactions are similar to those of the honeycomb Kitaev spin liquid. The ferromagnetic ground state of CoNb2O6 arises from the compromise between two axes. Owing to this frustration, even at zero field domain walls have quantum motion, which is described by the celebrated Su–Schriefer–Heeger model of polyacetylene and shows rich behaviour as a function of field. Nevertheless, close to the critical field, this model enters a universal regime in the Ising universality class. We observe that the excitation gap in the ferromagnet closes at a rate twice that of the paramagnet. This universal ratio originates in the Kramers–Wannier duality between domain walls and spin flips, and in the topological conservation of domain wall parity. Our work also shows that Co2+ magnets are fertile ground in the search for quantum spin liquids.
This is a preview of subscription content, access via your institution
Subscribe to Nature+
Get immediate online access to Nature and 55 other Nature journal
Subscribe to Journal
Get full journal access for 1 year
only $8.25 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
All data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
Onsager, L. Crystal statistics. I. A two-dimensional model with an order–disorder transition. Phys. Rev. 65, 117–149 (1944).
Kramers, H. A. & Wannier, G. H. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60, 252–262 (1941).
Sachdev, S. Quantum Phase Transitions 2nd edn (Cambridge Univ. Press, 2011).
Mussardo, G. Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics 1st edn (Oxford Univ. Press, 2010).
Pfeuty, P. The one-dimensional Ising model with a transverse field. Ann. Phys. 57, 79–90 (1970).
McCoy, B. M. & Wu, T. T. Two-dimensional Ising field theory in a magnetic field: breakup of the cut in the two-point function. Phys. Rev. D. 18, 1259–1267 (1978).
Zamolodchikov, A. B. Integrals of motion and S-matrix of the (scaled) T = Tc Ising model with magnetic field. Int. J. Mod. Phys. A 4, 4235–4248 (1989).
Polyakov, A. M. Gauge Fields and Strings (CRC Press, 1987).
Polchinski, J. String Theory Vol. 1 (Cambridge Univ. Press, 2005).
Fisher, M. P. A. in Strong Interactions in Low Dimensions (eds Baeriswyl, D. & Degiorgi, L.) 419–438 (Springer Netherlands, 2004).
Heid, C. et al. Magnetic phase diagram of CoNb2O6: a neutron diffraction study. J. Magn. Magn. Mater. 151, 123–131 (1995).
Kobayashi, S., Mitsuda, S. & Prokes, K. Low-temperature magnetic phase transitions of the geometrically frustrated isosceles triangular Ising antiferromagnet CoNb2O6. Phys. Rev. B 63, 024415 (2000).
Coldea, R. et al. Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry. Science 327, 177–180 (2010).
Morris, C. M. et al. Hierarchy of bound states in the one-dimensional ferromagnetic Ising chain CoNb2O6 investigated by high-resolution time-domain terahertz spectroscopy. Phys. Rev. Lett. 112, 137403 (2014).
Kinross, A. W. et al. Evolution of quantum fluctuations near the quantum critical point of the transverse field Ising chain system CoNb2O6. Phys. Rev. X 4, 031008 (2014).
Steinberg, J., Armitage, N. P., Essler, F. H. L. & Sachdev, S. NMR relaxation in Ising spin chains. Phys. Rev. B 99, 035156 (2019).
Liang, T. et al. Heat capacity peak at the quantum critical point of the transverse Ising magnet CoNb2O6. Nat. Commun. 6, 7611 (2015).
Amelin, K. et al. Experimental observation of quantum many-body excitations of E8 symmetry in the Ising chain ferromagnet CoNb2O6. Phys. Rev. B 102, 104431 (2020).
Kjäll, J. A., Pollmann, F. & Moore, J. E. Bound states and E8 symmetry effects in perturbed quantum Ising chains. Phys. Rev. B 83, 020407 (2011).
Robinson, N. J., Essler, F. H. L., Cabrera, I. & Coldea, R. Quasiparticle breakdown in the quasi-one-dimensional Ising ferromagnet CoNb2O6. Phys. Rev. B 90, 174406 (2014).
Fava, M., Coldea, R. & Parameswaran, S. A. Glide symmetry breaking and Ising criticality in the quasi-1D magnet CoNb2O6. Proc. Natl Acad. Sci. USA 117, 25219–25224 (2020).
Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).
Liu, H. & Khaliullin, G. Pseudospin exchange interactions in d7 cobalt compounds: possible realization of the Kitaev model. Phys. Rev. B 97, 014407 (2018).
Sano, R., Kato, Y. & Motome, Y. Kitaev–Heisenberg Hamiltonian for high-spin d7 Mott insulators. Phys. Rev. B 97, 014408 (2018).
Liu, H., Chaloupka, J. & Khaliullin, G. Kitaev spin liquid in 3d transition metal compounds. Phys. Rev. Lett. 125, 047201 (2020).
Kugel, K. I. & Khomskiĭ, D. I. The Jahn–Teller effect and magnetism: transition metal compounds. Sov. Phys. Usp. 25, 231–256 (1982).
Nussinov, Z. & van den Brink, J. Compass models: theory and physical motivations. Rev. Mod. Phys. 87, 1–59 (2015).
You, W.-L., Horsch, P. & Oleś, A. M. Quantum phase transitions in exactly solvable one-dimensional compass models. Phys. Rev. B 89, 104425 (2014).
Heeger, A. J., Kivelson, S., Schrieffer, J. R. & Su, W.-P. Solitons in conducting polymers. Rev. Mod. Phys. 60, 781–850 (1988).
Fishman, M., White, S. R. & Stoudenmire, E. M. The ITensor Software Library for tensor network calculations. Preprint at http://arxiv.org/abs/2007.14822 (2020).
Zhong, R., Gao, T., Ong, N. P. & Cava, R. J. Weak-field induced nonmagnetic state in a Co-based honeycomb. Sci. Adv. 6, eaay6953 (2020).
Vivanco, H. K., Trump, B. A., Brown, C. M. & McQueen, T. M. Competing antiferromagnetic-ferromagnetic states in d7 Kitaev honeycomb magnet. Phys. Rev. B 102, 224411 (2020).
Work at Johns Hopkins University and Princeton University was supported as part of the Institute for Quantum Matter, an Energy Frontier Research Center funded by the Office of Basic Energy Sciences of the Department of Energy, under grant no. DE-SC0019331. Work at the University of Kentucky was supported by National Science Foundation award no. DMR-1611161. The work at the National Institute of Chemical Physics and Biophysics was supported by institutional research funding grant no. IUT23-3 of the Estonian Ministry of Education and Research and by European Regional Development Fund project no. TK134.
The authors declare no competing interests.
Peer review information Nature Physics thanks Jan Ravnik and other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Morris, C.M., Desai, N., Viirok, J. et al. Duality and domain wall dynamics in a twisted Kitaev chain. Nat. Phys. 17, 832–836 (2021). https://doi.org/10.1038/s41567-021-01208-0
This article is cited by
Nature Materials (2023)