Frustration refers to competition between different interactions that cannot be simultaneously satisfied—a familiar feature in many magnetic solids. Strong frustration leads to highly degenerate ground states and a large suppression of ordering by fluctuations. Key challenges in frustrated magnetism include the characterization of the fluctuating spin-liquid regime and determination of the mechanism of eventual order at lower temperature. Here, we study a model of a diamond-lattice antiferromagnet appropriate for numerous spinel materials. With sufficiently strong frustration, a massive ground-state degeneracy develops amongst spirals whose propagation wavevectors reside on a continuous two-dimensional ‘spiral surface’ in momentum space. We argue that an important ordering mechanism is entropic splitting of the degenerate ground states, an elusive phenomenon called ‘order by disorder’. A broad spiral spin-liquid regime emerges at higher temperatures, where the underlying spiral surface can be directly revealed through spin correlations. We discuss the agreement between these predictions and the well-characterized spinel MnSc2S4.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
Scientific Reports Open Access 08 July 2023
Scientific Reports Open Access 31 May 2021
npj Quantum Materials Open Access 18 December 2019
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
Ramirez, A. P. Strongly geometrically frustrated magnets. Annu. Rev. Mater. Sci. 24, 453–480 (1994).
Lee, S.-H et al. Emergent excitations in a geometrically frustrated magnet. Nature 418, 856–858 (2002).
Isakov, S. V., Gregor, K., Moessner, R. & Sondhi, S. L. Dipolar spin correlations in classical pyrochlore magnets. Phys. Rev. Lett. 93, 167204 (2004).
Henley, C. L. Power-law spin correlations in pyrochlore antiferromagnets. Phys. Rev. B 71, 014424 (2005).
Hermele, M., Fisher, M. P. A. & Balents, L. Pyrochlore photons: The U(1) spin liquid in a s=1/2 three-dimensional frustrated magnet. Phys. Rev. B 69, 064404 (2004).
Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore materials. Science 294, 1495–1501 (2001).
Yamashita, Y. & Ueda, K. Spin-driven Jahn–Teller distortion in a pyrochlore system. Phys. Rev. Lett. 85, 4960–4963 (2000).
Veillette, M. Y., Chalker, J. T. & Coldea, R. Ground states of a frustrated spin-(1/2) antiferromagnet: Cs2CuCl4 in a magnetic field. Phys. Rev. B 71, 214426 (2005).
Villain, J., Bidaux, R., Carton, J. P. & Conte, R. Order as an effect of disorder. J. Physique 41, 1263–1272 (1980).
Rastelli, E. & Tassi, A. Order produced by quantum disorder in the Heisenberg rhombohedral antiferromagnet. J. Phys. C 20, L303–L306 (1987).
Henley, C. L. Ordering due to disorder in a frustrated vector antiferromagnet. Phys. Rev. Lett. 62, 2056–2059 (1989).
Chubukov, A. Order from disorder in a kagomé antiferromagnet. Phys. Rev. Lett. 69, 832–835 (1992).
Henley, C. L. Ordering by disorder: Ground-state selection in fcc vector antiferromagnets. J. Appl. Phys. 61, 3962–3964 (1987).
Gvozdikova, M. V. & Zhitomirsky, M. E. Monte Carlo study of first-order transition in Heisenberg fcc antiferromagnet. JETP Lett. 81, 236–240 (2005).
Reimers, J. N. Absence of long-range order in a three-dimensional geometrically frustrated antiferromagnet. Phys. Rev. B 45, 7287–7294 (1992).
Moessner, R. & Chalker, J. T. Low-temperature properties of classical geometrically frustrated antiferromagnets. Phys. Rev. B 58, 12049–12062 (1998).
Tristan, N. et al. Geometric frustration in the cubic spinels MAl2O4 (M=Co, Fe, and Mn). Phys. Rev. B 72, 174404 (2005).
Suzuki, T., Nagai, H., Nohara, M. & Takagi, H. Melting of antiferromagnetic ordering in spinel oxide CoAl2O4 . J. Phys. Condens. Matter 19, 145265 (2007).
Fritsch, V. et al. Spin and orbital frustration in MnSc2S4 and FeSc2S4 . Phys. Rev. Lett. 92, 116401 (2004).
Smart, J. S. Effective Field Theories of Magnetism (W. B. Saunders Company, Philadelphia, 1966).
Diep, H. T. & Kawamura, H. First-order phase transition in the fcc Heisenberg antiferromagnet. Phys. Rev. B 40, 7019–7022 (1989).
Minor, W. & Giebultowicz, T. Studies of FCC Heisenberg antiferromagnets by Monte Carlo simulation on large spin arrays. J. Phys. Colloq. 49, 1551 (1988).
Alonso, J. L. et al. Monte Carlo study of O(3) antiferromagnetic models in three dimensions. Phys. Rev. B 53, 2537–2545 (1996).
Roth, W. L. Magnetic properties of normal spinels with only A–A interactions. J. Physique 25, 507–515 (1964).
Krimmel, A. et al. Magnetic ordering and spin excitations in the frustrated magnet MnSc2S4 . Phys. Rev. B 73, 014413 (2006).
Lyons, D. H., Kaplan, T. A., Dwight, K. & Menyuk, N. Classical theory of the ground spin-state in cubic spinels. Phys. Rev. 126, 540–555 (1962).
Luttinger, J. M. & Tisza, L. Theory of dipole interaction in crystals. Phys. Rev. 70, 954–964 (1946).
Luttinger, J. M. A note on the ground state in antiferromagnetics. Phys. Rev. 81, 1015–1018 (1951).
Lyons, D. H. & Kaplan, T. A. Method for determining ground-state spin configurations. Phys. Rev. 120, 1580–1585 (1960).
Rastelli, E. & Tassi, A. The rhombohedral Heisenberg antiferromagnet: infinite degeneracy of the ground state and magnetic properties of solid oxygen. J. Phys. C 19, L423–L428 (1986).
Garanin, D. A. & Canals, B. Classical spin liquid: Exact solution for the infinite-component antiferromagnetic model on the kagomé lattice. Phys. Rev. B 59, 443–456 (1999).
Mucksch, M. et al. Multi-step magnetic ordering in frustrated thiospinel MnSc2S4 . J. Phys. Condens. Matter 19, 145262 (2007).
We would like to acknowledge R. Shindou, Z. Wang, C. Henley and M. P. A. Fisher for illuminating discussions, as well as T. Suzuki, M. Muecksch and A. Krimmel for sharing their unpublished results. This work was supported by the Packard Foundation (D.B. and L.B.) and the National Science Foundation through grants DMR-0529399 (J.A.) and DMR04-57440 (D.B. and L.B.).
The authors declare no competing financial interests.
About this article
Cite this article
Bergman, D., Alicea, J., Gull, E. et al. Order-by-disorder and spiral spin-liquid in frustrated diamond-lattice antiferromagnets. Nature Phys 3, 487–491 (2007). https://doi.org/10.1038/nphys622
This article is cited by
Scientific Reports (2023)
Scientific Reports (2021)
Frontiers of Physics (2021)
Nature Nanotechnology (2020)