Ferromagnetism of the semi-simple cubic lattice
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Ferromagnetism of the semi-simple cubic lattice M. D. Kuz’min1,a , R. O. Kuzian2,3 , J. Richter4,5 1 IM2NP-CNRS UMR 7334, Aix-Marseille Université, Campus St. Jérôme, Case 142, 13397 Marseille,
France
2 Institute for Problems of Materials Science NASU, Krzhizhanovskogo 3, Kiev 03180, Ukraine 3 Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, San Sebastian/Donostia 20018,
Basque Country, Spain
4 Institut für Physik, Otto-von-Guericke-Universität Magdeburg, PF 4120, 39016 Magdeburg, Germany 5 Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
Received: 21 July 2020 / Accepted: 27 August 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Heisenberg ferromagnet on a lattice with a low coordination number, Z = 3, has been studied by means of high-temperature series and harmonic spin-wave expansion. The lattice is constructed by removing every second bond from the simple cubic lattice and therefore called ’semi-simple cubic’; it is topologically similar to the Laves graph, alias K 4 crystal. The openness of the lattice does not prevent ferromagnetic ordering and the thermal dependence of spontaneous magnetization differs little from that of other common lattices with higher Z . The study extends naturally toward a more general model where the bonds previously removed are now reinstated but endowed with a distinct exchange integral, J2 . We concentrate on the more interesting frustrated case, J2 < 0 < J1 , and a first prediction √ in this direction is that ferromagnetism disappears at J2 /J1 = 2 2 − 3 = −0.172, giving way to a long-wavelength spiral structure propagating along [111].
1 Introduction A Heisenberg model ferromagnet cannot possess a nonzero Curie temperature, TC , unless the supporting lattice is more than two-dimensional [1]. Excluding for the moment exotic systems whose dimensionality is fractional and/or greater than three, one can say that all Heisenberg ferromagnets are three-dimensional (3D). Three-dimensionality is a necessary but not a sufficient condition of a nonzero TC ; one cannot a priori rule out the existence of very open 3D structures that do not order magnetically. A suitable object to check would be a 3D lattice with a low coordination number, Z = 3. All 3D lattices having Z ≥ 4 can host Heisenberg ferromagnets. However, for Z = 4 (diamond lattice) the usual techniques of determining TC based on high-temperature series converge very slowly, especially for S = 1/2 [2,3]. This makes one speculate that perhaps a more open system with Z = 3 would not order at all. In any case, a lattice with Z = 2 cannot be but a linear chain, with no ferromagnetic order possible at T > 0. So the system with Z = 3 is an interesting non-trivial border case. (Here Z is used as a measure of openness, which should not to be confused with
a Corresponding author
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Eur. Phys. J. Plus
(2020) 135:750
Fig. 1 Semi-simple cubic lattice
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