Dzyaloshinskii-Moriya interaction and magnetic ordering in 1D and 2D at nonzero T

The inclusion of a Dzyaloshinskii-Moriya short-range antisymmetric interaction in the Heisenberg Hamiltonian induces spontaneous magnetization, at nonzero temperatures, in one and two dimensions. It is shown that quantum fluctuations are reduced by the DzyaloshinskiiMoriya interaction, but short-range correlations are increased, thereby allowing the existence of long-range magnetic order in these low-dimensional systems. Copyright c © EPLA, 2014 Introduction. – In 1966 Mermin and Wagner [1] proved a most relevant theorem, namely that “For oneor two-dimensional Heisenberg systems with isotropic interactions, and such that the interactions are short ranged, namely, which satisfy the condition ∑ R R2|J(R)| < +∞, (1) there can be no spontaneous ferroor antiferromagnetic long-range order at T > 0”. With only very few rigorous results available this theorem constitutes a most valuable piece of knowledge, especially to test the validity of the usual approximate results. The validity of the theorem was extended, also using the Bogoliubov inequality [2], to classical interacting particles by Mermin [3], and to fermion and boson systems by Hohenberg [4]. In 2001 Bruno [5] extended these results even further, to long-range RKKY interactions. More precisely, as formulated by Bruno as a corollary, “A D-dimensional (D = 1 or 2) Heisenberg or XY system with interactions monotonically decaying as |J(R)| ∝ R−α, with α ≥ 2D, cannot be ferroor antiferromagnetic”. Physically, it is the fluctuations that prevent the onset of long-range order competing against the correlations induced by short-range interactions. Therefore, the long-range magnetic order observed in oneor two-dimensional systems could be due, for instance, to magnetic anisotropies or external magnetic fields. In this paper we present an alternative approach, where spontaneous ordering of low-dimensional magnetic systems is due to the symmetry breaking that the shortrange Dzyaloshinskii-Moriya (DM) interaction [6] generates, and which to the best of our knowledge has not been reported in the literature. The physical basis for the Mermin-Wagner theorem seems to be the existence of degrees of freedom that are not constrained by an interaction, which makes the fluctuations strong enough to prevent long-range order. However, in spite of the fact that the Mermin-Wagner theorem excludes the possibility of ordering for a wide range of finite-range interactions, we prove below that the DM interaction for the Heisenberg Hamiltonian, in spite of being of short range, leads to spontaneous magnetic order in one and two dimensions, at nonzero temperatures. In fact the DM interaction, by reducing the spin fluctuations, yields a canted spin arrangement which turns out to be stable in one and two dimensions. The Hamiltonian. – The Heisenberg Hamiltonian H0, including a weak Zeeman term, is given by H0 = − ∑ R,R′ J(R−R′)S(R) · S(R′)− h ∑