Solid State Communications, Vol. 28, pp. 667-670. © Pergamon Press Ltd. 1978. Printed in Great Britain. 0038-1098/78/1122-0667 $02.00•0 ABSENCE OF FERROMAGNETIC LONG RANGE ORDER IN RANDOM ISOTROPIC DIPOLAR MAGNETS AND IN SIMILAR SYSTEMS* A. Aharony Department of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Israel (Received 29 August 1978 by I41.Low) It is shown that random off-diagonal exchange interaction coefficients destroy ferromagnetic long range order in isotropic systems with less than four spatial dimensions. A special case is that of random isotropic systems with dipole-dipole interactions. The proof breaks down when (cubic or hexagonal) symmetry breaking terms are introduced. However, in some cases such terms turn the transition into the ferromagnetic phase first order, with a possible spin glass phase above it. THE EFFECTS of random impurities on the critical properties of magnets have recently drawn much attention [1 ]. In some cases, the existence of impurities leads to the drastic result that ferromagnetic long range order, which exists in the absence of impurities, must completely disappear once such impurities are introduced. This has been shown to be the case when random magnetic fields are introduced in systems with rotational invariance (described by Heisenberg or XY models) in d < 4 dimensions [2]. Very recently, the same result was shown also to be true [3-5] for sys- tems with random uniaxial anisotropy, i.e. with a single ion term I~ i (ni • Si) 2 in the Hamiltonian, where n i is a unit vector of random direction and Si is an n-component spin vector at the site i in the system [6, 7]. The exper- imental situations with such systems is far from clear, since (a) no magnetic systems with random fields are known and no detailed experiments on other such sys- tems are available, (b) the experiments on amorphous rare earth-iron alloys [6], to which the model of ran- dom uniaxial anisotropy is assumed to apply, are not very clear, and (c) the assumption of complete rotational invariance, needed in the proofs of [2-4], is usually not satisfied. The present paper contains two new results, which are aimed at resolving these difficul- ties. First, it is shown that the random uniaxial aniso- tropy is only a very special case of a large variety of systems in which long range order is eliminated. Con- sider a general spin system, with the Hamiltonian n 1 Z E (1) Jf = -2 ia ~,o=I * Supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. with random exchange coefficeints j~t3, such that the configurational averages [J~t~]av maintain the n- dimensional spin space isotropy. We show, that any ran- dom off-diagonal exchange coefficient, J~ with ~ 4: f3, will prevent the system from having ferromagnetic long range order for d < 4. A very important example, which is described by (1) is that of a random isotropic dipolar magnet, with [8] -- (d-- 2)gigiu (8 2 = -- dxijxo/ro)/rli, (2) where r o = r i -- ri and gi/sB is the magnetic moment of the ion at site i. Both the moments gila~ and the locations r i may be random, leading to random values of Ji~#. For isotropic distributions of these random vari- ables, the averages [JJ]av will describe the usual iso- tropic dipolar system [8]. The effects of randomness on this system were recently studied, in d = 4 -- e dimen- sions [9]. It was found, that the isotropic pure dipolar fixed point is unstable with respect to randomness. A "random" stable dipolar fixed point was found, but it was not clear whether this fixed point can be reached from physical initial Hamiltonians. Our present results show, that this fixed point cannot be reached, since no ferromagnetic long range order is possible. Many other examples exist. For example, quenched random anisotropic strains, e7 n , may induce terms in the Hamiltonian of the form [10] Z g e?6s sf, (3) agq6 leading to similar results. It is hoped that this large variety of possible systems will make a detailed exper- imental study of the general statements easier. The second result relates to the assumption of rotational invariance. Real crystals usually have sym- metry breaking terms, e.g. of cubic or "hexagonal" sym- metry [11,121, 667