Z. Phys. B Condensed Matter 84, 285-293 (1991) Condensed Zeitsehrift Matter fLir Physik B ?9 Springer-Verlag 1991 Quantum Heisenberg S= 1 spin glass: effect of anisotropy and ferromagnetic interaction T.K. Kopek* and G. Biittner Theoretische Tieftemperaturphysik, Universit/it Duisburg, Lotharstrasse l, W-4100 Duisburg 1, Federal Republic of Germany Received October 4, 1990; revised version January 22, 1991 A theoretical analysis is given for the phase structure of the infinite-range quantum Heisenberg spin glass with uniaxial anisotropy (D) and coaxial external magnetic field (h) in the presence of ferromagnetic interaction. The thermofield dynamics is used as a substitute for the n- replica trick, including a generalisation of the de Almei- da-Thouless stability analysis. Within a mean-field theory a multiplicity of spin glass phases has been found for the spin value S= 1, including: longitudinal, trans- verse and mixed spin glass phases. Additionally, phases with long range order turn out to exist: collinear and randomly canted ferromagnetic states (ferromagnet and spin glass). Phase transition points and lines are found in the temperature-field plane for various values of the ferromagnetic coupling (Jo) together with irreversibility crossovers. 1. Introduction The last few years saw some advances in determining the influence of a strong anisotropy on the magnetic susceptibility in a number of hexagonal metallic spin glass systems [1-3]. These systems behave either Ising- like or Heisenberg-like depending on the sign and the size of the energy splitting of the magnetic moment ground state, and they may be described by a model in which in addition to the random isotropic Heisenberg exchange interaction a single spin uniaxial anisotropy energy --D(Sz) 2 is added [4, 5], where S= denotes the z component of a spin operator. Theoretically, anisotropic agencies bring about sever- al new features which have been investigated for classical spin models both with the presence of magnetic field [6 8] and local uniaxial anisotropy [9-11] where a mul- tiplicity of phases has been found. In the quantum limit, * Permanent address: Institute for Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 937, PL-50-950 Wroclaw 2, Poland for a large negative anisotropy, D one expects for integer valued spins at low temperatures a condensation in the Sz = 0 state resulting in a non-magnetic spin state accom- panied by the destruction of the spin glass state. More- over the system undergoes a crossover from the Ising-like longitudinal freezing to the X-Y-like freezing of the transverse component both as a function of field and anisotropy as has been demonstrated recently [12] where uniaxial anisotropy leads to a strong modification of the phase diagram of a vector spin glass. For D > 1, ordering is preferred in the longitudinal direction where- as transverse ordering is favoured for D < 0. While deal- ing with uniaxial anisotropy two qualitatively distinct cases have to be distinguished according to the sign of the energy splitting D. For large positive D the system is of the Ising type while in the opposite case the spins prefer to lie in the basal plane, but no direction is then preferred due to the rotational X-u symmetry. The continuous crossover behaviour occurs if one looks at the field-temperature phase diagram of the system for different values of the constant D [12]. For large positive D the system exhibits a typical Almeida-Thouless [13] (AT) behaviour characteristic for the freezing of the lon- gitudinal spin components. For negative D, in turn, the system is of the X- Y type and for small magnetic fields one observes the field-temperature dependence corre- sponding to the Gabay-Toulouse [6, 7] (GT) line indi- cating the freezing of the transverse spin components. At large fields, however, the transverse spin components order and the system crosses over to the longitudinal behaviour determined by the AT-like line. The purpose of this work is to extend our previous analysis of a Heisenberg model with exchange random- ness and both uniaxial anisotropy and applied external magnetic field to the case where both spin glass and ferromagnetic exchange couplings are present. Consider- ation is given to the predictions of the stability analysis, together with their consequences for various susceptibili- ties and order parameters. We are interested in the quan- tum limit and we consider, as previously, the case corre- sponding to the spin dimensionality S= t. In order to
