Numerical Study of Competing Spin-Glass and Ferromagnetic Order

Two and three dimensional random Ising models with a Gaussian distribution of couplings with variance J and non-vanishing mean value J0 are studied using the zero-temperature domain-wall renormalization group (DWRG). The DWRG trajectories in the (J0, J) plane after rescaling can be collapsed on two curves: one for J0/J > rc and other for J0/J < rc. In the first case the DWRG flows are toward the ferromagnetic fixed point both in two and three dimensions while in the second case flows are towards a paramagnetic fixed point and spin-glass fixed point in two and three dimensions respectively. No evidence for an extra phase is found. In some range of concentration of magnetic impurities in a non-magnetic host one observes a competition between spin-glass and ferromagnetic order [1]. The phenomenon can be described by an Ising model in which couplings are distributed randomly with some non-vanishing mean value J0 and width J . This issue has been addressed by Sherrington and Kirkpatrick [2] in the case of a soluble infinite-range model. For the case of insoluble short range model Migdal-Kadanoff real-space rescaling [3], [4] and computer simulations [5]-[13] have been employed. It is established [8],[9] that there is no finite-temperature phase transition in a two-dimensional (2d) Ising spin-glass (random Ising model with J0 = 0). McMillan [7] investigated the 2d random Ising model for the general case, J0 not necessarily zero, using finite-temperature domain-wall renormalization group (DWRG). He found at small temperatures only two phases: ferromagnetic and paramagnetic. However, some later studies [12], [13] have found a finite temperature transition to a “random antiphase state”, which has similar properties to a spin glass. Therefore more study of the subject is desirable. To my knowledge no DWRG study of three dimensional (3d) random Ising model (with J0 6= 0) have been done. In this paper the 2d and 3d random Ising models are investigated using the zero-temperature DWRG [7]-[9]. In particular no evidence for the “random antiphase state” is found. The system is the Edwars-Anderson model [14] of an Ising spin glass with Hamiltonian: