PHYSICAL REVIEW B VOLUME 45, NUMBER 2 1 JANUARY 1992-II Computer-simulation study of a disordered classical spin system in one dimension with lang-range anisotropic ferromagnetic interactions S. Romano Dipartirnento di Fisica "A. Volta" Universita di Pavia, via A. Bassi 6, I-87100 Pavia, Italy (Received 4 April 1991) The present paper considers a classical system, consisting of n-component unit vectors (n = 2 or 3), associated with a one-dimensional lattice (uq~k g Z}, and interacting via a translation- ally invariant pair potential of the long-range, ferromagnetic and anisotropic form W = W~~ (au&, ~us, + biz u~ zua z). Here e is a positive quantity setting energy and tem- perature scales (i.e. , T' = kaT/e), a and b are positive numbers, the larger of which can be set to 1, and uA, g denotes the Cartesian components of the unit vectors. According to the available rigorous results, the system disorders at all finite temperatures when a = b, or n = 3, a = 0, and possesses an ordering transition at finite temperature when b = 0. Approximate arguments and simulation re- sults suggest that the isotropic models (a = b) produce a transition to a low-temperature phase with infinite susceptibility and power-law decay of the correlation function. If this is true, the available correlation inequalities entail that it also happens in the anisotropic but O(2)-invariant case n = 3, 6 = 0. We report here Monte Carlo calculations for this latter potential model; simulation results were found to be consistent with this conjecture, and to suggest that T, ' = 0. 65 + 0. 01. I. INTRODUCTION Over the last twenty years, the study of spin systems associated with a low-dimensional lattice and interacting via long-range potentials has attracted significant theo- retical attention; the present paper continues along this line, addressing a case where the existence of a phase transition can be argued by approximate analytical ar- guments and numerical results. We consider a classical system, consisting of n- component unit vectors (classical spins, n = 2 or 3), associated with a one-dimensional lattice (ut~k 6 Z}, and interacting via a translationally invariant pair po- tential of the long-range, ferromagnetic and, in general, anisotropic form Here E is a positive quantity setting energy and temper- ature scales, (i. e. , T" = 1~T/c), a and b are positive numbers, the larger of which can be set to 1, p & 1, and uy g denotes the Cartesian components of the unit vec- tors. The condition p ) 1 is needed in order to avoid a ground state with an infinite energy per spin. When a = b, the interaction is isotropic in spin space, i.e. , it only depends on their relative orientations. A number of theoretical results are now known, con- cerning the existence, or absence, of a finite-temperature transition to a ferromagnetically ordered phase; it has been proved that the system disorders at all finite tem- peratures when p ) 2, even for n = 1 (Ising model), and that an ordering transition exists when 1 ( p ( 2; the borderline case p = 2 has also been extensively studied. For n = 1, the ordering transition survives4 up to p = 2; more recently, the critical exponents were proved to be mean-field-like for 1 & p & 2, also for more general single-spin distributions. In the isotropic (continuous- spin) cases, an ordering transitions ~~ is known to ex- ist for 1 & p & 2, but not for p ) 2; available corre- lation inequalities entail that this also happens in the extremely anisotropic case n = 3, a = 0. When p = 2, the same theorems imply the existence of an ordering transition in the other extreme case n = 2, 3, b = 0. The transition properties of the isotropic mod- els have also been investigated by other techniques, in- cluding renormalization group (e. g. , Refs. 16 and 17), simulation, 8 and spherical model treatment, which pre- dicts the existence of an ordering transition for 1 p ( 2; rigorous bounds have also been established for correlation functions in the disordered phases. ~ About the interplay between anisotropy and di- mensionality, we point out that, in two dimensions, the nearest-neighbor counterparts of present anisotropic models can produce an ordering transition at finite temperature, whereas this happens for the isotropic models only if 2 ( p ( 4; on the other hand, when n = 2 and p & 4, the system disorders at all finite temper- atures, but possesses a transition to a low-temperature phase with infinite susceptibility, as can be proved us- ing correlation inequalities and the known existence of a Ikosterlitz- Thouless transition for the nearest-neighbor model ~2 As for the isotropic and antiferromagnetic counterparts of the present models [i. e. , + sign in Eq. (1)j, associated with a d-dimensional lattice (d = 1, 2), available rigorous results (e. g. , Refs. 1 and 8 — 11) imply orientational disor- 45 1037 1992 The American Physical Society