Scaling behavior for finite O(n) systems with long-range interaction

Abstract

A detailed investigation of the scaling properties of the fully finite O(n) systems, under periodic boundary conditions, with long-range interaction, decaying algebraically with the interparticle distance r like r(-d-sigma), below their upper critical dimension, is presented. The computation of the scaling functions is done to one loop order in the nonzero modes. The results are obtained in an expansion of powers of sqrt[epsilon], where epsilon=2sigma-d up to O(epsilon(3/2)). The thermodynamic functions are found to depend upon the scaling variable z=RU(-1/2)L(2-eta-epsilon/2), where R and U are the coupling constants of the constructed effective theory, and L is the linear size of the system. Some simple universal results are obtained.

Critical Behavior of Systems With Long-Range Interaction in Restricted Geometry

“…The function F d,σ (y) is well known [2] in the theory of FSS. Finally, let us notice that one can easily see that the expression (6) for z as a function of y verifies the FSS hypotheses and, consequently, all the thermodynamic functions, which indeed are z dependent, do [45]. At the critical temperature T c (i.e.…”

“…It has been found that the scaling properties of the system with SR interaction remains valid also in the case of LR interaction (for a review see e.g Chapters 4 and 5 in [2] and references therein). For finite n a limited number of recent numerical results [30,32,34], as well as few analytical works [32,34,44,45,46] became available.…”

“…Analytically the nonclassical case has been studied first for n = 1 [32] and for n 1 [45,46]. It has been found that, as for the bulk systems, the critical behavior depends on the small parameter ǫ = 2σ − d. The scaling properties of the finite system are not altered by the presence of the LR interaction and can be written as…”

“…Here we present the analytical result for B (B = 1 − M 4 /3M 2 2 , where M 2n is the n-th moment; for some details, see [42] and also [32,45]). Close to the critical point we have…”

“…(A9), that does not reduce the d-dimensional problem to the effective one-dimensional one. The approach proposed in [45] based on the parametrization (A10) is more efficient in the sense that the corresponding final expressions can be handled by analytical means [49]. It has been found [32], using the Monte Carlo method, that the amplitude ratio Q = M 2 2 /M 4 (which is related to B) is a linear function of ǫ (see also Fig.1 and Fig.2), while the analytical result (8) shows an expansion in powers of √ ǫ [32,45].…”