Renormalization group flow of the two-dimensional Ising model at T<Tc

Fritz Haake, Maciej Lewenstein, Martin Wilkens

Dima E. Feldman, A V Izyumov, Виктор Степанович Доценко

We study the critical properties of the weakly disordered two-dimensional Ising and Baxter models in terms of the renormalization group (RG) theory generalized to take into account replica symmetry breaking (RSB) effects. Recently it has been shown that the traditional replica-symmetric RG flows in the dimension are unstable with respect to the RSB potentials and a new spin-glass type critical phenomena has been discovered (Dotsenko Vik S, Harris B, Sherrington D and Stinchbombe R 1995 J. Phys. A: Math. Gen. 28 3093; Dotsenko Vik S and Feldman D E 1995 J. Phys. A: Math. Gen. 28 5183). In contrast, here it is demonstrated that in the considered two-dimensional systems the renormalization-group flows are stable with respect to the RSB modes. It is shown that the solution of the renormalization group equations with arbitrary starting RSB coupling matrix exhibits asymptotic approach to the traditional replica-symmetric ones. Thus, to leading order the non-perturbative RSB degrees of freedom do not affect the critical phenomena in the two-dimensional weakly disordered Ising and Baxter models studied earlier.

Douglas F. de Albuquerque, I. P. Fittipaldi

A unified effective-field renormalization-group framework (EFRG) for both quenched bond- and site-diluted Ising models is herein developed by extending recent works. The method, as in the previous works, follows up the same strategy of the mean-field renormalization-group scheme (MFRG), and is achieved by introducing an alternative way for constructing classical effective-field equations of state, based on rigorous Ising spin identities. The concentration dependence of the critical temperature, Tc(p), and the critical concentrations of magnetic atoms, pc, at which the transition temperature goes to zero, are evaluated for several two- and three-dimensional lattice structures. The obtained values of Tc and pc and the resulting phase diagrams for both bond and site cases are much more accurate than those estimated by the standard MFRG approach. Although preserving the same level of simplicity as the MFRG, it is shown that the present EFRG method, even by considering its simplest size-cluster version, provides results that correctly distinguishes those lattices that have the same coordination number, but differ in dimensionality or geometry.

Wei-Tou Ni

John P. Bartel

Erik Schwartz Sørensen

Using the mapping of the partition function of the two-dimensional Ising model onto a pfaffian we evaluate the domain wall free energy difference for the pure and disordered Ising model close to the pure fixed point. Using this method very large lattices can be studied exactly and we confirm that disorder even including frustrating interactions indeed are irrelevant close to the pure fixed point. The finite-size renormalization group flow shows a power-law behavior modified by a logarithmic term that dominates for small lattice sizes.