On the possible existence of modulated phases in the random chessboard model

Abstract

Abstract The random chessboard Hamiltonian describes a spin system in D = 2 dimensions with two Ising-like variables per site that interact via a next-nearest neighbor ( k 2 ) and a four-spin (Γ 0 ) coupling constants. In this work we present an improvement of the early considered renormalization group-flow equations (RGFE) for this model. We introduce an unsteady basis of six marginal operators that diagonalize the Kadanoff operator algebra coefficients with the instantaneous four-spin Hamiltonian. The equations of motion are formulated in this ‘noninertial’ frame in a standard way and afterwards they are written using a fixed frame of marginal operators. The resulting nonlinear first-order differential equation is studied. We find that after a ‘time’ l = l c , the renormalized coupling become complex numbers. For l →∞ the renormalized Hamiltonian iterates to a double eight-vertex fixed line. As the approach to this limit occurs with oscillations of ground frequency ω ≅ Γ 0 , we deduce the possible existence of a modulated phase. On the other hand, analyzing the properties of the full set of solutions of RGFE we obtain that the model could have at least eight nonuniversal ordered phases in the small Γ 0 limit, four of these being modulated and the remaining four nonmodulated.