Critical behavior of two-dimensional models with spatially modulated phases: Analytic results

Abstract

The two-dimensional Elliott [or axial next-nearest-neighbor Ising (ANNNI)] model is mapped into an eight-vertex model with direct and staggered fields. With the use of the transfer-matrix approach it is shown that the dual of the ANNNI model belongs to the universality class of the one-dimensional quantum $\mathrm{XY}$ model in a staggered field at $T=0\mathrm{}$. The phase structure is investigated by high- and low-temperature expansions of the correlation length and by spin-wave-like approximations valid in first order at low and high temperatures, respectively. The fact that the phase diagram obtained at low temperatures agrees qualitatively with recent results by Villain and Bak and by Coppersmith et al. shows that the paramagnetic phase extends until $T=0\mathrm{}$. The role of the umklapp scattering in determining the critical wave vector in the modulated phase and in stabilizing the $〈2〉$ antiphase is pointed out. In the eight-vertex representation the critical indices are identified in the floating, massless phase. The dislocations destabilizing this incommensurate phase correspond to the energy operator of the eight-vertex model. Finally, it is argued that the apparent contradiction between the low-temperature results on one hand, and the Monte Carlo simulations and high-temperature-expansion results on the other hand, is probably due to the strong oscillatory behavior of spin-spin correlation functions in the massive paramagnetic region.