Effective lagrangian for the two-dimensional [formula omitted]P 2 model at low temperatures

Abstract

Recent Monte Carlo simulations for the two-dimensional R P N−1 models have shown that the matrix non-linear sigma models (NL σM) have a long-distance behavior which is different from that for the corresponding vector models (S N−1 ). This discrepancy cannot be explained by the renormalization group (RG) applied to the naive continuum lagrangians (O( N)-(vector NL σ) model). Motivated by this fact, we study the low-temperature long-distance effective lagrangian for the two-dimensional R P 2 model by means of the saddle-point (SP) plus renormalization methods. Introducing a matrix order parameter Q via the Hubbard-Stratonovich transformation we analyze the system of a macroscopic effective continuum theory for the Q-field. After taking the O(1)×O(2)-invariant SP for the potential we renormalize the massive fluctuations suppressed by the cut-off. This procedure allows us to find, in addition to the O(3)-model, a new term for the massless modes. Thanks to the new term, it is expected that the infrared RG behavior for the R P 2 model could be distinguished from the S 2 model. The effective lagrangian itself turns out to be perturbatively non-renormalizable. The RG analysis is formulated in the space of general O(2)-invariant non-linear lagrangians with an infinite number of coupling constants. The first-order RG analysis using the same field-renormalization constant as of the O(3)-model suggests that at long distances the effective lagrangian in the vicinity of zero temperature deviates from the O(3)-model.