Classical XY Model in 1.99 Dimensions.

We consider the classical XY model (O(2) nonlinear σ-model) on a class of lattices with the (fractal) dimensions 1 < D < 2. The Berezinskii’s harmonic approximation suggests that the model undergoes a phase transition in which the low temperature phase is characterized by stretched exponential decay of correlations. We prove an exponentially decaying upper bound for the two-point correlation functions at nonzero temperatures, thus excluding the possibility of such a phase transition. The classical XY model (or, equivalently, O(2) nonlinear σ-model) has been a subject of considerable interest in the contexts of statistical physics and relativistic field theory. The model exhibits a standard ferromagnetic phase transition in dimensions d > 2, while it undergoes an exotic phase transition called the Berezinskii-Kosterlitz-Thouless transition [1, 2, 3] in d = 2. The existence of a phase transition in two dimensions is in a remarkable contrast with the O(n)-rotator models (σ-models) with n ≥ 3 such as the classical Heisenberg model, which are conventionally believed to be asymptotically free and have no phase transitions in two dimensions [4]. In one dimension, general arguments guarantee that the XY model (like all the other short range spin systems) has no phase transitions. It is now common to regard the dimension d as a continuous parameter, having in mind lattices with fractal structures for example. Then a natural question is whether the classical XY model exhibits a (finite temperature) phase transition in the intermediate dimensions 1 < d < 2. Although there have been no publications directly devoted to this problem as far as we know, we believe that the problem is important and is worth settling. To illustrate that the problem is nontrivial, denote by Tc(d) the critical temperature (at which the susceptibility diverges) of the classical XY model on the d-dimension hypercubic lattice. The rigorously established facts are that Tc(d) is finite for d = 2, 3, 4, · · ·, and is vanishing for d = 1. If one naively regards Tc(d) as a function of the continuous parameter d, then the most natural “interpolation” may be that Tc(d) continuously decreases below d = 2, and vanishes continuously at d = 1 (or some dimension 1 < d < 2). A much stronger support for this naive guess comes from the observation that a straightforward extension of the Berezinskii’s harmonic approximation [1] (which correctly predicts the existence of the Kosterlitz-Thouless transition in d = 2) indicates that, in 1 < d < 2, there can be an exotic low temperature phase characterized by stretched exponential decay of correlations. One who came across with this argument might conjecture the existence of a phase transition in 1 < d < 2. 1 Published in Phys. Rev. Lett. 74, 3916 (1995).