Phases of Two-Dimensional O(N) Models

Abstract

In this thesis we study the phase structures of classical O(N) spin models on a two-dimensional lattice, focusing on the cases N ≤ 3. In the case of N = 1, there are two phases separated by a second order phase transition at a critical inverse temperature βc. For β < β_c, the model is disordered (the magnetization vanishes), while for β > β_c, the model exhibits long range order (LRO) (the magnetization is non-zero, and the spin-spin correlation approaches a non-zero constant at large separation). The models with N = 2, 3 do not exhibit LRO for any temperature as a result of the Mermin-Wagner theorem. When N = 2, there are two distinct phases separated by critical inverse temperature β_BKT. The high temperature phase β < β_BKT exhibits finite correlation length, which diverges as β → β_BKT , while the low temperature phase β > β_BKT (known as the Berezinskii-Kosterlitz- Thouless - or BKT - phase) exhibits an infinite correlation length. For N = 3, it is generally believed that there is only one phase in which the correlation length is always finite. Some authors have questioned this conclusion, arguing that the O(3) model has a BKT phase similar to the O(2) model. In this thesis we look closely at this issue by studying the O(2) and O(3) models — as well as some discretized versions of these models — numerically. Our results provide a picture of the phase structure in these models. Our work provides strong evidence for the prevailing theory that there is no phase transition in the O(3) model.