Séminaire de Physique Théorique

The Classical XY Model – Vortex- and Random Walk Representations

by Prof. Jürg FRÖHLICH (ETH Zurich, Suisse & IHES)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


35, route de Chartres, F-91440 Bures-sur-Yvette (France)

A review of results concerning the classical XY model in various dimensions is presented.


I start by showing that the XY model does not exhibit any phase transitions in a non-vanishing external magnetic field, and that connected spin-correlations have exponential decay. These results can be derived from the Lee-Yang theorem.


Subsequently, I study the XY model in zero magnetic field: The McBryan-Spencer upper bound on spin-spin correlations in two dimensions is derived. The XY model is then reformulated as a gas of vortices of integer vorticity (Kramers-Wannier duality). This representation is used to explain some essential ideas underlying the proof of existence of the Kosterlitz-Thouless transition in the two-dimensional XY model. Remarks on the existence of phase transitions accompanied by continuous symmetry breaking and the appearance of Goldstone modes in dimension three or higher come next.


Finally, I sketch the random-walk representation of the XY model and explain some consequences thereof – such as convergence to a Gaussian fixed point in the scaling limit, provided the dimension is > 4; and the behaviour of the inverse correlation length as a function of the external magnetic field.