The elementary excitations in the XY model are spin waves and vortices. The work of Vitelli and Ari Turner reveals that the vortices on a curved surface are coupled to the curvature in a way that reflects the conformal anomaly [1]. In this talk, I will first review the basics of the XY model and some essential concepts of differential geometry. Then, I will construct the XY model on an arbitrarily curved surface [2,3]. Finally, I will analyze the effects of curvature on vortices and discuss the underlying physics.

Main idea

[1] Vincenzo Vitelli and Ari M. Turner. Anomalous Coupling Between Topological Defects and Curvature. (PRL 93, 215301 (2004))

Details

[2] Vincenzo Vitelli and David R. Nelson. Defect generation and deconfinement on corrugated topographies. (PRE 70, 051105(2004))

[3] D. R. Nelson and L. Peliti. Fluctuations in membranes with crystalline and hexatic order. (Journal de Physique, 48, 7(1987))