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SUMMARY:The Classical XY Model – Vortex- and Random Walk Representations
DTSTART;VALUE=DATE-TIME:20170210T100000Z
DTEND;VALUE=DATE-TIME:20170210T110000Z
DTSTAMP;VALUE=DATE-TIME:20210421T170537Z
UID:indico-event-2155@indico.math.cnrs.fr
DESCRIPTION:A review of results concerning the classical XY model in vario
us dimensions is presented.\n\n \n\nI start by showing that the XY model
does not exhibit any phase transitions in a non-vanishing external magneti
c field\, and that connected spin-correlations have exponential decay. The
se results can be derived from the Lee-Yang theorem.\n\n \n\nSubsequently
\, 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-W
annier duality). This representation is used to explain some essential ide
as underlying the proof of existence of the Kosterlitz-Thouless transition
in the two-dimensional XY model. Remarks on the existence of phase transi
tions accompanied by continuous symmetry breaking and the appearance of Go
ldstone modes in dimension three or higher come next.\n\n \n\nFinally\, I
sketch the random-walk representation of the XY model and explain some co
nsequences thereof – such as convergence to a Gaussian fixed point in th
e scaling limit\, provided the dimension is > 4\; and the behaviour of the
inverse correlation length as a function of the external magnetic field.\
n\n\n\n \n\nhttps://indico.math.cnrs.fr/event/2155/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/2155/
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