Classical XY Model - General Properties

General Properties

The existence of the thermodynamic limit for the free energy and spin correlations were proved by Ginibre, extending to this case the Griffiths inequality.

Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon proved that the two point spin correlation of the ferromagnetics XY model in dimension, coupling and inverse temperature is dominated by (i.e. has upper bound given by) the two point correlation of the ferromagnetic Ising model in dimension, coupling and inverse temperature

$\langle \mathbf{s}_i\cdot \mathbf{s}_j\rangle_{J,2\beta} \le \langle \sigma_i\sigma_j\rangle_{J,\beta}$

Hence the critical of the XY model cannot be smaller than the double of the critical temperature of the Ising model

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