Spherical Model - Critical Behaviour

Critical Behaviour

For the critical temperature occurs at absolute zero, resulting in no phase transition for the spherical model. For d greater than 2, the spherical model exhibits the typical ferromagnetic behaviour, with a finite Curie temperature where ferromagnetism ceases. The critical behaviour of the spherical model was derived in the completely general circumstances that the dimension d may be a real non-integer dimension.

The critical exponents and in the zero-field case which dictate the behaviour of the system close to were derived to be

$\alpha = \begin{cases} - \frac{4-d}{d-2} & \ \mathrm{if} \ 2<d<4 \\ 0 & \ if \ d > 4 \end{cases}$

$\gamma = \begin{cases} \frac{2}{d-2} & \ if \ 2 <d<4 \\ 1 & if \ d > 4 \end{cases}$

$\delta = \begin{cases} \frac{d+2}{d-2} & \ if \ 2 < d < 4 \\ 3 & if \ d > 4 \end{cases}$

which are independent of the dimension of d when it is greater than four, the dimension being able to take any real value.

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