Spin Glass - Infinite-range Model

Infinite-range Model

The infinite-range model is a generalization of the Sherrington–Kirkpatrik model where we not only consider two spin interactions but -spin interactions, where and is the total number of spins. Unlike the Edwards–Anderson model, similar to the SK model, the interaction range is still infinite. The Hamiltonian for this model is described by:

H = -\sum_{i_1 < i_2 < \cdots < i_r} J_{i_1 \dots i_r} S_{i_1}\cdots S_{i_r}

where have similar meanings as in the EA model. The limit of this model is known as the Random energy model. In this limit, it can be seen that the probability of the spin glass existing in a particular state, depends only on the energy of that state and not on the individual spin configurations in it. A gaussian distribution of magnetic bonds across the lattice is assumed usually to solve this model. Any other distribution is expected to give the same result, as a consequence of the central limit theorem. The gaussian distribution function, with mean and variance, is given as:

P(J_{i_1\cdots i_r}) = \sqrt{\dfrac{N^{r-1}}{J^2 \pi r!}} \exp\left\{-\dfrac{N^{r-1}}{J^2 r!}\left(J_{i_1\cdots i_r} - \dfrac{J_0 r!}{2N^{r-1}}\right)\right\}

The order parameters for this system are given by the magnetization and the two point spin correlation between spins at the same site, in two different replicas, which are the same as for the SK model. This infinite range model can be solved explicitly for the free energy in terms of and, under the assumption of replica symmetry as well as 1-Replica Symmetry Breaking.

\begin{align} \beta f &= \dfrac{\beta^2 J^2 q^r}{4} - \dfrac{r\beta^2 J^2 q^r}{2} - \dfrac{\beta^2 J^2}{4} + \dfrac{\beta J_0 r m^r}{2} + \dfrac{r\beta^2 J^2 q^{r-1}}{4\sqrt{2\pi}} \\ &\qquad + \int \exp\left(-\frac{z^2}{2}\right)\log \left(2\cosh\left(\beta Jz\sqrt{\dfrac{rq^{r-1}}{2}} + \dfrac{\beta J_0 r m^{r-1}}{2}\right)\right) \, \mathrm{d}z \end{align}

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