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The Ising model on $\mathbb{Z}^2$ is given by the Hamiltonian $$ H(\sigma)=-\sum_{\{x,y\}}\sigma_x\sigma_y $$ and the Gibbs measure as $$ \frac{\exp(-\beta H(\sigma))}{Z_\beta}\,. $$

There exists an exact solution found by Onsager and re-derived later on by others. My question is:

  1. On Wikipedia there are explicit expressions for internal energy per site and the spontaneous magnetization (below the critical temperature). Are there any explicit formulas for the infinite-volume two point function $$ \mathbb{E}_\beta[\sigma_x\sigma_y]$$ below, at and above the critical temperature $$ \beta_c = \frac{1}{2}\log(1+\sqrt{2})?$$

  2. In case the explicit expressions are too complicated (or too complicated to make conclusions from), I would just like to verify a few things: Above the critical temperature we have exponential decay with respect to $\|x-y\|$. Below it we have long range order and hence $$ \mathbb{E}_\beta[\sigma_x\sigma_y] \geq C \qquad(\beta>\beta_c)\,. $$ What is this constant and is it temperature dependent? Clearly it shouldn't be larger than 1. Moreover, at the critical temperature, it seems like there should be polynomial behavior $$ \mathbb{E}_{\beta_c}[\sigma_x\sigma_y] \approx \|x-y\|^{-1/{2\pi\beta_c}} \approx \|x-y\|^{-0.361152}\,. $$ Is this correct? Is there another expression?

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Are there any explicit formulas for the infinite-volume two point function?

There are indeed explicit expressions. The standard reference is the book by McCoy and Wu; see also this page.

Concerning your other questions:

Above the critical temperature we have exponential decay with respect to $\|x−y\|$.

This is correct. The explicit expression for the rate of exponential decay can be found in the links given above.

What is this constant and is it temperature dependent?

The constant $C$ can be taken to be the square of the spontaneous magnetization; it is indeed temperature dependent (it is a decreasing function of $T$, that tends to $0$ as $T\uparrow T_c$ and to $1$ as $T\downarrow 0$).

Note, however, that the truncated two-point function $\langle\sigma_x\sigma_y\rangle - \langle\sigma_x\rangle\langle\sigma_y\rangle$ decays exponentially for all $T\neq T_c$. Again, the rate of exponential decay os given in the above links.

At the critical temperature, it seems like there should be polynomial behavior

There is indeed algebraic decay, but your exponent is wrong: the correct exponent is $-1/4$.

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  • $\begingroup$ Thanks! So the lower bound on the two point function at high temperature is a simple consequence of the Griffiths inequality I suppose. $\endgroup$
    – PPR
    Aug 31, 2020 at 6:23
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    $\begingroup$ Yes, exactly (it hold at all temperatures, but is interesting at low temperatures). And it is sharp (it is the correct limit as $\|y-x\|\to \infty$. $\endgroup$ Aug 31, 2020 at 6:26

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