1
$\begingroup$

I'm interested in simulating the continuum limit of the 2D Ising model $$H=J\sum_{\langle i j\rangle} s_i s_j+ h \sum_i s_i$$

In one dimension I can fix average magnetization $m=\langle s\rangle$ and correlation length $\xi=a\Delta i$, $a$ being the lattice spacing, and take $a\to 0$. Because everything is exactly solvable, I can solve for $\beta J$ and $\beta h$ and expand about $a=0$ to find:

$$\beta J=-\frac{1}{2}\log(a)-\frac{1}{4}\log\left(\frac{1-m^2}{4\xi^2}\right)+O(a^2)$$ $$\beta h=a \frac{m}{2\xi}+O(a^3)$$

As I take $a\to 0$ I get a continuum theory with fixed correlation length and average magnetization. The logarithm is because the fixed point is at $(\beta h,\beta J)=(0,\infty)$.

In two dimensions, we have critical point $(\beta h,\beta J)=(0,J_c)$ and lots of great information given to us by critical exponent relationships like $\xi\propto \tau^{-\nu}$. But I haven't been able to figure out how to do the same thing.

How can I take the $a\to 0$ limit with fixed (not necessarily known) $m$ and $\xi$? Is this essentially the same question as asking how to take the $\xi\to \infty$ limit at fixed $\langle s\rangle$? I assume the relationships $\xi\propto \tau^{-\nu}$, $m\propto(-\tau)^\beta$, and $h\propto m^\delta|_{\tau=0}$ will be needed (where this takes the viewpoint that $\xi\to\infty$ instead of the equivalent $a\to 0$ picture).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.