Ising entropy of a finite L_1 x L_2 lattice - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:56:36Z http://mathoverflow.net/feeds/question/103962 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103962/ising-entropy-of-a-finite-l-1-x-l-2-lattice Ising entropy of a finite L_1 x L_2 lattice Peter Grassberger 2012-08-04T16:57:20Z 2012-09-28T15:33:16Z <p>We know the entropy per site of the 2-d Ising model from Onsager's solution. Has anybody also calculated the entropy for a finite rectangle of size L_1 x L_2 with periodic boundary conditions (i.e. on a torus)? In particular, what is the entropy (per site) on an infinitely long cylinder of circumference L?</p> http://mathoverflow.net/questions/103962/ising-entropy-of-a-finite-l-1-x-l-2-lattice/108353#108353 Answer by jc for Ising entropy of a finite L_1 x L_2 lattice jc 2012-09-28T15:32:42Z 2012-09-28T15:32:42Z <p>I had a fairly useless comment posted earlier which I apologize for. It turns out much is known though. Below everything is restricted to zero applied field.</p> <p>For a finite square grid of size $m$ by $n$ with periodic boundary conditions, the expression for the partition function $Z$ (from which the entropy per site can be worked out) was given already by <a href="http://prola.aps.org/abstract/PR/v76/i8/p1232_1" rel="nofollow">Kaufmann</a> (note that you can find freely accessible copies of this paper through google) very shortly after Onsager's solution:</p> <p><code>$Z=\frac{1}{2}(2\sinh(2H))^{mn/2}\left\{\prod_{r=1}^n\left(2\cosh(\frac{m}{2}\gamma_{2r-1})\right)+\prod_{r=1}^n\left(2\sinh(\frac{m}{2}\gamma_{2r-1})\right)+\prod_{r=1}^n\left(2\cosh(\frac{m}{2}\gamma_{2r})\right)+\prod_{r=1}^n\left(2\sinh(\frac{m}{2}\gamma_{2r})\right)\right\}$</code></p> <p>where, $H=J/k_BT$, $H'=J'/k_BT$, $\cosh\gamma_j=\cosh(2H)\cosh(2H')-\sinh(2H)\sinh(2H')\cos(\pi j/n)$, with $J,J'$ being the coupling constants in the horizontal and vertical directions, respectively.</p> <p>A fuller explanation is in chapter IV of McCoy and Wu's book "The Two-dimensional Ising model". Another way to derive the partition function which has proved useful for <a href="http://stacks.iop.org/ja/35/5189" rel="nofollow">generalizations to antiperiodic boundary conditions</a> was worked out in <a href="http://www.springerlink.com/content/t277405172q6n523/" rel="nofollow">this paper by V.N. Plechko</a>.</p> <p>On an infinitely long cylinder with circumference $L$ I was able to find in a <a href="http://arxiv.org/abs/cond-mat/0407731v3" rel="nofollow">2004 paper by Huang et al</a> an expression for the free energy (with $J=J'=1$):</p> <p>$f=-\frac{1}{2}\ln(4z)-\frac{1}{2L}\sum_p\int_0^{2\pi}\frac{d\phi}{2\pi}\ln\left[z+z^{-1}-\Phi_p(\phi)\right]$</p> <p>where now $z=\sinh(2\beta)$ and $\Phi_p(\phi)=\cos\phi+\cos\frac{2\pi p}{L}$.</p>