I'm looking for derivation of explicit form of partition function for $SO(N)$ one-dimensional lattice model. The initial expression is $$ Z = \int \limits_{-\infty}^{\infty}d\sigma_{1}...d\sigma_{N}\text{exp}\left[-\beta\sum_{x = 1}^{L}\sum_{k = 1}^{N}\sigma_{k}(x)\sigma_{k}(x+1)\right]\delta(1-\sum_{k}\sigma_{k}^{2}) $$ I've heard that it is possible to obtain explicit expression for this sum, but don't know how to do that?


1 Answer 1


Yes, this can be done as an application of the transfer ``matrix'' method. You can, for example, find an explicit computation for the XY model (N=2) in the book The theory of magnetism II: Thermodynamics and Statistical Mechanics by Daniel Mattis (p. 75 ff.). Another reference, in which the general case is treated, is Section 9.2 in the book Elements of Phase Transitions and Critical Phenomena by Nishimori and Ortiz. The result is (well, you should replace $\beta$ by $-\beta$ in your case, since you seem to be interested in the antiferromagnetic case ): $$ Z_L^{\mathrm{free}} = \Bigl\{\bigl(\tfrac\beta2\bigr)^{1-N/2}I_{\tfrac N2-1}(\beta)\Bigr\}^{L-1}\,, $$ where $I_{N/2-1}$ is the modified Bessel function of the first kind.

Let me illustrate an alternative approach in the case of the XY model. In that case, writing the spins $\sigma_i=(\cos\theta_i,\sin\theta_i)$, the partition function with free boundary condition can be written $$ Z_L^{\rm free} = (2\pi)^{-L}\int_{[-\pi,\pi]^L} \mathrm{d}\theta_1\cdots\mathrm{d}\theta_L \prod_{i=1}^{L-1} e^{-\beta\cos(\theta_{i+1}-\theta_i)}\,. $$ Now, making the change of variables $\tau_i=\theta_{i+1}-\theta_i$ (say, with the additional variable $\tau_L=\theta_L$), the integrand factorizes and we get $$ Z_L^{\rm free} = \Bigl\{ \tfrac1{2\pi}\int_{[-\pi,\pi]} \mathrm{d}\tau\, e^{-\beta\cos\tau} \Bigr\}^{L-1}\,. $$ Now, you need to remember the definition of the modified Bessel function of the first kind: $$ I_0(x) = \tfrac1{2\pi} \int_{-\pi}^{\pi} e^{x\cos t} \mathrm{d} t\,. $$ The conclusion is $$ Z_L^{\rm free} = \bigl( I_0(-\beta) \bigr)^{L-1}\,. $$ (Of course, you should check the above computation, I might have made some mistakes.)

  • $\begingroup$ Thank You! Yes, the sign is minus. But usually I see the series over the Bessel function. For example, for $O(N )$ model it is $$ Z_{N} = \sum_{l = 0}^{\infty}\frac{N+2l-2}{l!}\frac{\Gamma(N+l-2)}{\Gamma (N-1)}\times b_{l}^{N-1}, $$ where $$ b_{l} = \Gamma \left( \frac{N}{2}\right)\left(\frac{2}{\beta}\right)^{\frac{N}{2}-1}I_{l+\frac{N}{2}-1}(\beta) $$ Why there is only $I_{0}$? $\endgroup$
    – Name YYY
    Jun 5, 2016 at 14:22
  • $\begingroup$ And how about delta-function? $\endgroup$
    – Name YYY
    Jun 5, 2016 at 14:31
  • $\begingroup$ Well, your delta function is just to force the spin to have norm $1$, which is automatic in my computation in terms of the angles. Concerning your first question, I don't know. Where did you read that? The solution I give above coincides with the one in the reference by Mattis I mentioned (obtained using the transfer matrix method), up to an irrelevant constant (he chooses a different high temperature limit), so it seems to be correct. $\endgroup$ Jun 5, 2016 at 14:43
  • $\begingroup$ I know that this result (from my comment) is for $O(N)$ (my professor told me). I'm surprised that the diffeence is so huge. $\endgroup$
    – Name YYY
    Jun 5, 2016 at 14:54
  • $\begingroup$ I added an alternative reference and stated the general-$N$ result, which is established there (and coincides with the formula I derive for the $N=2$ case above). I also made my notations match yours (using $L$ for the number of spins and reserving $N$ to parametrize the dimension of the spins). $\endgroup$ Jun 5, 2016 at 15:17

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