Partition sum for $SO(N)$ one-dimensional lattice model 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?
 A: 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.)
