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I am looking for an exact way of evaluating the partition function of the $q$-state Potts model on a $2D$ grid of size $N \times H$:

$$ \mathcal{Z}(\beta) = \sum_{\left(\sigma_{i, j}\right)_{\substack{1 \leq i \leq N\\1 \leq j \leq H}} \in \left\{1,\,\ldots,\,q\right\}^{N \times H}}\exp\left(\beta\left\{\sum_{\substack{1 \leq i < N\\1 \leq j \leq H}}\mathbf{1}\left[\sigma_{i, j} = \sigma_{i + 1, j}\right] + \sum_{\substack{1 \leq i \leq N\\1 \leq j < H}}\mathbf{1}\left[\sigma_{i, j} = \sigma_{i, j + 1}\right]\right\}\right) $$

More specifically, I am interested in the regime where $N$ is arbitrarily large but $H$ is held constant. The requirement for the exact analytic methods are the following:

  • It should have complexity polynomial in $N$.
  • It should have complexity independent of $q$.
  • However, let us say the complexity in $h$ does not matter in first approximation (it may be exponential or worse).

As a non-expert in analytic methods of statistical physics, I have currently come up with the following:

  • Apply the naive transfer matrix method along the large, size $N$, dimension. This involves matrices of size $q^H \times q^H$, hence does not satisfy the requirement of independence in $q$.
  • By exploiting the symmetries of the transfer matrix, I could reduce that to matrices of size at most $2^HB_H^2$, where $B_H$ (Bell number) is the number of partitions of a set of $H$. This is now independent of the number of states $q$, though quite huge as a function of $H$!

Is is the best one can do indeed? As a non-expert, I have good reasons to doubt that and I would appreciate if more knowledgeable people could share their thoughts!

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