An FPTAS for the square lattice six-vertex and eight-vertex models at low temperatures

We give the first efficient approximate counting and sampling algorithms for the six-vertex model and the eight-vertex model on regions of the square lattice ℤ2 in the low temperature regime. All previous algorithms for these problems are for high temperature settings, and rely on the rapid mixing of Markov chains. We prove that these natural Markov chains are torpidly mixing (exponentially slowly) in the low temperature settings. Rather than depending on rapid mixing MCMC, our algorithms are obtained by defining a special edge-2-coloring model, and showing an equivalence to (a linear combination of) abstract polymer models. We then prove the convergence of the cluster expansion of these polymer models. This allows us to employ the approach recently developed by Helmuth, Perkins, and Regts [25]. This combined with Barvinok's method [3, 42] via Taylor expansion (zero-free region of log partition function) gives the approximate counting and sampling algorithms. Significantly, these results provide the first counting problems that admit a fully polynomial time approximation scheme (FPTAS) on square lattice graphs but NP-hard to approximate even on bipartite graphs (rather than the weaker #BIS-hardness.)