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.)

We initiate a study of the classification of approximation complexity of the eight-vertex model defined over 4-regular graphs. The eight-vertex model, together with its special case the six-vertex model, is one of the most extensively studied models in statistical physics, and can be stated as a problem of counting weighted orientations in graph theory. Our result concerns the approximability of the partition function on all 4-regular graphs, classified according to the parameters of the model. Our complexity results conform to the phase transition phenomenon from physics. We introduce a quantum decomposition of the eight-vertex model and prove a set of closure properties in various regions of the parameter space. Furthermore, we show that there are extra closure properties on 4-regular planar graphs. These regions of the parameter space are concordant with the phase transition threshold. Using these closure properties, we derive polynomial time approximation algorithms via Markov chain Monte Carlo. We also show that the eight-vertex model is NP-hard to approximate on the other side of the phase transition threshold.

We study the approximation complexity of the partition function of the eight-vertex model on general 4-regular graphs. For the first time, we relate the approximability of the eight-vertex model to the complexity of approximately counting perfect matchings, a central open problem in this field. Our results extend those in arXiv:1811.03126 [cs.CC]. In a region of the parameter space where no previous approximation complexity was known, we show that approximating the partition function is at least as hard as approximately counting perfect matchings via approximation-preserving reductions. In another region of the parameter space which is larger than the previously known FPRASable region, we show that computing the partition function can be reduced to (with or without approximation) counting perfect matchings. Moreover, we give a complete characterization of nonnegatively weighted (not necessarily planar) 4-ary matchgates, which has been open for several years. The key ingredient of our proof is a geometric lemma. We also identify a region of the parameter space where approximating the partition function on planar 4-regular graphs is feasible but on general 4-regular graphs is equivalent to approximately counting perfect matchings. To our best knowledge, these are the first problems of this kind.

We take the first step toward a classification of the approximation complexity of the six-vertex model. This is a subject of extensive research in statistical physics. Our result concerns the approximability of the partition function on 4-regular graphs, classified according to the parameters of the model. Our complexity results conform to the phase transition phenomenon from physics. We show that the approximation complexity of the six-vertex model behaves dramatically differently on the two sides separated by the phase transition threshold. Furthermore, we present structural properties of the six-vertex model on planar graphs for parameter settings that have known relations to the Tutte polynomial T(G; x, y).

In this paper, we study the mixing time of two widely used Markov chain algorithms for the six-vertex model, Glauber dynamics and the directed-loop algorithm, on the square lattice Z^2. We prove, for the first time that, on finite regions of the square lattice these Markov chains are torpidly mixing under parameter settings in the ferroelectric phase and the anti-ferroelectric phase.