The complexity of approximating the complex-valued Ising model on bounded degree graphs

We study the complexity of approximating the partition function $Z_{\mathrm{Ising}}(G; \beta)$ of the Ising model in terms of the relation between the edge interaction $\beta$ and a parameter $\Delta$ which is an upper bound on the maximum degree of the input graph $G$. Following recent trends in both statistical physics and algorithmic research, we allow the edge interaction $\beta$ to be any complex number. Many recent partition function results focus on complex parameters, both because of physical relevance and because of the key role of the complex case in delineating the tractability/intractability phase transition of the approximation problem. In this work we establish both new tractability results and new intractability results. Our tractability results show that $Z_{\mathrm{Ising}}(-; \beta)$ has an FPTAS when $\lvert \beta - 1 \rvert / \lvert \beta + 1 \rvert < \tan(\pi / (4 \Delta - 4))$. The core of the proof is showing that there are no inputs~$G$ that make the partition function $0$ when $\beta$ is in this range. Our result significantly extends the known zero-free region of the Ising model (and hence the known approximation results). Our intractability results show that it is $\mathrm{\#P}$-hard to multiplicatively approximate the norm and to additively approximate the argument of $Z_{\mathrm{Ising}}(-; \beta)$ when $\beta \in \mathbb{C}$ is an algebraic number such that $\beta \not \in \mathbb{R} \cup \{i, -i\}$ and $\lvert \beta - 1\rvert / \lvert \beta + 1 \rvert > 1 / \sqrt{\Delta - 1}$. These are the first results to show intractability of approximating $Z_{\mathrm{Ising}}(-, \beta)$ on bounded degree graphs with complex $\beta$. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model.