Lee-Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs

We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter $\lambda$ on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic approximation scheme for all $|\lambda|\neq 1$ by Liu, Sinclair, and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle, and on the tantalising question of what happens around $\lambda=1$, where on one hand the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability, and on the other hand the presence of Lee-Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point $\lambda=1$, and more generally on the entire unit circle. For an integer $\Delta\geq 3$ and edge interaction parameter $b\in (0,1)$ we show #P-hardness for approximating the partition function on graphs of maximum degree $\Delta$ on the arc of the unit circle where the Lee-Yang zeros are dense. This result contrasts with known approximation algorithms when $|\lambda|\neq 1$ or when $\lambda$ is in the complementary arc around $1$ of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee-Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs.