LeeYang zeros and the complexity of the ferromagnetic Ising Model on boundeddegree graphs
Abstract
We study the computational complexity of approximating the partition function of the ferromagnetic Ising model in the LeeYang circle of zeros given by $\lambda=1$, where $\lambda$ is the external field of the model. Complexvalued 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 in the circular arc 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 LeeYang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point $\lambda=1$; in fact, our techniques apply more generally to the whole unit circle $\lambda=1$. We show #Phardness for approximating the partition function on graphs of maximum degree $\Delta$ when $b$, the edgeinteraction parameter, is in the interval $(0,\frac{\Delta2}{\Delta}]$ and $\lambda$ is a nonreal on the unit circle. This result contrasts with known approximation algorithms when $\lambda\neq 1$ or $b\in (\frac{\Delta2}{\Delta},1)$, and shows that the LeeYang circle of zeros is computationally intractable, even on boundeddegree graphs.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.14828
 Bibcode:
 2020arXiv200614828B
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics
 EPrint:
 38 pages, 1 figure