Mapping between Spin-Glass Three-Dimensional (3D) Ising Model and Boolean Satisfiability Problem

TL;DR

This paper establishes a detailed mapping between the complex spin-glass 3D Ising model and the Boolean satisfiability problem, revealing deep connections related to topological structures and entanglements.

Contribution

It proves the existence of the absolute minimum core in the 3D Ising model and demonstrates its equivalence to K-SAT problems for specific K values.

Findings

The AMC model exists in the 3D Ising model.

The 3D Ising model maps to K-SAT for K ≥ 4.

The AMC model is equivalent to K-SAT for K=3.

Abstract

The common feature for a nontrivial hard problem is the existence of nontrivial topological structures, non-planarity graphs, nonlocalities, or long-range spin entanglements in a model system with randomness. For instance, the Boolean satisfiability (K-SAT) problems are nontrivial, due to the existence of non-planarity graphs, nonlocalities, and the randomness. In this work, the relation between a spin-glass three-dimensional (3D) Ising model with the lattice size N = mnl and the K-SAT problems is investigated in detail. With the Clifford algebra representation, it is easy to reveal the existence of the long-range entanglements between Ising spins in the spin-glass 3D Ising lattice. The internal factors in the transfer matrices of the spin-glass 3D Ising model lead to the nontrivial topological structures and the nonlocalities. At first, we prove that the absolute minimum core (AMC) model exists in the spin-glass 3D Ising model, which is defined as a spin-glass 2D Ising model interacting with its nearest neighboring plane. Any algorithms, which use any approximations and/or break the long-range spin entanglements of the AMC model, cannot result in the exact solution of the spin-glass 3D Ising model. Second, we prove that the dual transformation between the spin-glass 3D Ising model and the spin-glass 3D Z2 lattice gauge model shows that it can be mapped to a K-SAT problem for K > = 4 also in the consideration of random interactions and frustrations. Third, we prove that the AMC model is equivalent to the K-SAT problem for K = 3.