Mapping all classical spin models to a lattice gauge theory

Abstract

In our recent work [Phys. Rev. Lett. 102, 230502 (2009)] we showed that the partition function of all classical spin models, including all discrete standard statistical models and all Abelian discrete lattice gauge theories (LGTs), can be expressed as a special instance of the partition function of a 4-dimensional pure LGT with gauge group Z_2 (4D Z_2 LGT). This provides a unification of models with apparently very different features into a single complete model. The result uses an equality between the Hamilton function of any classical spin model and the Hamilton function of a model with all possible k-body Ising-type interactions, for all k, which we also prove. Here, we elaborate on the proof of the result, and we illustrate it by computing quantities of a specific model as a function of the partition function of the 4D Z_2 LGT. The result also allows one to establish a new method to compute the mean-field theory of Z_2 LGTs with d > 3, and to show that computing the partition function of the 4D Z_2 LGT is computationally hard (#P hard). The proof uses techniques from quantum information.

21 pages, 21 figures; published version

References in corpus (3)

• Error Threshold for Color Codes and Random 3-Body Ising Models
• Completeness of the classical 2D Ising model and universal quantum computation
• Unifying all classical spin models in a Lattice Gauge Theory

Cited by in corpus (14)

• Towards analog quantum simulations of lattice gauge theories with trapped ions
• Coarse graining methods for spin net and spin foam models
• Quantum algorithms for classical lattice models
• Trapped surfaces and emergent curved space in the Bose-Hubbard model
• Computational Overhead of Locality Reduction in Binary Optimization Problems
• A quantum algorithm for additive approximation of Ising partition functions
• The U(1) Lattice Gauge Theory Universally Connects All Classical Models with Continuous Variables, Including Background Gravity
• An algorithmic proof for the completeness of two-dimensional Ising model
• Completeness of classical ${\varphi }^4$ theory on 2D lattices
• A quantum information approach to statistical mechanics
• Quantum simulation of classical thermal states
• Systematic study of the completeness of two-dimensional classical ${\varphi }^4$ theory
• Ising models and topological codes: classical algorithms and quantum simulation
• Quantum information and statistical mechanics: an introduction to frontier