Unifying all classical spin models in a lattice gauge theory

Area of Science:

• Statistical Mechanics
• Lattice Gauge Theory
• Quantum Information Theory

Background:

• Classical spin models and Abelian discrete lattice gauge theories (LGTs) exhibit diverse behaviors.
• A unified theoretical framework for these models is lacking.
• Understanding the computational complexity of LGTs is crucial for theoretical advancements.

Purpose of the Study:

• To unify diverse classical spin models and Abelian lattice gauge theories (LGTs) into a single framework.
• To develop a novel method for computing the mean-field theory of Abelian LGTs.
• To determine the computational complexity of the 4D Z2 LGT partition function.

Main Methods:

• Expressing the partition function of various classical models as a specific case of the 4D Z2 LGT partition function.
• Applying quantum information techniques for theoretical proofs.
• Utilizing computational complexity theory to analyze the partition function.

Main Results:

• All classical spin models and Abelian LGTs are shown to be special instances of the 4D Z2 LGT.
• A new method for computing the mean-field theory of Abelian LGTs (d >= 4) is established.
• Computing the 4D Z2 LGT partition function is proven to be computationally hard (#P-hard).
• The 4D Z2 LGT is demonstrated to be approximately complete for Abelian continuous models.

Conclusions:

• The 4D Z2 LGT provides a universal model unifying classical statistical and lattice gauge theories.
• The established method offers new computational possibilities for Abelian LGTs.
• The #P-hardness result highlights fundamental computational limitations in analyzing these models.