Current expansion and couplings for Ising lattice gauge theory

TL;DR

The paper develops a random current expansion for Ising lattice gauge theory on $\mathbb{Z}^n$ that yields surface-based current representations on plaquettes. It introduces a surface-analogue switching lemma and constructs couplings with the high-temperature expansion and the random-cluster model, enabling Wilson loop expectations to be expressed as probabilities in graphical representations. The results include monotonicity and positivity properties, Griffiths' inequalities, an area-law bound for small $\beta$, and exponential decay of correlations in both extremes of the coupling parameter, with broader implications for the understanding of lattice gauge theories via surface-based combinatorial representations. The work also clarifies connections to independent developments and provides a versatile set of tools for analyzing Wilson loop observables through multiple graphical frameworks.

Abstract

In this note, we discuss a random current expansion and a switching lemma for Ising lattice gauge theory at all choices of inverse temperature $β$, leading to summation over surfaces. We also describe couplings of this expansion with other representations, including the high-temperature expansion and the cluster expansion. We deduce some simple consequences, including several expressions for the Wilson loop expectation (at any $β$), a new proof of the area law estimate for sufficiently small \( β\), and a proof of exponential decay of correlations for small and large \( β. \) We also derive a few results analogous to corresponding results for the Ising model. In particular, we show that the Wilson loop expectation is non-negative at any $β$ and give an alternative short proof of Griffith's second inequality and, as a consequence, show that the Wilson loop expectations are increasing in \( β\) for all $β$.

Figures (1)

• Figure 1: Above, we draw a graph summarizing the couplings between Ising lattice gauge theory, the random current model, the high-temperature expansion, and the random cluster model.

Theorems & Definitions (27)

• Remark
• Theorem 1.1: Current expansion
• Lemma 1.2: Switching lemma
• Remark
• Remark
• Theorem 1.3
• Remark
• Remark
• proof : Proof of Theorem \ref{['proposition: random currents ALGT Z2']}
• proof : Proof of Lemma \ref{['lemma: the switching lemma']}