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Ornstein-Zernike decay of Wilson line observables in the free phase of the \( \mathbb{Z}_2 \) lattice Higgs model

Malin Palö Forsström

TL;DR

This work rigorously establishes Ornstein-Zernike-type decay for Wilson line observables in the free phase of the Z2 lattice Higgs model. By combining a high-temperature expansion in the Higgs coupling with a detailed cluster expansion and bounds on vortex configurations, the authors show that in a dilute gas regime (κ small, β large in relation to line length), the Wilson line expectation decays as $\langle W_{\gamma_n} \rangle_{\beta_n,\kappa} \sim \frac{C_{\beta_n,\kappa} e^{-c_{\kappa} |\gamma_n|}}{|\gamma_n|^{\sqrt{m-1}}}$, where $c_{\kappa}$ coincides with the Ising spin–spin decay exponent. The result relies on carefully controlling contributions from minimal vortices, employing Ursell-function factorizations, and proving that error terms are negligible in the dilute regime, thereby connecting the lattice Higgs model behavior to the classical Ornstein-Zernike decay of Ising-type systems. The findings sharpen the understanding of phase structure in lattice gauge theories, providing a rigorous bridge between dilute-gas analyses and Ising OZ asymptotics with explicit polynomial corrections. This has potential implications for binding versus unbinding interpretations of dynamical quarks and for rigorous quantification of correlation decay in lattice gauge observables.

Abstract

In the physics literature, the Wilson line observable is believed to have a phase transition between a region with pure exponential decay and a region with Ornstein-Zernike type corrections. In~\cite{f2024b}, we confirmed the first part of this prediction. In this paper, we complement these results by showing that if \( κ\) is small and \( β\) large compared to the length of the line, then Wilson line expectations have exponential decay with Ornstein-Zernike type behavior.

Ornstein-Zernike decay of Wilson line observables in the free phase of the \( \mathbb{Z}_2 \) lattice Higgs model

TL;DR

This work rigorously establishes Ornstein-Zernike-type decay for Wilson line observables in the free phase of the Z2 lattice Higgs model. By combining a high-temperature expansion in the Higgs coupling with a detailed cluster expansion and bounds on vortex configurations, the authors show that in a dilute gas regime (κ small, β large in relation to line length), the Wilson line expectation decays as , where coincides with the Ising spin–spin decay exponent. The result relies on carefully controlling contributions from minimal vortices, employing Ursell-function factorizations, and proving that error terms are negligible in the dilute regime, thereby connecting the lattice Higgs model behavior to the classical Ornstein-Zernike decay of Ising-type systems. The findings sharpen the understanding of phase structure in lattice gauge theories, providing a rigorous bridge between dilute-gas analyses and Ising OZ asymptotics with explicit polynomial corrections. This has potential implications for binding versus unbinding interpretations of dynamical quarks and for rigorous quantification of correlation decay in lattice gauge observables.

Abstract

In the physics literature, the Wilson line observable is believed to have a phase transition between a region with pure exponential decay and a region with Ornstein-Zernike type corrections. In~\cite{f2024b}, we confirmed the first part of this prediction. In this paper, we complement these results by showing that if is small and large compared to the length of the line, then Wilson line expectations have exponential decay with Ornstein-Zernike type behavior.

Paper Structure

This paper contains 18 sections, 15 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related papers
  3. Structure of paper
  4. Funding
  5. Background
  6. Notation
  7. Discrete exterior calculus
  8. Paths
  9. Vortices
  10. The abelian lattice Higgs model
  11. The high temperature expansion
  12. The cluster expansion
  13. Ornstein-Zernike decay for the Ising model
  14. Useful upper and lower bounds
  15. Proof of Theorem \ref{['theorem: dilute free phase']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assume that $\kappa >0$ is sufficiently small, and let $(\beta_n)_{n\geqslant 1}$ be a sequence such that $\lim_{n \geqslant 1} |\gamma_n| e^{-8(m-1)\beta_n} < \infty.$ Then, there is $C_{\beta_n,\kappa}$, defined in eq: C def ad with $0< \liminf_{n \to \infty} C_{\beta_n,\kappa} \leqslant \limsup_{ where $c_\kappa$ is the same constant as that for the spin-spin correlation of the Ising model for

Figures (1)

  • Figure 1: The natural bijection between graphs $G \in \mathcal{G}^{|\mathcal{C}|}$ such that $(\eta,\omega),(\eta',\omega),(\eta,\eta') \in E(G)$ and graphs $G' \in \mathcal{G}^{|\mathcal{C}|}$ with $(\eta,\omega),(\eta',\omega) \in E(G')$ and $(\eta,\eta') \notin E(G')$ in the proof of Lemma \ref{['lemma: minimal ursell']}.

Theorems & Definitions (29)

  • Theorem 1.1
  • Proposition 2.1: Proposition 5.8 in f2024
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 19 more