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A Cellular Representation of the Potts Lattice Higgs Model

Summer Eldridge, Malin P. Forsström, Benjamin Schweinhart

Abstract

The $i$-dimensional Potts lattice Higgs model is a random assignment of spins in $\mathbb{Z}_q$ to the $i$-dimensional cells of a cell complex induced by a Hamiltonian with a Potts interaction on the $(i+1)$-cells and an additional term playing the role of an external field. We develop a representation of this model as a pair of dependent plaquette percolations, and prove that Wilson line expectations can be expressed in terms of the probability of a topological event. As an application, we prove the existence of a phase transition for the Marcu--Fredenhagen ratio in the Potts lattice Higgs model on $\mathbb{Z}^d$ when $i=1.$

A Cellular Representation of the Potts Lattice Higgs Model

Abstract

The -dimensional Potts lattice Higgs model is a random assignment of spins in to the -dimensional cells of a cell complex induced by a Hamiltonian with a Potts interaction on the -cells and an additional term playing the role of an external field. We develop a representation of this model as a pair of dependent plaquette percolations, and prove that Wilson line expectations can be expressed in terms of the probability of a topological event. As an application, we prove the existence of a phase transition for the Marcu--Fredenhagen ratio in the Potts lattice Higgs model on when
Paper Structure (20 sections, 27 theorems, 146 equations, 5 figures)

This paper contains 20 sections, 27 theorems, 146 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions and Notation
  3. Main Results
  4. Topological Terminology and Techniques
  5. Absolute Homology and Cohomology
  6. Relative Homology and Cohomology
  7. Mayer--Vietoris Sequences
  8. Alexander Duality
  9. The coupling between the CPP and the Potts lattice Higgs model
  10. Derivation of the Coupling
  11. Topological Interpretation of Wilson Line Observables
  12. Potts Lattice Higgs in General Gauge
  13. Equivalence with the Ghost Vertex Construction
  14. Properties of the CPP
  15. Special Cases of the CPP
  16. ...and 5 more sections

Key Result

Theorem 5

Let $X$ be a finite cell complex, $q$ be a prime integer, and $\beta_2,\beta_1\geq 0.$ If $p_2=1-e^{-{\beta_2}},$$p_1=1-e^{-{\beta_1}}$, $k_2=\frac{p_2}{1-p_2}=e^{\beta_2}-1,$ and $k_1=\frac{p_1}{1-p_1}=e^{\beta_1}-1$, then there is a coupling $\kappa=\kappa_{k_2,k_1,q,i,X}$, ${\kappa \colon C^i(X;\ which satisfies the following.

Figures (5)

  • Figure 1: Two illustrations of the event $V_{\gamma}.$ On the left, $P_2$ is depicted by the collection of light blue squares and $P_1$ by the orange bonds. The events $V_{\gamma_1}$ and $V_{\gamma_2}$ both occur, where $\gamma_1$ and $\gamma_2$ are the loop and path shown in dark gray. On the right, $P_2$ is shown by the blue surface and $P_1$ by the orange line segment. $\gamma$ is depicted in black. Here, $V_{\gamma}$ occurs when homology is taken with coefficients in $\mathbb{Z}_2$ but the plaquettes in the orange surface cannot be compatibly oriented to yield a null-homology when coefficients are taken in a field of odd characteristic. The figure on the right was adapted from Figure 1 of duncan2025sharp which was in turn inspired by Figure 1 of aizenman1983sharp.
  • Figure 2: In the figures above, we draw the paths and surfaces used in the definitions of the Marcu--Fredenhagen ratio for $i = 1$ and $i = 2$ respectively.
  • Figure 3: Conjectured limiting behavior of $R\left( \beta_2,\beta_1,n \right)$. We will not address the difference between the Confinement phase and the Higgs phase here. We propose to rigorously establish the diagram in the blue regions.
  • Figure 4: A cell complex with six vertices, seven edges, and one face.
  • Figure :

Theorems & Definitions (61)

  • Definition 1: Potts lattice Higgs Model
  • Definition 2: Percolation Subcomplex
  • Definition 3
  • Definition 4: Coupled Plaquette Percolation (CPP)
  • Theorem 5
  • Definition 6: $V_\gamma$ and $W_\gamma$
  • Theorem 7
  • Definition 8: The Marcu--Fredenhagen ratio for $i = 1$
  • Definition 9: The topological Marcu--Fredenhagen ratio
  • Theorem 10
  • ...and 51 more