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Phase transitions in the charged compact abelian lattice Higgs model

Malin Palö Forsström

Abstract

We consider the (compact) abelian lattice Higgs model with charge \( k \geq 1 \) and show, using charged Wilson~loop observables and charged versions of the Marcu--Fredenhagen ratio, that this model exhibits several distinct phase transitions. In particular, we show that if \(k = 2\), then the Marcu--Fredenhagen ratio and Wilson~loop observables together can distinguish among three distinct phases of the parameter space, and hence both can be used as order parameters for the model.

Phase transitions in the charged compact abelian lattice Higgs model

Abstract

We consider the (compact) abelian lattice Higgs model with charge and show, using charged Wilson~loop observables and charged versions of the Marcu--Fredenhagen ratio, that this model exhibits several distinct phase transitions. In particular, we show that if , then the Marcu--Fredenhagen ratio and Wilson~loop observables together can distinguish among three distinct phases of the parameter space, and hence both can be used as order parameters for the model.
Paper Structure (18 sections, 16 theorems, 107 equations, 4 figures)

This paper contains 18 sections, 16 theorems, 107 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Contributions of this paper
  4. Structure of paper
  5. Acknowledgements
  6. Preliminaries
  7. Notation and standing assumptions
  8. Polymer expansions
  9. Current expansions
  10. A polymer expansion for the Higgs/confinement phase
  11. Charge 1
  12. Charge $\geqslant 2$
  13. Proof of main results
  14. Upper and lower bounds of Wilson lines and Wilson loops
  15. The Marcu--Fredenhagen ratio
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $m \geqslant 4,$ let $\gamma_n$ and $\gamma_n'$ be as in Figure figure: Wilson lines MF, let $R_n = R n$ and $T_n = T n$, and consider the Marcu--Fredenhagen ratio $r_n$ as defined in eq: ratio MF. Then the following holds.

Figures (4)

  • Figure 1: The paths $\gamma_n$ (solid) and $\gamma_n'$ (dashed) used in the definition of the Marcu--Fredenhagen ratio.
  • Figure 2: A summary of the reuslts of Theorem \ref{['theorem: main result charge nondiv']} and Theorem \ref{['theorem: main result charge div']} in the special case $k = 2 .$
  • Figure 3: The figure above shows the level sets of the function on the right hand side of the function $(e^{\beta}-1)e^{4\kappa}$, which equivalently is the level sets of the upper bound in Proposition \ref{['proposition: confinement']}.
  • Figure 4: The Figure above shows the level sets of the function on the right hand side of \ref{['eq: upper Holder bound 1']}, which is an upper bound of the quantity that needs to be small for Proposition \ref{['proposition: higgs regime']} to be applicable.

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 29 more