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A quantum N-dimer model

Daniel C. Douglas, Richard Kenyon, Nicholas Ovenhouse, Samuel Panitch, Sri Tata

TL;DR

This work develops a quantum variant of the $n$-dimer model by merging Reshetikhin–Turaev invariants with planar dimer combinatorics on bipartite ribbon graphs. It builds a two- and three-dimensional framework: in 3D, a quantum trace on $n$-webs yields a polynomial invariant, while in 2D planar graphs it produces a palindromic, sign-stable quantum partition function $\mathbf{Z}_q$ via quantum connections and a quantum identity connection. A central result is the $q$-Kasteleyn determinant formula $\mathrm{Kdet}_q(\Phi_q)$ giving $\mathbf{Z}_q$ (up to sign) in planar settings, generalizing the classical determinant approach to higher rank dimers. The paper also defines a local twist variable $X_n(m)$, analyzes the $n=2$ case to obtain loop-density results for the infinite honeycomb, and provides explicit examples (cycle and snake graphs) illustrating generating functions and integral representations. Overall, the framework links topology, representation theory, and statistical mechanics, offering a new route to compute loop statistics and to study higher-rank dimer-like models via determinant–type formulas and skein-theoretic invariants.

Abstract

We study a quantum version of the $n$-dimer model from statistical mechanics, based on the formalism from quantum topology developed by Reshetikhin and Turaev (the latter which, in particular, can be used to construct the Jones polynomial of a knot in $\mathbb{R}^3$). We apply this machinery to construct an isotopy invariant polynomial for knotted bipartite ribbon graphs in $\mathbb{R}^3$, giving, in the planar setting, a quantum $n$-dimer partition function. As one application, we compute the expected number of loops in the (classical) double dimer model for planar bipartite graphs.

A quantum N-dimer model

TL;DR

This work develops a quantum variant of the -dimer model by merging Reshetikhin–Turaev invariants with planar dimer combinatorics on bipartite ribbon graphs. It builds a two- and three-dimensional framework: in 3D, a quantum trace on -webs yields a polynomial invariant, while in 2D planar graphs it produces a palindromic, sign-stable quantum partition function via quantum connections and a quantum identity connection. A central result is the -Kasteleyn determinant formula giving (up to sign) in planar settings, generalizing the classical determinant approach to higher rank dimers. The paper also defines a local twist variable , analyzes the case to obtain loop-density results for the infinite honeycomb, and provides explicit examples (cycle and snake graphs) illustrating generating functions and integral representations. Overall, the framework links topology, representation theory, and statistical mechanics, offering a new route to compute loop statistics and to study higher-rank dimer-like models via determinant–type formulas and skein-theoretic invariants.

Abstract

We study a quantum version of the -dimer model from statistical mechanics, based on the formalism from quantum topology developed by Reshetikhin and Turaev (the latter which, in particular, can be used to construct the Jones polynomial of a knot in ). We apply this machinery to construct an isotopy invariant polynomial for knotted bipartite ribbon graphs in , giving, in the planar setting, a quantum -dimer partition function. As one application, we compute the expected number of loops in the (classical) double dimer model for planar bipartite graphs.

Paper Structure

This paper contains 39 sections, 28 theorems, 123 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Background on quantum topology and the Jones polynomial
  3. The dimer model
  4. The classical (q=1) case
  5. The q=q case: three dimensions
  6. The q=q case: two dimensions
  7. Probability
  8. Preliminaries
  9. Ribbon graphs
  10. Webs and traces
  11. Multiwebs
  12. Proof of Theorem \ref{['theo:polynomial']}
  13. Planar setting
  14. Planar graphs
  15. Quantum connections and local definition of trace
  16. ...and 24 more sections

Key Result

Theorem 1

The quantum trace $\mathbf{tr}_q(m)$ of an $n$-multiweb $m$ in a bipartite (ciliated) ribbon graph $\mathbf{G}$ in $\mathbb{R}^3$ is a Laurent polynomial in $q^\frac{1}{n}$.

Figures (13)

  • Figure 1: Left: Ciliated ribbon graph with a right handed twist. Middle: Ciliated ribbon graph with a positive kink and the blackboard framing. Right: Diagram of a ciliated ribbon graph.
  • Figure 2: Ciliated framed Reidemeister moves.
  • Figure 3: Crossing change relation.
  • Figure 4: Positive kink removing relation.
  • Figure 5: Source-sink removing relation.
  • ...and 8 more figures

Theorems & Definitions (100)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 90 more