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Modified rational six vertex model on a rectangular lattice : new formula, homogeneous and thermodynamic limits

Matthieu Cornillault, Samuel Belliard

TL;DR

This work extends determinant-based representations of the partition function for the modified rational six-vertex (MR6V) model with general boundary conditions to rectangular lattices. By adapting Foda–Wheeler methods, the authors derive a new determinant expression that blends modified Izergin and Vandermonde structures, enabling the homogeneous limit and a detailed thermodynamic analysis. In the thermodynamic limit, the boundary parameter $β$ and the mismatch $d=|n-m|$ shape the bulk free energy and associated physical quantities, revealing explicit $β$-dependent regimes and boundary effects that persist in finite systems. The results bridge inhomogeneous/pDWBC determinants with homogeneous and infinite-lattice limits, and point toward extensions to trigonometric and elliptic vertex models via Modified Bethe Ansatz structures.

Abstract

We continue the work of Belliard, Pimenta and Slavnov (2024) studying the modified rational six vertex model. We find another formula of the partition function for the inhomogeneous model, in terms of a determinant that mix the modified Izergin one and a Vandermonde one. This expression enables us to compute the partition function in the homogeneous limit for the rectangular lattice, and then to study the thermodynamic limit. It leads to a new result, we obtain the first order of free energy with boundary effects in the thermodynamic limit.

Modified rational six vertex model on a rectangular lattice : new formula, homogeneous and thermodynamic limits

TL;DR

This work extends determinant-based representations of the partition function for the modified rational six-vertex (MR6V) model with general boundary conditions to rectangular lattices. By adapting Foda–Wheeler methods, the authors derive a new determinant expression that blends modified Izergin and Vandermonde structures, enabling the homogeneous limit and a detailed thermodynamic analysis. In the thermodynamic limit, the boundary parameter and the mismatch shape the bulk free energy and associated physical quantities, revealing explicit -dependent regimes and boundary effects that persist in finite systems. The results bridge inhomogeneous/pDWBC determinants with homogeneous and infinite-lattice limits, and point toward extensions to trigonometric and elliptic vertex models via Modified Bethe Ansatz structures.

Abstract

We continue the work of Belliard, Pimenta and Slavnov (2024) studying the modified rational six vertex model. We find another formula of the partition function for the inhomogeneous model, in terms of a determinant that mix the modified Izergin one and a Vandermonde one. This expression enables us to compute the partition function in the homogeneous limit for the rectangular lattice, and then to study the thermodynamic limit. It leads to a new result, we obtain the first order of free energy with boundary effects in the thermodynamic limit.

Paper Structure

This paper contains 37 sections, 15 theorems, 225 equations, 10 figures.

Table of Contents

  1. Notation
  2. Modified rational six vertex (MR6V) model
  3. Definition
  4. Known results
  5. A new expression for rectangular lattice
  6. Applications
  7. Homogeneous limit
  8. Physical interpretation for completely homogeneous finite lattice
  9. Thermodynamic limit
  10. The semi-infinite lattice
  11. The infinite lattice
  12. Proofs of the new formula for the partition function
  13. Using the method of Foda and Wheeler
  14. An alternative proof
  15. The partial Cauchy matrix and its inverse
  16. ...and 22 more sections

Key Result

Proposition 1.1

Belliard2024 The partition function of the inhomogeneous rational six-vertex model with general boundary conditions is with The modified Izergin determinant can be expressed with two formulae Belliard2018 For the special case of a square lattice ($m = n$), we have an additional formula Belliard2024

Figures (10)

  • Figure 1: Modelisation of the lattice and vertices. The open arrows indicate the orientation of the lines.
  • Figure 2: The six different vertices in the QISM.
  • Figure 3: The $n \times m$ rectangular lattice with general boundary conditions.
  • Figure 4: Graphs of $\tilde{F}$ for different values of $\tilde{\beta}$. "xt" corresponds to $\tilde{x}$ and "Ft" to $\tilde{F}$.
  • Figure 5: Graphs of the physical characteristics for $k_BT = 1$ and $\tilde{\beta} \geqslant 0$. The green, red and dark-red curve are respectively for $\tilde{\beta} = 0, 1$ and $10$.
  • ...and 5 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Remark 1.1
  • Definition 1.2
  • Proposition 1.1
  • Remark 1.2
  • Proposition 1.2
  • Proposition 1.3
  • Remark 1.3
  • Definition 2.1
  • Remark 2.1
  • ...and 32 more