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Sum rule for the eight-vertex model on its combinatorial line
Mathematical Physics · Submitted 20 Feb 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We compute the partition function of the model on an infinite cylinder with certain restrictions on the inhomogeneities, and taking the homogeneous limit, we obtain an expression for the squared norm of the ground state of the XYZ spin chain as a solution of a differential recurrence relation.
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We investigate the conjectured ground state eigenvector of the 8-vertex model inhomogeneous transfer matrix on its combinatorial line, i.e., at $\eta=\pi/3$, where it acquires a particularly simple form. We compute the partition function of the model on an infinite cylinder with certain restrictions on the inhomogeneities, and taking the homogeneous limit, we obtain an expression for the squared norm of the ground state of the XYZ spin chain as a solution of a differential recurrence relation.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the conjectured ground state eigenvector of the 8-vertex model inhomogeneous transfer matrix on its combinatorial line, i.e., at $\eta=\pi/3$, where it acquires a particularly simple form. | contribution 0.71 | 0.56 | 0.09 | speculative 0.49 |
| s1 | We compute the partition function of the model on an infinite cylinder with certain restrictions on the inhomogeneities, and taking the homogeneous limit, we obtain an expression for the squared norm of the ground state of the XYZ spin chain as a solution of a differential recurrence relation. | result 0.71 | 0.95 | 0.62 | asserted 1.00 |
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The paper
Original abstract
We investigate the conjectured ground state eigenvector of the 8-vertex model inhomogeneous transfer matrix on its combinatorial line, i.e., at $\eta=\pi/3$, where it acquires a particularly simple form. We compute the partition function of the model on an infinite cylinder with certain restrictions on the inhomogeneities, and taking the homogeneous limit, we obtain an expression for the squared norm of the ground state of the XYZ spin chain as a solution of a differential recurrence relation.
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How the literature uses it
7 arXiv papers cite this in 16 sentences
Papers citing it · 8 sentences without a confident label
Compares with it 2 papers
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A form of this result in which the odd orthogonal characters on the RHSs of [ref] and [ref] are replaced by certain symplectic characters was obtained previously by Zinn-Justin [ref] .
Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions · Partitions of double-staircase shape · 0.88
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Furthermore, in the current work, we are able to provide explicit formulae for the entries of the solution of the qKZB system, leaving the inhomogeneities free – in [ref] , they were “half-specialised” to pairs of opposite values, resulting in expressions involving only even theta functions and therefore a rational parameterisation; here no such parameterisation is available.
Theta function solutions of the qKZB equation for a face model · Discussion · 1.00
Cites it as background 3 papers
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The polynomials $q_n(z)$ have the symmetries [ref] [equation] Zinn-Justin [ref] observed that the polynomials $q_n(z)$ seem to be given by specializing the variables in a determinant equivalent to [ref].
A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain · Filali's determinant in terms of the polynomials $q_n$ · 0.81
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The supersymmetry singlets of the XYZ Hamiltonian [ref] were studied in [ref] .
Spin chains with dynamical lattice supersymmetry · An example: the XYZ chain · 0.82
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Zinn-Justin [ref] investigated a family of polynomials equivalent to Rosengren's polynomials $T$.
A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. II. The Polynomials $p_n$ · Formulas for $p_n(z)$ · 0.91
What it cites · 44 sentences without a confident label
Builds on 2 papers
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Consider the following elliptic version of Tsuchiya's determinant [ref] : (see a similar determinant in [ref] ) [equation] Conventionally, $_̋0=1$.
Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end · Further factorization as determinants · 0.93
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Consider the following elliptic version of Tsuchiya's determinant [ref] : (see a similar determinant in [ref] ) [equation] Conventionally, $_̋0=1$.
Symmetry classes of alternating-sign matrices under one roof · Further factorization as determinants · 0.93
Uses the method or tool of 3 papers
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A standard application of the Jacobi–Desnanot identity (see [ref] for the simpler case of the Izergin–Korepin determinant) to the determinant in the right hand side produces the Toda lattice equation: [equation] The left hand side involves first and second derivative of $H_2m(u,…,v,…)$, which at $u=v=0$ can be reexpressed in terms of derivatives w.r.t. $α$ thanks to Eq. [ref].
Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions · Bilinar recurrence relations · 0.85
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Here we follow the same general method as in [ref] (see also appendix B of [ref] ).
A new way to deal with Izergin-Korepin determinant at root of unity · Alternative determinant formula · 1.00
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Here we follow the same general method as in [ref] (see also appendix B of [ref] ).
Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule · Alternative determinant formula · 1.00
Compares with 6 papers
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If we recombine the $H_2m$ in pairs to form $A_n$ and $B_n$, we recognize the reciprocal polynomials of the $s_n$ of [ref] : (see their appendix A) [equation] from which we conclude [equation] which coincides with the expression given in Conjecture 1 of [ref] up to the factor of two (which is due to our slightly different way of lifting the two-fold degeneracy: $F_∗Ψ_L=(-1)^nΨ_L$ effectively duplicates every entry of $Ψ_L$ compared to [ref] ).
Eight-vertex model and Painlev\'e VI equation. II. Eigenvector results · Summary · 1.00
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We now recognize the Pfaffian $A_n$ (Eq. [ref]) in even size: [equation] which up to some prefactors is exactly the Pfaffian given in [ref] (Eq. (3.27)) for the square of the partition function $Z_UASM$ of U-turn symmetric ASMs of [ref] .
Symmetry classes of alternating-sign matrices under one roof · Pfaffians · 1.00
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We now recognize the Pfaffian $A_n$ (Eq. [ref]) in even size: [equation] which up to some prefactors is exactly the Pfaffian given in [ref] (Eq. (3.27)) for the square of the partition function $Z_UASM$ of U-turn symmetric ASMs of [ref] .
Inhomogeneous loop models with open boundaries · Pfaffians · 1.00
Cites as background 5 papers
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Many remarkable observations were made on the corresonding eigenvector, $Ψ_L$ in [ref] .
Eight-vertex model and Painlev\'e VI equation. II. Eigenvector results · Homogeneous limit · 0.88
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Many remarkable observations were made on the corresonding eigenvector, $Ψ_L$ in [ref] .
A possible combinatorial point for XYZ-spin chain · Homogeneous limit · 0.88
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More recently, Razumov and Stroganov [ref] and Bazhanov and Mangazeev [ref] studied the model with such conditions.
Eight-vertex model and non-stationary Lame equation · Combinatorial line · 0.96
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