On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

Hagendorf, Christian; Liénardy, Jean

doi:10.1088/1742-5468/ab7748

Cited by 7 publications

(14 citation statements)

References 34 publications

(105 reference statements)

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“…In the trigonometric limit, these correlation functions are known [35,37]. Finally, we mention that there is a conjecture for a simple eigenvalue of the transfer matrix of the inhomogeneous supersymmetric eight-vertex model with open boundary conditions [38]. A characterisation of its eigenspace, similar to Conjecture 2.1, is still to be found.…”

Section: Discussionmentioning

confidence: 89%

Sum rules for the supersymmetric eight-vertex model

Brasseur,

Hagendorf

2020

Preprint

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The eight-vertex model on the square lattice with vertex weights a, b, c, d obeying the relation (a 2 + ab)(b 2 + ab) = (c 2 + ab)(d 2 + ab) is considered. Its transfer matrix with L = 2n + 1, n 0, vertical lines and periodic boundary conditions along the horizontal direction has the doubly-degenerate eigenvalue Θn = (a+b) 2n+1 . A basis of the corresponding eigenspace is investigated. Several scalar products involving the basis vectors are computed in terms of a family of polynomials introduced by Rosengren and Zinn-Justin. These scalar products are used to find explicit expressions for particular entries of the vectors. The proofs of these results are based on the generalisation of the eigenvalue problem for Θn to the inhomogeneous eight-vertex model.

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Section: Discussionmentioning

confidence: 89%

Sum rules for the supersymmetric eight-vertex model

Brasseur,

Hagendorf

2020

Preprint

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“…First, an obvious generalisation is the extension of the present results to other boundary conditions. In a companion paper [35], we consider the supersymmetric eight-vertex model on a strip and the related XYZ chain with open boundary conditions. We classify all boundary terms that are compatible with a lattice supersymmetry.…”

Section: Resultsmentioning

confidence: 99%

“…The first term on the right-hand side of this equality is 4 n λ n . The second term vanishes because of (35). This leads to λ n = 1 4 n Φ n (ζ −1 )|Ψ n .…”

Section: Zero-energy Statesmentioning

confidence: 99%

On the transfer matrix of the supersymmetric eight-vertex model. I. Periodic boundary conditions

Hagendorf¹,

Liénardy²

2018

J. Stat. Mech.

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The square-lattice eight-vertex model with vertex weights a, b, c, d obeying the relation (a 2 + ab)(b 2 + ab) = (c 2 + ab)(d 2 + ab) and periodic boundary conditions is considered. It is shown that the transfer matrix of the model for L = 2n + 1 vertical lines and periodic boundary conditions along the horizontal direction possesses the doubly degenerate eigenvalue Θn = (a + b) 2n+1 . This proves a conjecture by Stroganov from 2001. The proof uses the supersymmetry of a related XYZ spin-chain Hamiltonian. The eigenstates of the transfer matrix corresponding to Θn are shown to be the ground states of the spin-chain Hamiltonian. Moreover, for positive vertex weights Θn is the largest eigenvalue of the transfer matrix.

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“…It follows from (37) that it obeys the braid version of the Yang-Baxter equation Ř1,2 (z/w) Ř2,3 (z) Ř1,2 (w) = Ř2,3 (w) Ř1,2 (z) Ř2,3 (z/w).…”

Section: The Exchange Relationsmentioning

confidence: 99%

“…for each 1 i, j N . These commutation relations follow from the Yang-Baxter equation (37) and the boundary Yang-Baxter equation (57).…”

Section: The Boundary Quantum Knizhnik-zamolodchikov Equationsmentioning

confidence: 99%

The open XXZ chain at $Δ=-1/2$ and the boundary quantum Knizhnik-Zamolodchikov equations

Hagendorf,

Liénardy

2020

Preprint

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The open XXZ spin chain with the anisotropy parameter ∆ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik-Zamolodchikov equations associated with the R-matrix and diagonal K-matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.1 For N = 1 sites, the bulk interaction term is absent by convention. The Hamiltonian reduces to H = (p + p)σ z .

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On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

Cited by 7 publications

References 34 publications

Sum rules for the supersymmetric eight-vertex model

Sum rules for the supersymmetric eight-vertex model

On the transfer matrix of the supersymmetric eight-vertex model. I. Periodic boundary conditions

The open XXZ chain at $Δ=-1/2$ and the boundary quantum Knizhnik-Zamolodchikov equations

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