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arXiv:1911.03348 · Journal of Statistical Mechanics Theory and Experiment

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

Christian Hagendorf, Jean Liénardy

Published Mar 1, 2020• 6 citations
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Abstract

Abstract The transfer matrix of the square-lattice eight-vertex model on a strip with vertical lines and open boundary conditions is investigated. It is shown that for vertex weights that obey the relation and appropriately chosen K -matrices this transfer matrix possesses the remarkably simple, non-degenerate eigenvalue . For positive vertex weights, is shown to be the largest transfer-matrix eigenvalue. The corresponding eigenspace is equal to the space of the ground states of the Hamiltonian of a related XYZ spin chain. An essential ingredient in the proofs is the supersymmetry of this Hamiltonian.

DOI
10.1088/1742-5468/ab7748

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States with $v_1 = \lambda, v_2 = -\lambda$ and reciprocal equations in the six-vertex model(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

M. J. Rodríguez-Plaza

The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of $n$ constants $v_{1},$ $\ldots,$ $v_{n}$ , the zeros of the function $Q(v)$ .A peculiar class of eigenvalues are those in which two of the constants $v_{1},$ $v_{2}$ are equal to $\lambda,$ $-\lambda$ , with $\Delta=-\cosh$ A and $\Delta$ related to the Boltzmann weights of the six-vertex model by the usual combination $\Delta=(a^{2}+b^{2}-c^{2})/2ab$ .The eigenvectors associated to these eigenvalues are Bethe states (although they seem not).We count the number of such states (eigenvectors) for $n=2,3,4,5$ when $N$ , the columns in a row of a square lattice, is arbitrary.The number obtained is independent of the value of $\Delta$ , but depends on $N$ .We give the explicit expres- sion of the eigenvalues in terms of $a,$ $b,$ $c$ (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary $N$ in the blocks $n$ considered.For real $a,$ $b,c$ such eigenvalues are real.

数理解析研究所講究録•Apr 1, 2006•0 citations
Abstract The transfer matrix of the square-lattice eight-vertex model on a strip with vertical lines and open boundary conditions is investigated. It is shown that for vertex weights that obey the relation and appropriately chosen K -matrices this transfer matrix possesses the remarkably simple, non-degenerate eigenvalue . For positive vertex weights, is shown to be the largest transfer-matrix eigenvalue. The corresponding eigenspace is equal to the space of the ground states of the Hamiltonian of a related XYZ spin chain. An essential ingredient in the proofs is the supersymmetry of this Hamiltonian.
Abstract The square-lattice eight-vertex model with vertex weights a , b , c , d obeying the relation ( a 2 + a b ) ( b 2 + a b ) = ( c 2 + a b ) ( d 2 + a b ) and periodic boundary conditions is considered. It is shown that the transfer matrix of the model for L = 2 n + 1 vertical lines and periodic boundary conditions along the horizontal direction possesses the doubly degenerate eigenvalue Θ n = ( a + b ) 2 n + 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 Θ n are shown to be the ground states of the spin-chain Hamiltonian. Moreover, for positive vertex weights Θ n is the largest eigenvalue of the transfer matrix.
We show that the XYZ spin chain along the special line of couplings J x J y +J x J z +J y J z =0 possesses a hidden mathcalN=(2,2)\\mathcal{N}=(2,2)mathcalN=(2,2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain
It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit solution of its Bethe equations. The corresponding eigenvector is computed by means of the algebraic Bethe ansatz and both a simple component and its square norm are expressed in terms of the Izergin–Korepin determinant. In the homogeneous limit, the vector coincides with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown that in a natural polynomial normalisation scheme its square norm and the simple component coincide with generating functions for weighted enumeration of alternating sign matrices.
Abstract We study the inhomogeneous eight-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, -twisted, -twisted or -twisted. We show that in all these cases but the periodic one with an even number of sites , the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical six-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin’s separation of variables approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical six-vertex model, the twisted eight-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the eight-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical six-vertex antiperiodic transfer matrix. In all these cases we can adapt to the eight-vertex case the reformulations of the dynamical six-vertex transfer matrix spectrum and eigenstates that had been obtained by T – Q functional equations, where the Q -functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enable one to characterize the eight-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.
Abstract Sutherland showed that the XYZ quantum spin-chain Hamiltonian commutes with the eight-vertex model transfer matrix, so that Baxter’s subsequent tour de force proves the integrability of both. The proof requires parametrising the Boltzmann weights using elliptic theta functions and showing they satisfy the Yang-Baxter equation. We here give a simpler derivation of the integrability of the XYZ chain by explicitly constructing an extensive sequence of conserved charges from a matrix-product operator. We show that they commute with the XYZ Hamiltonian with periodic boundary conditions or an arbitrary boundary magnetic field. A straightforward generalisation yields impurity interactions that preserve the integrability. Placing such an impurity at the edge gives an integrable generalisation of the Kondo problem with a gapped bulk. We make contact with the traditional approach by relating our matrix-product operator to products of the eight-vertex model transfer matrix.
States with v1=λ,v2=−λv_1 = \lambda, v_2 = -\lambdav1​=λ,v2​=−λ and reciprocal equations in the six-vertex model(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )
The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of nnn constants v1,v_{1},v1​, …,\ldots,…, vnv_{n}vn​ , the zeros of the function Q(v)Q(v)Q(v) .A peculiar class of eigenvalues are those in which two of the constants v1,v_{1},v1​, v2v_{2}v2​ are equal to λ,\lambda,λ, −λ-\lambda−λ , with Δ=−cosh⁡\Delta=-\coshΔ=−cosh A and Δ\DeltaΔ related to the Boltzmann weights of the six-vertex model by the usual combination Δ=(a2+b2−c2)/2ab\Delta=(a^{2}+b^{2}-c^{2})/2abΔ=(a2+b2−c2)/2ab .The eigenvectors associated to these eigenvalues are Bethe states (although they seem not).We count the number of such states (eigenvectors) for n=2,3,4,5n=2,3,4,5n=2,3,4,5 when NNN , the columns in a row of a square lattice, is arbitrary.The number obtained is independent of the value of Δ\DeltaΔ , but depends on NNN .We give the explicit expres- sion of the eigenvalues in terms of a,a,a, b,b,b, ccc (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary NNN in the blocks nnn considered.For real a,a,a, b,cb,cb,c such eigenvalues are real.