The 8-vertex model with quasi-periodic boundary conditions

Abstract

We study the inhomogeneous 8-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, ${\sigma }^x$-twisted, ${\sigma }^y$-twisted or ${\sigma }^z$-twisted. We show that in all these cases but the periodic one with an even number of sites $\mathsf{N}$, the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical 6-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin's Separation of Variables (SOV) 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 6-vertex model, the twisted 8-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for $\mathsf{N}$ odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the 8-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 6-vertex antiperiodic transfer matrix. In all these cases we can adapt to the 8-vertex case the reformulations of the dynamical 6-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 enables one to characterize the 8-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.

35 pages

References in corpus (7)

• Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors
• Antiperiodic dynamical 6-vertex model I: Complete spectrum by SOV, matrix elements of the identity on separate states and connections to the periodic 8-vertex model
• SOS model partition function and the elliptic weight functions
• Analytic theory of the eight-vertex model
• A new Q-matrix in the Eight-Vertex Model
• New Q matrices and their functional equations for the eight vertex model at elliptic roots of unity
• The TQ equation of the 8 vertex model for complex elliptic roots of unity

Cited by in corpus (19)

• The open XXX spin chain in the SoV framework: scalar product of separate states
• On quantum separation of variables
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I
• Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables
• Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors
• The open XXZ spin chain in the SoV framework: scalar product of separate states
• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II
• On Scalar Products in Higher Rank Quantum Separation of Variables
• On quantum separation of variables beyond fundamental representations
• Complete spectrum of quantum integrable lattice models associated to ${\mathcal{U}}_q(\widehat{g{l}_n})$ by separation of variables
• Separation of variables bases for integrable $g{l}_{\mathcal{M}|\mathcal{N}}$ and Hubbard models
• Scalar products of Bethe vectors in the 8-vertex model
• On Separation of Variables for Reflection Algebras
• Correlation functions for open XXZ spin 1/2 quantum chains with unparallel boundary magnetic fields
• Correlation functions by Separation of Variables: the XXX spin chain
• Current mean values in the XYZ model
• On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
• Exactly Solved Models and Beyond: a special issue in honour of R J Baxter's 75th birthday
• The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint