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arXiv:cond-mat/0610481 · Journal of Physics A Mathematical and Theoretical

A newQ-matrix in the eight-vertex model

K. Fabricius

Published Mar 23, 2007• 26 citations
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Abstract

We construct a $Q$-matrix for the eight-vertex model at roots of unity for crossing parameter $η=2mK/L$ with odd $L$, a case for which the existing constructions do not work. The new $Q$-matrix $\Q$ depends as usual on the spectral parameter and also on a free parameter $t$. For $t=0$ $\Q$ has the standard properties. For $t\neq 0$, however, it does not commute with the operator $S$ and not with itself for different values of the spectral parameter. We show that the six-vertex limit of $\Q(v,t=iK'/2)$ exists.

DOI
10.1088/1751-8113/40/15/002

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We construct a QQQ-matrix for the eight-vertex model at roots of unity for crossing parameter η=2mK/Lη=2mK/Lη=2mK/L with odd LLL, a case for which the existing constructions do not work. The new QQQ-matrix \Q\Q\Q depends as usual on the spectral parameter and also on a free parameter ttt. For t=0t=0t=0 \Q\Q\Q has the standard properties. For t≠0t\neq 0t=0, however, it does not commute with the operator SSS and not with itself for different values of the spectral parameter. We show that the six-vertex limit of \Q(v,t=iK′/2)\Q(v,t=iK'/2)\Q(v,t=iK′/2) exists.
Following Baxter's method of producing Q 72 -operator, we construct the Q -operator of the root-of-unity eight-vertex model for the crossing parameter with odd N where Q 72 does not exist. We use this new Q -operator to study the functional relations in the Fabricius–McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N -state chiral Potts model. By the compatibility of the constructed Q -operator with the structure of Baxter's eight-vertex (solid-on-solid) SOS model, we verify the set of functional relations of the root-of-unity eight-vertex model using the explicit form of the Q -operator and fusion weights of the SOS model.
The construction of creation operators of exact strings in eigenvectors of the eight-vertex model at elliptic roots of unity of the crossing parameter which allow the generation of the complete set of degenerate eigenstates is based on the conjecture that the ‘naïve’ string operator vanishes. In this paper, we present a proof of this conjecture. Furthermore, we show that for chains of odd length the string operator is either proportional to the symmetry operator S or vanishes depending on the precise form of the crossing parameter.
We study the transfer matrix of the 8 vertex model with an odd number of lattice sites N. For systems at the root of unity points η = mK/L with m odd the transfer matrix is known to satisfy the famous “TQ ” equation where Q(v) is a specifically known matrix. We demonstrate that the location of the zeroes of this Q(v) matrix is qualitatively different from the case of even N and in particular they satisfy a previously unknown equation which is more general than what is often called “Bethe’s equation. ” For the case of even m where no Q(v) matrix is known we demonstrate that there are many states which are not obtained from the formalism of the SOS model but which do satisfy the TQ equation. The ground state for the particular case of η = 2K/3 and N odd is investigated in detail.
We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at q=εq=\varepsilonq=ε, where ε\varepsilonε is a primitive NNNth root of unity with NNN odd. For the commuting layer transfer matrices T(y)\mathbb{T}(y)T(y) with mixed boundary conditions, we establish the quantum Frobenius property T(y)T(εy)⋯T(εN−1y)=PN(yN)I, \mathbb{T}(y)\mathbb{T}(\varepsilon y)\cdots \mathbb{T}(\varepsilon^{N-1}y) = \mathscr{P}_N(y^N)\mathbb{I}, T(y)T(εy)⋯T(εN−1y)=PN​(yN)I, where PN\mathscr{P}_NPN​ is a scalar Laurent polynomial obtained directly from the monomial expansion of T(y)\mathbb{T}(y)T(y). Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting τ  2τ_{\; 2}τ2​ model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.
At roots of unity the NNN-state integrable chiral Potts model and the six-vertex model descend from each other with the τ2τ_2τ2​ model as the intermediate. We shall discuss how different gauge choices in the six-vertex model lead to two different quantum group constructions with different qqq-Pochhammer symbols, one construction only working well for NNN odd, the other equally well for all NNN. We also address the generalization based on the sl(m,n)(m,n)(m,n) vertex model.