At elliptic roots of unity of the crossing parameter, the transfer matrix of the eight-vertex model (equivalently the spin-1/2 XYZ chain) develops exponentially large degeneracies of its eigenvalues. For the six-vertex model the analogous degeneracies are fully explained by an sl2sl_2 loop algebra whose Chevalley generators were explicitly constructed by Deguchi, Fabricius and McCoy, with degenerate multiplets forming irreducible highest-weight representations labeled by Drinfeld polynomials. For the eight-vertex model no such structure is known: Fabricius (2011), after proving the vanishing conjecture for the naive string operator, concluded that the symmetry operators which generalize the six-vertex Chevalley operators are still elusive. The problem is to construct those symmetry operators — explicit operators commuting with the eight-vertex transfer matrix that act non-trivially within its degenerate eigenspaces, for both chain-length parities and both root-of-unity classes — and to identify the full symmetry algebra (generators, relations, and representation theory, presumably with an elliptic analogue of the Drinfeld polynomial) whose irreducible representations account for the observed degeneracy pattern.

Background

The eight-vertex model of Baxter is a two-dimensional lattice model in which arrow (Ising-like) variables ±1\pm1 live on the edges of a square lattice and each vertex carries one of eight arrow configurations, paired by simultaneous arrow reversal into four classes with Boltzmann weights a,b,c,da,b,c,d; the weights are Jacobi theta functions of a spectral parameter λ\lambda and depend on a crossing parameter η\eta; KK and K′K' denote the half-periods of the underlying elliptic functions. Its row-to-row transfer matrix T8(λ)T_8(\lambda) generates the partition function, and the model is equivalent to the spin-1/2 XYZ chain, the fully anisotropic generalization of the Heisenberg chain. Baxter (1972) computed its free energy, and the eigenvectors of T8T_8 are constructed by the algebraic Bethe ansatz of Takhtadzhan and Faddeev: one writes a 2×22\times2 monodromy matrix with operator entries A,B,C,DA,B,C,D and obtains eigenvectors as products of Bk,l(λ)B_{k,l}(\lambda) operators acting on reference states, a construction that for the eight-vertex model necessarily involves two free parameters s,ts,t.

When the crossing parameter is an elliptic root of unity, η=2m1K/L+i m2K′/L\eta=2m_1K/L+i\,m_2K'/L with integers m1,m2,Lm_1,m_2,L (the analysis of the string operator in Fabricius (2011) restricts, for simplicity, to the case m2=0m_2=0), the spectrum develops extensive degeneracies: complete strings of Bethe roots (a string is a set of roots spaced by the crossing parameter, λk=λc−2(k−1)η\lambda_k=\lambda_c-2(k-1)\eta, k=1,…,Lsk=1,\ldots,L_s; for m2=0m_2=0 the string length is Ls=LL_s=L for odd LL and Ls=L/2L_s=L/2 for even LL) appear together with companion strings shifted by iK′iK', each complete string doubling the eigenvalue multiplicity, so an eigenvalue carrying nLn_L complete strings is 2nL2^{n_L}-fold degenerate, and some degenerate eigenspaces have dimension L 2N/LL\,2^{N/L} on NN sites when NN is an even multiple of LL.

For the six-vertex model (the trigonometric limit, where the quantum-group deformation parameter qq is a root of unity) the analogous phenomenon is completely understood. Deguchi, Fabricius and McCoy (2001) showed that the transfer matrix commutes with all generators of an sl2sl_2 loop algebra — the infinite-dimensional Lie algebra sl2⊗C[t,t−1]sl_2\otimes\mathbb{C}[t,t^{-1}], spanned by one sl2sl_2 Chevalley triple (em,fm,hm)(e_m,f_m,h_m), i.e. generators obeying [h,e]=2e[h,e]=2e, [h,f]=−2f[h,f]=-2f, [e,f]=h[e,f]=h, for each integer mode mm — realized by explicit operators on the chain, built from monodromy-matrix entries evaluated at shifted spectral parameters. The degenerate multiplets are irreducible highest-weight representations of this loop algebra, labeled by Drinfeld polynomials (a polynomial invariant whose roots are the free evaluation parameters of the degenerate states), and Fabricius and McCoy (2002) constructed the generating current whose modes both produce the loop-algebra generators and create the exact Bethe strings.

For the eight-vertex model the parallel structure is missing. Fabricius and McCoy (2006) constructed the string-creation operator (the elliptic current) — a product of BB operators corrected by derivatives with respect to η\eta and λ\lambda — which creates the complete strings and generates all missing degenerate eigenstates, enlarging the free parameter set {s,t}\{s,t\} by arbitrary string centers λc\lambda_c; the naive, uncorrected string operator BsB_s vanishes, as conjectured in 2006 and proved by Fabricius (2011) for even NN, and for odd NN when 2Lsη≡0(mod4K)2L_s\eta\equiv0\pmod{4K}, while for odd NN with 2Lsη≡2K(mod4K)2L_s\eta\equiv2K\pmod{4K} the naive operator is proportional to the symmetry operator S=σ3⊗σ3⊗⋯⊗σ3S=\sigma_3\otimes\sigma_3\otimes\cdots\otimes\sigma_3 (the third Pauli matrix on every site). The corrected string operator is itself a symmetry operator in the sense that it maps each degenerate eigenspace onto itself. But unlike the six-vertex current, the elliptic current is not nilpotent (an arbitrary number of exact strings can be appended), the chain has no conserved total SzS^z with which to organize sectors, and the cyclic (root-of-unity) representation theory of the quantum affine algebra Uq(sl^2)U_q(\widehat{sl}_2) that underlies the six-vertex construction is not available elliptically. Consequently the explicit operators generalizing the six-vertex Chevalley generators, and the algebra they generate with its relations and representation theory, are not known — a gap stated already in the 2004 review of Fabricius and McCoy (for the eight-vertex model the symmetry algebra is not known even though such an algebra must exist, and the expected object should involve an elliptic generalization of the Drinfeld polynomial) and restated in the conclusions of Fabricius (2011). Natural candidate structures exist — the elliptic quantum group Eτ,η(sl2)E_{\tau,\eta}(sl_2) of Felder and Varchenko (a dynamical, elliptic deformation of the quantum affine algebra Uq(sl^2)U_q(\widehat{sl}_2), whose algebraic Bethe ansatz governs the eight-vertex eigenvectors); the central elements of the elliptic Yang-Baxter algebra (the algebra generated by the monodromy-matrix entries) constructed by Belavin and Jimbo at odd roots of unity; the Sklyanin algebra (the quadratic associative algebra generated by the entries of the eight-vertex Lax operator, whose representation theory encodes the XYZ chain) at roots of unity — but none has yet produced the commutant of the eight-vertex transfer matrix or a representation theory matching the observed degeneracies.

Problem Statement

Construct the symmetry operators that generalize, for the eight-vertex model (equivalently the spin-1/2 XYZ chain) at elliptic roots of unity of the crossing parameter, the Chevalley generators of the six-vertex sl2sl_2 loop algebra: explicit operators commuting with the eight-vertex transfer matrix T8(λ)T_8(\lambda) that act non-trivially within its degenerate eigenspaces, valid for both parities of the chain length NN and for the two classes of elliptic roots of unity distinguished for m2=0m_2=0 (namely 2Lsη≡0 mod 4K2L_s\eta\equiv0\bmod 4K and 2Lsη≡2K mod 4K2L_s\eta\equiv2K\bmod 4K). Identify the symmetry algebra generated by these operators — its defining relations and its finite-dimensional representation theory, presumably with an elliptic analogue of the Drinfeld polynomial — and show that the decomposition of the 2N2^N-dimensional state space into irreducible representations of this algebra accounts for the observed degeneracy pattern of T8T_8: the exponentially growing multiplicities, the factor of two contributed by each complete LsL_s-string and its companion string shifted by iK′iK', and the free parameters ss, tt and string centers λc\lambda_c that label states within one degenerate multiplet — in the same way that the sl2sl_2 loop algebra and its Drinfeld polynomials account for the degeneracies of the six-vertex model at roots of unity. All of the parts stated above are jointly required, and no proper subset of them constitutes a solution: the symmetry operators, valid for both parities of the chain length NN and for both of the m2=0m_2=0 root-of-unity classes; the defining relations of the algebra they generate; its finite-dimensional representation theory; and the exhibition that the decomposition of the 2N2^N-dimensional state space reproduces the degeneracy pattern listed above. The elliptic analogue of the Drinfeld polynomial is the expected form of that representation-theory labeling, not a separate additional deliverable.

References

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