The exact thermodynamic-limit solution of the fundamental Sp(2n) integrable vertex model rests on an analyticity structure that has been verified only for ranks n=2 and n=3 and is explicitly assumed for general n: inside the analyticity strip -1/2-Delta < Re(lambda) < 1/2 (Delta = n+1), only the leading eigenvalue of the last fusion-level transfer matrix is free of zeros, while the leading eigenvalues of all lower fusion levels carry zeros on the central line Re(lambda) = -Delta/2 whose number grows linearly with the lattice circumference, condensing into an extended singularity as the system size goes to infinity. This lost analyticity is what defeats the standard inversion-relation method, and it generates CDD-type integral terms in the thermodynamic solution whose closed gamma-function representation is known only at low rank. The problem asked here is to prove or refute this zero-structure statement for arbitrary rank n, and to determine whether the CDD-type integral terms admit a closed gamma-product representation uniform in n, or whether no such representation exists.

Background

An integrable vertex model is a statistical-mechanical system on the square lattice whose Boltzmann weights are built from a solution R(λ)R(\lambda) of the Yang-Baxter equation, where λ\lambda is the spectral parameter. Its row-to-row transfer matrix T(λ)T(\lambda) generates the partition function, and integrability yields an infinite family of fused transfer matrices T(m)(λ)T^{(m)}(\lambda) acting on higher representations of the underlying Lie algebra, obtained by composing R-matrices with fusion projectors. These satisfy the fusion hierarchy, a system of functional relations among the T(m)T^{(m)} (Kirillov-Reshetikhin; Bazhanov-Reshetikhin) which for the models considered here truncates to finitely many levels exactly in the thermodynamic limit. The standard functional method for the partition function per site (Baxter; Klümper) combines inversion relations with the analyticity of the leading eigenvalue Λ0(λ)\Lambda_0(\lambda) - the eigenvalue of largest magnitude at fixed λ\lambda, which dominates the thermodynamic limit - inside an analyticity strip of the complex spectral-parameter plane. The method tolerates at most finitely many zeros, removable by so-called CDD factors, the standard term (borrowed from factorized-scattering theory) for extra multiplicative factors that satisfy the same functional equations as the minimal solution.

Ribeiro, Klümper and Pearce (J. Stat. Mech. 2022) found the first exception among rational vertex models. For the fundamental Sp(4)Sp(4) vertex model - the symplectic-group model whose R-matrix acts on the 4-dimensional fundamental representation, R12(λ)=λ(λ+Δ)I12+(λ+Δ)P12+λE12,R_{12}(\lambda)=\lambda(\lambda+\Delta)I_{12}+(\lambda+\Delta)P_{12}+\lambda E_{12}, with Δ=n+1\Delta=n+1 the crossing parameter, II and PP the identity and permutation operators, and EE a projector built from the symplectic form - the leading eigenvalue of the fundamental transfer matrix has zeros lying exactly on the central line of the analyticity strip, in number growing linearly with the lattice circumference. In the thermodynamic limit this vertical line of zeros condenses into an extended singularity that splits the strip in two, so the ordinary inversion relation fails; the authors showed that the fusion relations nevertheless connect the two sides of the strip and determine the partition function per site, whose solution then contains CDD-like integral terms in addition to the expected gamma-function part.

Ribeiro (J. Stat. Mech. 2023) extended the computation to the general Sp(2n)Sp(2n) vertex model. The fusion levels are m=1,…,nm=1,\dots,n (for Sp(6)Sp(6): the 6-, 14- and 14-prime-dimensional representations, the last level {n}\{n\} being the terminal representation of the truncated hierarchy), the analyticity strip is −12−Δ<Re(λ)<12-\frac{1}{2}-\Delta<\mathrm{Re}(\lambda)<\frac{1}{2} with Δ=n+1\Delta=n+1, and the central line is Re(λ)=−Δ2\mathrm{Re}(\lambda)=-\frac{\Delta}{2}. The zero analysis itself was carried out explicitly only for n=2n=2 and n=3n=3 (finite sizes L=6,12L=6,12 for Sp(6)Sp(6), where Λ0(6)\Lambda_0^{(6)} and Λ0(14)\Lambda_0^{(14)} have zeros on Re(λ)=−2\mathrm{Re}(\lambda)=-2 while Λ0(14′)\Lambda_0^{(14')} is zero-free); for general nn the paper states that it infers and assumes that only the last fusion-level leading eigenvalue Λ0({n})(λ)\Lambda_0^{(\{n\})}(\lambda) is free of zeros in the strip while every other level's leading eigenvalue carries zeros on the central line, with density growing linearly in LL. On this assumption the logarithmic derivatives ω(m)(λ)=ddλlog⁡κ(m)(λ)\omega^{(m)}(\lambda)=\frac{d}{d\lambda}\log\kappa^{(m)}(\lambda) of the thermodynamic eigenvalues κ(m)(λ)\kappa^{(m)}(\lambda) (the L→∞L\to\infty limit of (Λ0(m)(λ))1/L(\Lambda_0^{(m)}(\lambda))^{1/L}) satisfy a closed system of functional equations, solved by Fourier-Laplace transform; the answer (Eqs. (56)-(57) of the source) splits into a gamma-function-product term - what the solution would be if the eigenvalues were analytic throughout the strip - plus CDD-type integral terms that encode the lost analyticity, for instance ∫0∞e−nt(1−e−t)(1−e−nt)(1+e−(n+1)t)(eλt+e−λt)dt\int_0^\infty \frac{e^{-nt}(1-e^{-t})}{(1-e^{-nt})(1+e^{-(n+1)t})}\left(e^{\lambda t}+e^{-\lambda t}\right)dt in ωII(1)(λ)\omega_{II}^{(1)}(\lambda), where the subscripts I and II label the two sides of the cut line Re(λ)=−Δ2\mathrm{Re}(\lambda)=-\frac{\Delta}{2}. The free energy so obtained agrees with the nested Bethe-ansatz string analysis of the Sp(2N)Sp(2N) spin chain (Martins, 2002) - the Bethe ansatz being the standard exact-solution method for quantum spin chains, in which complex roots organize into strings - so the thermodynamic result is independently cross-checked; the open questions concern its analytic foundations. At n=2n=2 (and similarly n=3n=3) the CDD integrals reduce by partial fractions to gamma- and digamma-function expressions, but the paper states that for arbitrary nn this reduction fails because the required partial-fraction expansion of the integrand changes greatly with nn. The partition function is also the stated platform for a correlation-function program: for the O(n)O(n) spin chain, two-site density matrices were previously obtained from quantum Knizhnik-Zamolodchikov (qKZ) type functional equations - difference equations in the spectral parameters whose solutions encode density matrices of integrable spin chains - and the source expects an analogous fused qKZ system for the Sp(2n)Sp(2n) chain, whose derivation it lists as a separate remaining challenge.

Problem Statement

Consider the fundamental Sp(2n)Sp(2n) integrable vertex model on the square lattice, with R-matrix R12(λ)=λ(λ+Δ)I12+(λ+Δ)P12+λE12R_{12}(\lambda)=\lambda(\lambda+\Delta)I_{12}+(\lambda+\Delta)P_{12}+\lambda E_{12} acting on the 2n2n-dimensional fundamental representation, crossing parameter Δ=n+1\Delta=n+1, and transfer matrices T(m)(λ)T^{(m)}(\lambda) fused to the levels m=1,…,nm=1,\dots,n of the truncated fusion hierarchy, the last level {n}\{n\} being its terminal representation. Let Λ0(m)(λ)\Lambda_0^{(m)}(\lambda) denote the leading (largest-magnitude) eigenvalue of T(m)(λ)T^{(m)}(\lambda) at spectral parameter λ\lambda, and let LL be the lattice circumference.

(1) Prove, or refute, the following statement for arbitrary rank n≥2n\ge 2: the analyticity strip is −12−Δ<Re(λ)<12-\frac{1}{2}-\Delta<\mathrm{Re}(\lambda)<\frac{1}{2}; within it, Λ0({n})(λ)\Lambda_0^{(\{n\})}(\lambda) has no zeros, while for every level m<nm<n the eigenvalue Λ0(m)(λ)\Lambda_0^{(m)}(\lambda) has zeros lying on the central line Re(λ)=−Δ2\mathrm{Re}(\lambda)=-\frac{\Delta}{2}, with the number of such zeros growing linearly in LL, so that as L→∞L\to\infty they condense into an extended singularity that splits the strip in two. (Currently this structure is verified by finite-size inspection only for n=2n=2 and n=3n=3 and is assumed for general nn in the derivation of the partition function.)

(2) Determine whether the CDD-type integral terms generated by this lost analyticity in the thermodynamic-limit solution, namely ∫0∞e−nt(1−e−t)(1−e−nt)(1+e−(n+1)t)(eλt+e−λt) dt\int_0^\infty \frac{e^{-nt}(1-e^{-t})}{(1-e^{-nt})(1+e^{-(n+1)t})}\left(e^{\lambda t}+e^{-\lambda t}\right)\,dt in ωII(1)(λ)\omega_{II}^{(1)}(\lambda) (Eq. (56) of the source; here ω(m)=ddλlog⁡κ(m)\omega^{(m)}=\frac{d}{d\lambda}\log\kappa^{(m)} is the logarithmic derivative of the thermodynamic partition function per site, and the subscripts I and II label the two sides of the cut line Re(λ)=−Δ2\mathrm{Re}(\lambda)=-\frac{\Delta}{2}) and the corresponding integrals for the remaining functions ωII(m)(λ)\omega_{II}^{(m)}(\lambda), m=2,…,n−1m=2,\dots,n-1, admit a closed representation as logarithmic derivatives of finite products of gamma (or digamma) functions given by a single formula uniform in nn - or establish that no such finite gamma-product representation exists. (At n=2n=2 and n=3n=3 such forms follow from partial-fraction expansions whose structure changes with nn.)

The two parts serve one objective: to place the exact thermodynamic-limit solution of the general-rank Sp(2n)Sp(2n) vertex model on the same rigorous, closed-form footing as the rank-two case. The source paper additionally lists, as separate open directions outside the scope of this problem, the derivation of fused quantum Knizhnik-Zamolodchikov equations that would completely determine correlation functions of the Sp(2n)Sp(2n) quantum spin chain, and the extension of the lost-analyticity functional method to other integrable models.

References

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