A universal formula expresses the normalized overlap between an integrable matrix product state (MPS) and an on-shell Bethe eigenstate of a rational (Yangian) spin chain as the product of an MPS-independent ratio of Gaudin determinants and an MPS-dependent prefactor built from eigenvalues of commuting operators constructed from the K-matrix defining the state. Gombor (arXiv:2505.20234, JHEP 2025) proved this formula in full representation-independent generality for glN\mathfrak{gl}_N-symmetric chains using only the algebraic Bethe ansatz and the KT-relation, and proposed its extension to soN\mathfrak{so}_N- and spN\mathfrak{sp}_N-symmetric chains: nested G- and F-operators were defined via algebra embeddings and low-rank isomorphisms, and the one-particle overlap functions were determined at every Bethe node under the assumption that the universal form holds — but the proof was explicitly left open, the author noting that it requires recursion, action and coproduct formulas for the off-shell Bethe vectors of these algebras, then available only for so2n+1\mathfrak{so}_{2n+1}. Those prerequisites have since been established for all classical series (Liashyk–Pakuliak–Ragoucy, 2026), yet no proof of the orthogonal/symplectic universal overlap formula has been published. The problem is to prove — or disprove by an explicit counterexample — the proposed universal overlap formula for soN\mathfrak{so}_N and spN\mathfrak{sp}_N rational spin chains in generality matching the glN\mathfrak{gl}_N theorem, including the Bethe-root selection rules and the cases where quasi-determinants of the K-matrix are ill-defined and a deformation-and-limit argument is required.

Background

Rational spin chains and Bethe states. A length-LL quantum spin chain is integrable when its sites carry representations of a symmetry algebra g\mathfrak{g} and its transfer matrix is built from an RR-matrix solving the Yang–Baxter equation. For the rational glN\mathfrak{gl}_N-invariant chain, R(u)=1+P/uR(u)=\mathbf{1}+P/u with PP the permutation operator; for orthogonal and symplectic chains, R(u)=1+P/u−Q/(u+κN)R(u)=\mathbf{1}+P/u-\mathbf{Q}/(u+\kappa_N), where Q\mathbf{Q} is constructed from the invariant bilinear form of soN\mathfrak{so}_N or spN\mathfrak{sp}_N (spN=sp2n\mathfrak{sp}_N=\mathfrak{sp}_{2n}) and κN=(N∓2)/2\kappa_N=(N\mp 2)/2 distinguishes the two cases. The transfer matrices generate the Yangian Y(gN)Y(\mathfrak{g}_N), an infinite-dimensional deformation of the enveloping algebra. In the algebraic Bethe ansatz (ABA), eigenstates (on-shell Bethe states) are created by monodromy-matrix elements acting on a pseudovacuum (the reference state annihilated by the lowering entries of the monodromy matrix) and are labelled by one set of rapidities (Bethe roots) per node of the Dynkin diagram of gN\mathfrak{g}_N; off-shell Bethe vectors are the same operator products evaluated at arbitrary rapidities. The Gaudin matrix is the Jacobian of the Bethe equations with respect to the roots; its determinants give Bethe-state norms and appear inside overlap formulas.

Integrable boundary states and the KT-relation. A matrix product state (MPS) has the form ⟨Ψ∣=∑tr(Ai1⋯AiL) ⟨i1⋯iL∣\langle\Psi|=\sum\mathrm{tr}(A_{i_1}\cdots A_{i_L})\,\langle i_1\cdots i_L|, where the matrices act on an auxiliary boundary space; a two-site product state is the degenerate case of a one-dimensional boundary space. The MPS is called integrable when its associated K-matrix solves the reflection equation, i.e. furnishes a representation of a twisted Yangian (reflection algebra) — a coideal subalgebra of the Yangian determined by the boundary symmetry. Such overlaps are the central building block of the quench-action method for non-equilibrium integrable dynamics (a saddle-point framework determining post-quench steady states from these overlaps) and of one-point functions in defect versions of N=4\mathcal{N}=4 super Yang–Mills theory and of ABJM theory (the Aharony–Bergman–Jafferis–Maldacena Chern–Simons matter theory, the other main integrable instance of the AdS/CFT correspondence). Gombor and Pozsgay (2021) introduced the KT-relation, K(u) ⟨Ψ∣T(u)=⟨Ψ∣T(−u) K(u)\mathbf{K}(u)\,\langle\Psi|T(u)=\langle\Psi|T(-u)\,\mathbf{K}(u) with T(u)T(u) the monodromy matrix, which reduces overlap computations to purely algebraic Bethe-ansatz manipulations that do not depend on which physical representation the chain sites carry.

The universal overlap formula and its status. Building on ABA overlap proofs for glN\mathfrak{gl}_N chains (Gombor 2022, 2024) and on the twisted-Yangian construction of MPS overlaps in the SO(6) chain (de Leeuw et al. 2020), Gombor (2024, published in Physical Review Letters 2025) determined the MPS-dependent prefactor for all integrable MPS of the glN\mathfrak{gl}_N, oN\mathfrak{o}_N and spN\mathfrak{sp}_N chains without complete proof, and then (Gombor 2025, the source of this problem) proved for glN\mathfrak{gl}_N the universal formula

⟨MPS∣uˉ⟩⟨uˉ∣uˉ⟩=(∑ℓ=1dBβℓ∏ν=1n+∏k=1rν+F~ℓ(ν)(ukν))×det⁡G+det⁡G−,\frac{\langle\mathrm{MPS}|\bar{u}\rangle}{\sqrt{\langle\bar{u}|\bar{u}\rangle}}=\Big(\sum_{\ell=1}^{d_B}\beta_{\ell}\prod_{\nu=1}^{n_+}\prod_{k=1}^{r_\nu^+}\tilde{\mathcal{F}}_{\ell}^{(\nu)}(u_k^\nu)\Big)\times\sqrt{\frac{\det G^+}{\det G^-}},

i.e. the normalized overlap factorizes into an MPS-dependent prefactor — eigenvalue products of mutually commuting F-operators built as quasi-determinants (determinant-like combinations K22−K21K11−1K12\mathbf{K}_{22}-\mathbf{K}_{21}\mathbf{K}_{11}^{-1}\mathbf{K}_{12} of blocks of the nested K-matrices), summed with eigenvalues βℓ\beta_\ell of a boundary-space operator — times an MPS-independent ratio of Gaudin determinants fixed by the Bethe equations alone. Section 6 of the same paper then proposes the same universal structure for soN\mathfrak{so}_N and spN\mathfrak{sp}_N chains: assuming the universal form, it determines the one-particle overlap functions F~(ν)(u)\tilde{\mathcal{F}}^{(\nu)}(u) at every Bethe node for the reflection algebras Y(soN,soM⊕soN−M)Y(\mathfrak{so}_N,\mathfrak{so}_M\oplus\mathfrak{so}_{N-M}), Y(sp2n,sp2m⊕sp2n−2m)Y(\mathfrak{sp}_{2n},\mathfrak{sp}_{2m}\oplus\mathfrak{sp}_{2n-2m}), Y(so2n,gln)Y(\mathfrak{so}_{2n},\mathfrak{gl}_n) and Y(sp2n,gln)Y(\mathfrak{sp}_{2n},\mathfrak{gl}_n), checks the low-rank cases through the isomorphisms sp2≅gl2\mathfrak{sp}_2\cong\mathfrak{gl}_2, so3≅gl2\mathfrak{so}_3\cong\mathfrak{gl}_2 (via fusion) and so4≅gl2⊕gl2\mathfrak{so}_4\cong\mathfrak{gl}_2\oplus\mathfrak{gl}_2, and handles K-matrices with non-invertible blocks by a continuous deformation restoring invertibility, based on Molev's theorem that finite-dimensional irreducible twisted-Yangian representations descend from Yangian ones. No proof is given. The author states that a proof requires recursion, action and coproduct formulas for the off-shell Bethe vectors of soN\mathfrak{so}_N/spN\mathfrak{sp}_N chains — at the time available only for so2n+1\mathfrak{so}_{2n+1} (Liashyk–Pakuliak–Ragoucy). Those formulas have since been established for all classical series (Liashyk–Pakuliak–Ragoucy 2026), which makes the missing proof a concrete, well-posed objective.

Problem Statement

Prove, or disprove by explicit counterexample, the proposed universal overlap formula for soN\mathfrak{so}_N- and spN\mathfrak{sp}_N-symmetric rational spin chains, in the same representation-independent generality in which the glN\mathfrak{gl}_N case is a theorem (Gombor, arXiv:2505.20234). Concretely: for integrable MPS whose K-matrices satisfy the reflection equation and generate the reflection algebras Y(soN,soM⊕soN−M)Y(\mathfrak{so}_N,\mathfrak{so}_M\oplus\mathfrak{so}_{N-M}), Y(sp2n,sp2m⊕sp2n−2m)Y(\mathfrak{sp}_{2n},\mathfrak{sp}_{2m}\oplus\mathfrak{sp}_{2n-2m}), Y(so2n,gln)Y(\mathfrak{so}_{2n},\mathfrak{gl}_n) or Y(sp2n,gln)Y(\mathfrak{sp}_{2n},\mathfrak{gl}_n) treated in Section 6 of the source, and for chains with arbitrary finite-dimensional physical representations, establish that the normalized overlap between the MPS and any on-shell Bethe state equals the product of (i) the MPS-dependent prefactor — the sum over boundary eigenvalues βℓ\beta_\ell of products of one-particle overlap functions F~ℓ(ν)(u)\tilde{\mathcal{F}}^{(\nu)}_\ell(u), i.e. eigenvalues of the commuting G-/F-operators defined through quasi-determinants of the nested K-matrices — and (ii) the MPS-independent Gaudin-determinant ratio det⁡G+/det⁡G−\sqrt{\det G^+/\det G^-}. The derivation must also establish the selection rules under which the overlap is non-vanishing — the pair structure of the Bethe roots as defined in the source (in the chiral case the roots at each node split into two subsets whose pair-structure limit is tk−,ν=−tk+,ν−νt^{-,\nu}_k=-t^{+,\nu}_k-\nu, while in the achiral case the pairing identifies the roots at node ν\nu with those at node N−νN-\nu), including the cases with extra selection rules identified in the source — and must cover the same K-matrix class as the glN\mathfrak{gl}_N theorem: all integrable MPS for which the relevant quasi-determinants of the nested K-matrices exist, together with the K-matrices whose relevant blocks are non-invertible and for which the source defines the universal quantities through the deformation-and-limit procedure. Following the source author's own assessment, the anticipated route is by algebraic-Bethe-ansatz methods — the KT-relation together with recursion, action and coproduct formulas for the off-shell Bethe vectors of the gN\mathfrak{g}_N chains — matching the rigor and generality of the glN\mathfrak{gl}_N proof. A disproof consists of an explicit integrable soN\mathfrak{so}_N or spN\mathfrak{sp}_N MPS within the class for which the source proposes the formula — a non-singular, irreducible K-matrix, so that the proposed right-hand side is defined — and an on-shell Bethe state whose true overlap differs from the proposed formula.

References

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