Completing the quantum inverse problem in separation of variables: reconstruction of all local operators and their form factors for the open spin-1/2 XXZ chain with non-diagonal boundaries

The open spin-1/2 XXZ quantum chain with integrable non-diagonal boundary matrices (physically, unparallel magnetic fields acting on the two ends of the chain) is exactly solved at the spectral level: quantum separation of variables (SoV) yields its complete transfer-matrix spectrum, eigenstates, and determinant scalar products, including for the most general boundaries. The operator side, however, is incomplete: the paper that founded this SoV program (Niccoli, J. Stat. Mech. 2012) explicitly states that its reconstruction program "is not yet completed", having reconstructed only special classes of quasi-local operators — notably strings of spin-lowering matrices — in terms of the quantum separate variables of the 6-vertex reflection algebra. The problem is the completion of that program in the class treated by the founding paper — one general non-diagonal boundary matrix with arbitrary complex parameters and one diagonal or triangular boundary matrix: for arbitrary chain length and generic, unconstrained boundary and inhomogeneity parameters within this class, reconstruct all local operators of the chain (equivalently, exhibit a complete convenient operator basis) in terms of the reflection-algebra separate variables, prove completeness, and compute closed-form matrix elements (form factors) of arbitrary local operators between transfer-matrix eigenstates — determinant formulae being the program's stated motivation in the founding paper — the standard input for computing arbitrary correlation functions. Correlation-function building blocks are known only for constrained boundary parameters, for one boundary fixed at a special value, for the rational XXX chain under one constraint, or for the part of the spectrum described by homogeneous Baxter TQ equations.

Background

The spin-1/2 XXZ chain is one of the canonical exactly solved lattice models of statistical mechanics: on $N$ sites its Hamiltonian is a nearest-neighbour anisotropic Heisenberg interaction, $H=\sum_n(\sigma_n^x\sigma_{n+1}^x+\sigma_n^y\sigma_{n+1}^y+\cosh\eta\,(\sigma_n^z\sigma_{n+1}^z-1))$, where $\sigma_n^{x,y,z}$ are Pauli matrices on site $n$ and the anisotropy parameter $\eta$ is taken generic (not a rational multiple of $i\pi$, the so-called non-root-of-unity regime). On an open (non-periodic) chain, integrability is compatible with boundary terms parametrized by two $2\times 2$ boundary matrices $K_\pm$, one attached to each chain end; physically these are magnetic fields at the two ends. Sklyanin's 1988 extension of the quantum inverse scattering method (QISM) realizes such models through the reflection algebra: a boundary monodromy matrix built from the 6-vertex R-matrix (the trigonometric solution of the Yang–Baxter equation, the cubic consistency relation guaranteeing integrability) and the K-matrices satisfies a quadratic exchange algebra, and its trace — the transfer matrix $\mathcal T(\lambda)$, with spectral parameter $\lambda$ — generates a commuting family of conserved quantities containing the Hamiltonian. In practice one works with the inhomogeneous chain, in which site-dependent shift parameters $\xi_n$ are inserted; the physical chain is recovered in the homogeneous limit $\xi_n\to 0$.

When both K-matrices are diagonal (boundary fields parallel to the anisotropy axis), a conserved $U(1)$ charge survives and the algebraic Bethe ansatz (ABA) applies. The correlation-function program could then be carried through completely: local operators are first reconstructed in terms of the monodromy-matrix entries — the quantum inverse scattering problem, solved for periodic chains by Göhmann and Korepin and for open chains with diagonal boundaries by the Lyon group — then their matrix elements between eigenstates (form factors) are computed, and correlation functions are assembled into multiple-integral representations (for the open XXZ chain with diagonal boundaries this was completed in 2007–2008 by Kitanine, Kozlowski, Maillet, Niccoli, Slavnov and Terras).

For general non-diagonal K-matrices (unparallel boundary fields) no conserved $U(1)$ charge remains and no ABA reference state exists, so this whole route breaks down. The spectral problem was nevertheless solved by Sklyanin's quantum separation of variables: the transfer matrix is represented on an SoV basis in which each of $N$ quantum separate variables (shifted eigenvalues of a chosen reflection-algebra generator, for instance the entry $\mathcal B$ of the boundary monodromy) is diagonal; eigenstates — so-called separate states — factorize into products of single-variable states, the spectrum is fixed by a finite discrete system of $N$ equations, and scalar products of separate states reduce to $N\times N$ determinants. In this framework Niccoli (J. Stat. Mech. 2012 P10025) treated the class with one general non-diagonal and one diagonal or triangular boundary matrix, proving completeness and simplicity of the transfer-matrix spectrum and deriving determinant scalar products.

Computing matrix elements of local operators in SoV requires one further ingredient: identities reconstructing each local operator through the boundary-algebra generators whose action on the SoV basis is known — the quantum inverse problem in the SoV framework. In the inhomogeneous chain the natural objects are quasi-local operators: operator strings carrying the site-dependent inhomogeneity structure, which reduce to ordinary local-operator strings in the homogeneous limit. Niccoli (2012) obtained such reconstructions only for particular classes, notably the strings $\sigma_1^-\cdots\sigma_n^-$ and $\sigma_n^-\cdots\sigma_N^-$ of spin-lowering matrices ($\sigma^\pm$ denote the spin-1/2 raising and lowering matrices), together with determinant formulae for their matrix elements on transfer-matrix eigenstates, and stated explicitly that the reconstruction program is not yet completed. The spectral side was later completed for the most general non-diagonal boundaries (Faldella–Kitanine–Niccoli 2014, including a proof that the SoV characterization is equivalent to an inhomogeneous Baxter $TQ$ functional equation — a second-order finite-difference equation relating the transfer-matrix eigenvalue $T$ to an auxiliary function $Q$), and determinant scalar products for general boundaries are also known. What is still missing, and is the subject of this problem, is the operator side: the reconstruction of all local operators, a complete operator basis, and the resulting form factors — the last missing step toward arbitrary correlation functions for the open XXZ chain with non-diagonal boundaries.

Problem Statement

For the open spin-1/2 XXZ quantum chain — equivalently, the transfer matrix of the 6-vertex reflection algebra in its general inhomogeneous spin-1/2 representation, with one general non-diagonal boundary matrix $K(\lambda;\zeta,\kappa,\tau)$ (arbitrary complex boundary parameters) at one chain end and a diagonal or triangular boundary matrix at the other, the class of models which in the homogeneous limit $\xi_n\to 0$ reproduces the open spin-1/2 XXZ quantum chains with non-diagonal boundary conditions — complete the local quantum inverse problem in the separation-of-variables framework. Specifically, for arbitrary chain length $N$ and generic, unconstrained values of the boundary and inhomogeneity parameters within this class:

1. Construct the reconstruction of all local operators of the chain in terms of the quantum separate variables of the reflection algebra — equivalently, identify a complete convenient basis of (quasi-)local operators whose action on the SoV basis, or on transfer-matrix eigenstates, is explicitly computable.
2. Prove completeness of the reconstruction: the reconstructed set spans all local operators (all operators acting on an arbitrary block of sites $1,\dots,m$ with $m\le N$) for arbitrary $N$ and generic parameters.
3. Use this reconstruction to compute closed-form expressions for the matrix elements (form factors) $\langle t'|O|t\rangle$ of arbitrary local operators $O$ between transfer-matrix eigenstates — determinant representations being the founding paper's stated motivation and the form of the existing results for special classes — so that arbitrary correlation functions of the open XXZ chain with non-diagonal boundaries become computable in principle, e.g. as form-factor (spectral) series. The thermodynamic-limit control needed to make such series quantitative at large $N$ — in particular of the ground-state distribution for general unconstrained boundaries — is a separately flagged difficulty of the same program and may be addressed within the answer, but does not replace the reconstruction, completeness and form-factor deliverables above.

The constrained boundary-parameter, special-boundary-value and $\sigma^\pm$-string sub-cases resolved in the follow-up literature constitute partial progress toward this program, not its completion.

References

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