Quantum inverse problem and determinant form factors for the antiperiodic XXZ spin-s chain by quantum separation of variables

The antiperiodic XXZ Heisenberg chain with arbitrary spins on its sites - the transfer matrix is twisted by the spin-flip matrix $\sigma^x$ - is the higher-spin trigonometric model whose transfer-matrix spectrum has been completely characterized within Sklyanin's quantum separation of variables (SOV), including its reformulation through Baxter-type T-Q functional equations. The neighbouring models of the same program - the antiperiodic spin-1/2 XXZ chain, the antiperiodic higher-spin XXX chain, and the antiperiodic dynamical six-vertex (SOS) model - additionally admit a solution of the quantum inverse problem, i.e. the reconstruction of all local spin operators from the quantum separate variables, and consequently closed one-determinant formulae for the form factors of local operators between transfer-matrix eigenstates, the elementary building blocks of correlation functions. For the antiperiodic XXZ chain with spin $s>1/2$ these two steps are missing: no reconstruction of the local spin generators in terms of the separate variables has been published, and no closed determinant formulae exist for the scalar products of separate states or for local-operator form factors of this model. The problem is to complete the SOV form-factor program for the antiperiodic higher-spin XXZ chain, the keystone being the quantum inverse problem. The model is a lattice discretization of the sine-Gordon field theory beyond the fundamental representation, so its exact form factors are the entry point to a microscopic construction of sine-Gordon correlation functions.

Background

The spin-$s$ Heisenberg chains are paradigmatic exactly solvable quantum lattice models: each site $n$ carries a $(2s+1)$-dimensional irreducible representation of the spin algebra, and neighbouring spins interact through

$H=\sum_n \left(S^x_n S^x_{n+1}+S^y_n S^y_{n+1}+\Delta\, S^z_n S^z_{n+1}\right)$

with $S^a_n$ the spin generators; $\Delta=1$ gives the isotropic XXX chain and $|\Delta|<1$ the anisotropic XXZ chain, whose fundamental ($s=1/2$) version is the six-vertex model. Integrability is encoded not in $H$ directly but in a commuting family of transfer matrices: traces over an auxiliary space of monodromy matrices, themselves products of local Lax operators satisfying the Yang-Baxter equation; the quadratic relations obeyed by the monodromy-matrix entries form the six-vertex Yang-Baxter algebra. Higher-spin chains are obtained by fusion, i.e. by composing several fundamental Lax operators in the auxiliary space and projecting, which produces Lax operators acting on $(2s+1)$-dimensional local spaces. 'Antiperiodic' boundary conditions mean that the model is generated by the transfer matrix $\bar T(\lambda)$ defined with a twist by the spin-flip matrix $\sigma^x$ (equivalently, the chain closes with a spin flip, so that one turn around the chain composes translation with spin reversal). Because this twist is non-diagonal, the total $S^z$ is not conserved, the algebraic Bethe ansatz has no simple reference state (pseudo-vacuum) to build on, and the determinant formula for scalar products of Bethe states available in the periodic case is not applicable: the standard technology breaks down.

Sklyanin's quantum separation of variables (SOV) is the method built for such situations. For the representations at hand, one monodromy-matrix element, conventionally $\hat B(\lambda)$, has a simple nondegenerate spectrum; its eigenbasis is the basis of quantum separate variables, in which the transfer matrix acts by elementary finite shifts and the spectral problem reduces to a discrete system of equations on the interpolation values of the eigenvalue function at the inhomogeneity parameters (spectral-shift parameters attached to the sites) of the chain. States with factorized wavefunctions in this basis - 'separate states', a class containing all transfer-matrix eigenstates - have scalar products given by the universal determinant formulae of the general SOV framework, whose matrix elements are sums over the spectrum of the separate variables; the explicit closed specialization of these formulae to the antiperiodic higher-spin XXZ model has not been constructed and is what item 2(a) of the Problem Statement requests. Two further ingredients turn this into correlation-function technology: the quantum inverse problem, i.e. the reconstruction of every local operator $X_n$ acting on site $n$ in terms of the monodromy-matrix entries and hence of the separate variables; and, through it, form factors $\langle t|X_n|t'\rangle$, the matrix elements of local operators between transfer-matrix eigenstates, which in the solved cases are given by closed one-determinant formulae. A complementary route to the spectrum is Baxter's T-Q functional equation $t(\lambda)Q(\lambda)=a(\lambda)Q(\lambda+\eta)+d(\lambda)Q(\lambda-\eta)$, whose solutions $Q$ are eigenvalues of a Q-operator; in the antiperiodic case the admissible $Q$-functions are trigonometric polynomials with doubled periodicity rather than ordinary product polynomials, so the associated Bethe equations and their completeness require separate proofs.

In Niccoli (2013) this whole program was carried out for the antiperiodic higher-spin XXX chain with arbitrary site spins $\{s_1,\dots,s_N\}$: complete characterization and simplicity of the $\bar T$-spectrum, the antiperiodic solution of the quantum inverse problem, determinant scalar products, and one-determinant form factors of the local spin generators. Its conclusions announce the extension to the antiperiodic XXZ spin-$s$ chains: the spectral characterization and the scalar-product formulae 'can be re-derived trivially along the same line', but 'the main technical point to address remains the solution of the inverse problem for the local spin generators in a similar form', from which the form factors would follow by the same steps; the author motivates the extension by the strict relation between the spin-$s$ XXZ chain and lattice discretizations of the sine-Gordon field theory.

Subsequent literature has settled the neighbouring cases: the antiperiodic spin-1/2 XXZ chain is fully solved by SOV including determinant form factors (Niccoli, 2013; Pei-Terras, 2022); the complete spectrum of the antiperiodic XXZ chain with arbitrary site spins, together with its exact reformulation in terms of Baxter-type T-Q equations with completeness proofs, is due to Niccoli-Terras (2015); the antiperiodic SOS (dynamical six-vertex) model, including determinant form factors of local operators, was solved by Niccoli (2013) and Levy-Bencheton, Niccoli and Terras (2016); a Q-operator for higher-spin quasi-periodic representations was constructed by Maillet-Niccoli (2021) in the rational case, the trigonometric homogeneous case going back to Yung-Batchelor (1995). What no later work supplies is the higher-spin ($s>1/2$) trigonometric core: the reconstruction of the local spin generators of the antiperiodic XXZ chain from the separate variables, the determinant scalar products for this model, and the determinant form factors that follow from them.

Problem Statement

For the inhomogeneous antiperiodic XXZ chain with arbitrary site-dependent spins $\{s_1,\dots,s_N\}$ - higher-spin highest-weight representations of the trigonometric six-vertex Yang-Baxter algebra, with antiperiodic transfer matrix $\bar T(\lambda)$ (twist by $\sigma^x$) - the SOV basis, the complete characterization of the transfer-matrix spectrum and eigenstates, and the equivalent Baxter-type T-Q functional-equation descriptions are known (Niccoli-Terras, 2015). Complete the SOV solution of this model, specifically:

1. (Quantum inverse problem) Reconstruct every local operator $X_n\in\mathrm{End}(V^{(s_n)}_n)$, where $V^{(s_n)}_n$ is the $(2s_n+1)$-dimensional local spin space on site $n$ - in particular the local spin generators $S^{\pm}_n$ and $S^z_n$ - as explicit operators in the quantum separate variables (equivalently, in the entries of the antiperiodic monodromy matrix), in a form analogous to the reconstruction available for the antiperiodic higher-spin XXX chain and for the antiperiodic spin-1/2 XXZ chain.

2. (Determinant formulae) Derive closed determinant formulae for (a) the scalar products of separate states, a class containing all $\bar T$-eigenstates, i.e. the explicit closed specialization of the universal SOV determinant to the antiperiodic higher-spin XXZ model, and (b) the form factors $\langle t|X_n|t'\rangle$ of local spin operators between transfer-matrix eigenstates, obtained from (a) through the reconstruction of item 1, following the pattern of the antiperiodic higher-spin XXX and antiperiodic spin-1/2 XXZ results.

Items 1 and 2 constitute a single program: the reconstruction is the keystone that converts the universal SOV scalar-product determinant into closed local-operator form factors, and completing it gives the antiperiodic XXZ chain with arbitrary spins the same exact finite-volume form-factor toolkit already available for the antiperiodic higher-spin XXX, antiperiodic spin-1/2 XXZ, and antiperiodic SOS models - the entry point to correlation functions of the higher-spin anisotropic chain and of its sine-Gordon continuum limits. The homogeneous chain may be treated either directly or as the limit of the inhomogeneous formulae, as in the neighbouring cases.

References

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