On correlation functions of the open XYZ spin 1/2 chain with boundary fields related by a constraint

TL;DR

This paper derives exact correlation function representations for the open XYZ spin-1/2 chain with boundary fields under a specific constraint, extending previous XXZ results using Sklyanin's separation of variables and Baxter's transformation.

Contribution

It generalizes correlation function calculations to the XYZ chain with boundary conditions related by a constraint, employing Sklyanin's separation of variables and scalar product formulas.

Findings

Exact elementary blocks for correlation functions are obtained.

Correlation blocks have similar structure across XYZ, XXZ, and XXX chains.

No non-local tail operators are needed in the correlation function expressions.

Abstract

In this paper, we consider the quantum XYZ open spin-1/2 chain with boundary fields. We focus on the particular case in which the six boundary parameters are related by a single constraint enabling us to describe part of the spectrum by standard Bethe equations. We derive for this model exact representations for a set of elementary blocks of correlation functions, hence generalising to XYZ the results obtained in the XXZ open case in arXiv:2208.10097. Our approach is also similar to the approach proposed in the XXZ case arXiv:2208.10097: we solve the model by Sklyanin's version of the quantum Separation of Variables, using Baxter's Vertex-IRF transformation; in this framework, we identify a basis of local operators with a relatively simple action on the transfer matrix eigenstates; we then use the solution of the quantum inverse problem and our recent formulae on scalar products of separate states arXiv:2402.04112 to compute some of the corresponding matrix elements, which can therefore be considered as elementary building blocks for the correlation functions. The latter are expressed in terms of multiple sums in the finite chain, and as multiple integrals in the thermodynamic limit. Our results evidence that, once the basis of local operators is properly chosen, the corresponding building blocks for correlation functions have a similar structure in the XXX/XXZ/XYZ open chains, and do not require any insertion of non-local "tail operators".