Separation of variables for the quantum SL(2,R) spin chain

TL;DR

The paper develops a complete Separation of Variables framework for the quantum $SL(2,\mathbb{R})$ spin chain, constructing the SoV representation, integral eigenfunctions, and the Sklyanin measure. It demonstrates that the SoV kernel is effectively a product of Baxter $\mathbb{Q}$-operators projected onto a reference state, and it establishes a multi-dimensional Baxter equation governing the separated-variable wavefunctions. A pyramid-diagram construction provides a transparent, diagrammatic route to the kernel and to the integration measure, while connecting SoV with the Algebraic Bethe Ansatz through explicit eigenfunctions and Bethe roots. The work paves the way for analyzing noncompact spin chains in QCD-like contexts and offers a framework extendable to inhomogeneous chains and related integrable systems.

Abstract

We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.

Figures (9)

• Figure 1: Diagrammatical representation of the function $\Lambda_u(z_1,\ldots,z_N|w_2,\ldots,w_N)$. The arrow with the index $\alpha$ that connects the points $\bar{w}$ and $z$ stands for $(z-\bar{w})^{-\alpha}$.
• Figure 2: Diagrammatical proof of the identity (\ref{['L-perm']}). One inserts two vertical lines with the indices $\pm\alpha=\pm i(x_2-x_1)$ into one of the rhombuses and displaces them in the directions indicated by arrows with a help of identities shown in Figures \ref{['comm-f']} and \ref{['amp1']}.
• Figure 3: Diagrammatical representation of the function $U_{\hbox{\boldmath${x}$}}(z;\bar{w}_N)$. The indices $\alpha_k=s-ix_k$ and $\beta_k=s+ix_k$ parameterize the corresponding factors entering (\ref{['La']}). The $SL(2,\mathbb{R})$ integration (\ref{['measure']}) over the position of internal vertices is implied.
• Figure 4: The scalar product of two pyramid diagrams at $N=2$.
• Figure 5: Diagrammatical calculation of the scalar product of two pyramids. The fat points indicate the vertices which can be integrated out using the chain relation shown in Figure \ref{['Chain']}.