[Paper Review] Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables

Nikolay Gromov, Fedor Levkovich-Maslyuk+1 more|arXiv (Cornell University)|Nov 16, 2020

Algebraic structures and combinatorial modelsMathematics98 references25 citations

TL;DR

This paper develops a determinant formula for correlation functions and wave function overlaps in high-rank integrable spin chains with gl(N) symmetry using the separation of variables (SoV) program. By explicitly computing the SoV measure matrix elements, the authors derive compact determinant representations for overlaps between on-shell and off-shell Bethe states, as well as between states with different twists—key results for AdS/CFT applications. The method extends previous work to general spin values and the non-compact case for the first time, providing a systematic framework for computing form factors and expectation values in higher-rank integrable models.

ABSTRACT

In this paper we take further steps towards developing the separation of variables program for integrable spin chains with gl(N) symmetry. By finding, for the first time, the matrix elements of the SoV measure explicitly we were able to compute correlation functions and wave function overlaps in a simple determinant form. In particular, we show how an overlap between on-shell and off-shell algebraic Bethe states can be written as a determinant. Another result, particularly useful for AdS/CFT applications, is an overlap between two Bethe states with different twists, which also takes a determinant form in our approach. Our results also extend our previous works in collaboration with A. Cavaglia and D. Volin to general values of the spin, including the SoV construction in the higher-rank non-compact case for the first time.

Motivation & Objective

• To extend the separation of variables (SoV) program to high-rank integrable spin chains with gl(N) symmetry, including non-compact and general spin representations.
• To compute the matrix elements of the SoV measure explicitly, enabling the evaluation of scalar products and overlaps in determinant form.
• To derive determinant representations for non-trivial observables such as overlaps between on-shell and off-shell Bethe states and between states with different twists.
• To generalize previous results for sl(2) and sl(3) to arbitrary gl(N) and the non-compact case, providing a unified framework for form factor computations.

Proposed method

• The authors use the SoV framework to construct separated variable bases for gl(N) spin chains, deriving wave functions that factorize into universal blocks.
• They compute the SoV measure matrix elements directly from integral representations and functional relations, using Baxter Q-functions and transfer matrices.
• A key technical advance is the derivation of a general formula for the measure in terms of Vandermonde determinants and Pochhammer-like products in the spectral parameters.
• The method employs a 'det-product' construction that maps the scalar product to a determinant via a novel algebraic structure involving quantum minors and the ˚-map.
• The approach is validated by showing consistency with known results in sl(2) and sl(3), and extended to the non-compact case using oscillator realizations of gl(N) generators.
• The formalism is implemented in Mathematica for explicit computation of measure elements and overlaps, with code provided in the appendix.

Experimental results

Research questions

• RQ1 How can the SoV measure be computed explicitly in high-rank gl(N) spin chains for arbitrary spin values, including non-compact representations?
• RQ2 Can overlaps between on-shell and off-shell Bethe states be expressed in a compact determinant form within the SoV framework?
• RQ3 What is the determinant representation of the overlap between two Bethe states with different twist parameters in gl(N) spin chains?
• RQ4 How does the SoV measure generalize to non-compact gl(N) models, and what role do oscillator representations of generators play in this construction?
• RQ5 Can the formalism be used to compute form factors of derivatives of transfer matrices, such as local spin expectation values, in determinant form?

Key findings

• The matrix elements of the SoV measure are derived in closed form for general gl(N) spin chains, with explicit expressions provided for sl(3) and general N.
• Overlaps between on-shell and off-shell Bethe states are shown to take a determinant form, generalizing known results from sl(2) to higher rank.
• Overlaps between Bethe states with different twist parameters are also expressed as determinants, a result previously unknown beyond sl(3).
• The method successfully extends to the non-compact case, including arbitrary spin s, with poles in the measure cancellation proven via functional relations.
• The local spin expectation value is computed as a determinant of a matrix built from the measure and transfer matrix derivatives, confirming consistency with known results.
• The Mathematica implementation provided allows for explicit computation of measure elements and overlaps, with example outputs shown for sl(4) and sl(3) chains.