Determinant form of correlators in high rank integrable spin chains via separation of variables

Abstract

In this paper we take further steps towards developing the separation of variables program for integrable spin chains with $$ \mathfrak{gl}(N) $$ 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.

Separation of variables in AdS/CFT: functional approach for the fishnet CFT

“…This type of orthogonality relations in the context of the spin chains are very useful, e.g. made it possible [7,13] to extract the SoV measure explicitly for higher-rank models.…”

“…Sever in [49]. Generalising this result to all states is currently an important open problem, but the general methods of [4,7,20] give a clear indication of how to construct the SoV transformation at least formally. The explicit construction of such map is referred to as the operatorial approach to SoV.…”

“…Here we introduce them for the first time for the single-trace operators in the fishnet model at finite coupling. Following [7,15], we show how variations of these parameters can be used to generate nontrivial operators acting on the spin chain, and to compute their expectation values in terms of Q-functions, as we discuss below.…”

“…The Separation of Variables (SoV), which traces its origin to the Hamilton-Jacobi approach and later to Sklyanin's works [1][2][3][4], is often regarded as the most powerful method to solve quantum integrable systems (for recent developments see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]). It is based on the expected property that eigenfunctions of the integrals of motion factorise when evaluated in a special basis x| labelled by the "separated variables" x ≡ {x n } L n=1 , where L is the number of degrees of freedom.…”

“…When it can be worked out, the SoV gives access not only to the spectrum and eigenfunctions of the integrals of motion (IM), but also usually leads to simple expressions for correlation functions (see e.g. [21,22] for examples in two-dimensional integrable field theories and recent work [7] for spin chains). The SoV paradigm is expected to hold true for quantum integrable systems carrying any representation of the symmetry algebra, in contrast e.g.…”