On scalar products in higher rank quantum separation of variables

Abstract

Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models , we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states, that are transfer matrix eigenstates or some simple generalization of them. In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector \langle\underline{\mathbf{h}}\rangle⟨𝐡̲⟩, there could be more than one non-vanishing overlap \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle⟨𝐡̲|𝐤̲⟩ with the vectors |{\underline{\mathbf{k}}}\rangle|𝐤̲⟩ of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the \mathcal{Y}(gl_3)𝒴(gl3) lattice model with NN sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality (by standard we mean \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle \propto \delta_{\underline{\mathbf{h}}, \underline{\mathbf{k}}}⟨𝐡̲|𝐤̲⟩∝δ𝐡̲,𝐤̲). In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the \mathcal{Y}(gl_2)𝒴(gl2) rank one case.

Correlation functions by separation of variables: the XXX spin chain

“…Given the recent notable achievements [6,88,[117][118][119][120][121][122][123][124][125][126][127][128][129] in their SoV description, with first results available also on compact scalar product formulae, the task to compute correlation functions in higher rank seems very promising nowadays. This is, in particular, the case if one can compute the action of the local operators in the simplified SoV basis introduced in [127] for the rank 2 models. Indeed, under this choice the SoV measure simplify considerably and rank 1 Slavnov's type determinants appear for the scalar product of separate states with transfer matrix eigenvectors.…”

“…Under the condition b = 0, one can apply Sklyanin's SoV approach [3,4], see also [114]. Here, we follow the presentation given in section 2 of [117] and in [127] for the diagonalization of the transfer matrix in this general twisted case. The separate variables are generated by the operator zeros of B (K) (λ).…”

“…while the SoV representation of C (K) (λ) follows from the above ones and the quantum determinant condition. Similarly, following Corollary B.2 of [127], the right SoV basis of H can be constructed as…”

“…4 A class of states with factorized wave-functions in the SoV basis, which notably includes the eigenstates of the transfer matrix. 5 Determinant formulae for scalar products of separate states have also emerged recently in [127] for the higher rank gl(3) case, under special choices of the conserved charges generating the SoV bases. Let us also mention the interesting and recent papers [128,129], also dealing with the computations of higher rank scalar products in a related SoV framework.…”

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

“…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.…”