On the properties of the density matrix of the $\mathfrak{sl}_{n+1}$-invariant model

Abstract: We present an ansatz of generalizing the construction of recursion relations for the correlation functions of the $\mathfrak{sl}_2$-invariant fundamental exchange model in the thermodynamic limit by Jimbo, Miwa, Smirnov, Takeyama and one of our present authors in 2004 for higher rank. Due to the structure of the correlators as functions of their inhomogeneity parameters, a recursion formula for the reduced density matrix was proven. In the case of $\mathfrak{sl}_3$, we use the explicit results of Kluemper and Ribeiro, and Nirov, Hutsalyuk and one of our present authors for the reduced density matrix of up to operator length three to verify whether it is possible to relate the residues of the density matrix of length $n$ to the density matrix of length smaller than $n$ as in $\mathfrak{sl}_2$. This is unclear, since the reduced quantum Knizhnik--Zamolodchikov equation splits into two parts for higher rank. In fact, we show two relations, one of which is a straightforward generalisation to the $\mathfrak{sl}_2$ case and one which is completely new. This allows us to construct an analogue of the operator $X_k$ which we call Snail Operator. In the $\mathfrak{sl}_2$-case, this operator has many nice properties including in particular the fact that only one irreducible representation of the Yangian $Y(\mathfrak{sl}_2)$, the Kirillov--Reshetikhin module $W_k$, contributed the residue at $\lambda_i-\lambda_j=-(k+1)$. Here, we give an overview of the mathematical background, T-systems, and show a new application of the extended T-systems introduced by Mukhin and Young in 2012 regarding the Snail Operator.