# Integrable quenches in nested spin chains II: fusion of boundary transfer matrices

## TL;DR

This paper analyzes the spectrum of integrable boundary transfer matrices in the $SU(3)$-invariant spin chain, deriving functional relations and applying them to compute the Loschmidt echo after quantum quenches.

## Contribution

It provides a detailed spectral analysis of fused boundary transfer matrices using functional relations, advancing understanding of integrable quenches in nested spin chains.

## Key findings

- Derived functional relations for eigenvalues of fused operators.
- Computed Loschmidt echo for various quench protocols.
- Connected boundary transfer matrix spectrum to physical observables.

## Abstract

We consider quantum quenches in the integrable $SU(3)$-invariant spin chain (Lai-Sutherland model), and focus on the family of integrable initial states. By means of a Quantum Transfer Matrix approach, these can be related to "soliton-non-preserving" boundary transfer matrices in an appropriate transverse direction. In this work, we provide a technical analysis of such integrable transfer matrices. In particular, we address the computation of their spectrum: this is achieved by deriving a set of functional relations between the eigenvalues of certain "fused operators" that are constructed starting from the soliton-non-preserving boundary transfer matrices (namely the $T$- and $Y$-systems). As a direct physical application of our analysis, we compute the Loschmidt echo for imaginary and real times after a quench from the integrable states. Our results are also relevant for the study of the spectrum of $SU(3)$-invariant Hamiltonians with open boundary conditions.