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

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.