Ground state and excitations of the \(C_2^{(1)}\) spin chain with non-diagonal boundaries

Abstract

Understanding boundary effects in higher-rank integrable quantum spin chains requires connecting exact spectral relations with the root configurations underlying their energies and excitations. With non-diagonal boundaries, determining these configurations remains challenging even when an exact spectral solution is available, hindering the direct extraction of thermodynamic properties. We address this problem for the rational $C_2^{(1)}$ chain by combining DMRG in the tensor-network framework with the quantum inverse scattering method. The numerically determined zeros of the fundamental and fused transfer-matrix eigenvalues reveal three bulk-string distributions, labeled by $x$, $y$, and $w$, together with boundary strings and additional roots. Substituting these configurations into the operator identities relating the two transfer matrices yields the ground-state energy and the surface energy in the thermodynamic limit under non-diagonal boundary conditions. The analytical surface energy agrees with the numerical results up to $O(\tfrac1N)$ corrections. From the regularity conditions of the operator identities, we derive a boundary-string selection rule and numerically identify allowed configurations beyond the ground state. For excitations preserving the boundary-string configuration, we identify three elementary hole excitations, $X$, $Y$, and $W$, in the $x$-, $y$-, and $w$-string distributions, respectively, and derive their energies. Based on the symmetry of the zero roots, we also present a bulk-excitation selection rule governing the allowed combinations of these elementary excitations in physical eigenstates. The framework developed here can therefore be used to systematically analyze boundary effects, bulk excitations, and related physical properties in other high-rank integrable quantum models with non-diagonal boundary conditions.