  Disturbances in gapless quantum many-body models are known to travel an
unlimited distance throughout the system. Here, we explore this phenomenon in
finite clusters with degenerate ground states. The specific model studied here
is the one-dimensional J1-J2 Heisenberg Hamiltonian at and close to the
Majumdar-Ghosh point. Both open and periodic boundary conditions are
considered. Quenches are performed using a local magnetic field. The degenerate
Majumdar-Ghosh ground state allows disturbances which carry quantum
entanglement to propagate throughout the system, and thus dephase the entire
system within the degenerate subspace. These disturbances can also carry
polarization, but not energy, as all energy is stored locally. The local
evolution of the part of the system where energy is stored drives the rest of
the system through long-range entanglement. We also examine approximations for
the ground state of this Hamiltonian in the strong field limit, and study how
couplings away from the Majumdar-Ghosh point affect the propagation of
disturbances. We find that even in the case of approximate degeneracy, a
disturbance can be propagated throughout a finite system.
