  We consider the spin $\frac{1}{2}$ Heisenberg chain with boundary magnetic
fields and analyze it using a combination of Bethe ansatz and density matrix
renormalization group (DMRG) techniques. We show that the system exhibits
several different ground states which depend on the orientation of the boundary
magnetic fields. When both the boundary fields take equal values greater than a
critical field strength, each edge in the ground state accumulates a fractional
spin which saturates to spin $\frac{1}{4}$, which is similar to systems
exhibiting symmetry protected topological phases (SPT). Unlike in SPT systems,
the fractional boundary spin in the Heisenberg spin chain is not a genuine
quantum number since the variance of the associated operator does not vanish,
this is due to the absence of a bulk gap. The system exhibits high energy bound
states when the boundary fields take values greater than the critical field.
All the excitations in the system can be sorted out into towers whose number
depends on the number of bound states exhibited by the system. As the boundary
fields are varied, in addition to the ground state phase transition, we find
that the system may undergo an eigenstate phase transition (EPT) where the
number of towers of the Hilbert space changes. We further inquire how the EPT
reflects itself on local ground state properties by computing the magnetization
profile $\langle S^z_j \rangle$ using DMRG. We identify a clear qualitative
change from low edge fields to high edge fields when crossing the critical
field. We though are unable to conclude on the basis of our data that EPT
corresponds to a genuine phase transition in the ground state.
