  We present a detailed analysis of topological properties of the valence bond
solid (VBS) states doped with fermionic holes. As concrete examples, we
consider the supersymmetric extension of the SU(2)- and the SO(5) VBS states,
dubbed UOSp(1|2) and UOSp(1|4) supersymmetric VBS states, respectively.
Specifically, we investigate the string-order parameters and the entanglement
spectra of these states to find that, even when the parent states (bosonic VBS
states) do not support the string order, they recover it when holes are doped
and the fermionic sector appears in the entanglement spectrum. These peculiar
properties are discussed in light of the symmetry-protected topological order.
To this end, we characterize a few typical classes of symmetry-protected
topological orders in terms of supermatrix-product states (SMPS). From this, we
see that the topological order in the bulk manifests itself in the
transformation properties of the SMPS in question and thereby affects the
structure of the entanglement spectrum. Then, we explicitly relate the
existence of the string order and the structure of the entanglement spectrum to
explain the recovery and the stabilization of the string order in the
supersymmetric systems.

Strongly Correlated Electrons;High Energy Physics - Theory;Quantum Physics