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We present a detailed analysis of topological properties of the valence bondsolid (VBS) states doped with fermionic holes. As concrete examples, weconsider 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 entanglementspectra of these states to find that, even when the parent states (bosonic VBSstates) do not support the string order, they recover it when holes are dopedand the fermionic sector appears in the entanglement spectrum. These peculiarproperties are discussed in light of the symmetry-protected topological order.To this end, we characterize a few typical classes of symmetry-protectedtopological orders in terms of supermatrix-product states (SMPS). From this, wesee that the topological order in the bulk manifests itself in thetransformation properties of the SMPS in question and thereby affects thestructure of the entanglement spectrum. Then, we explicitly relate theexistence of the string order and the structure of the entanglement spectrum toexplain the recovery and the stabilization of the string order in thesupersymmetric systems.

Quantum Entanglement and Topological Order in Hole-Doped Valence Bond Solid States