Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry

Recent vigorous investigations of topological order have not only discovered new topological states of matter but also shed new light to \already known" topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking but most typically manifest topological order known as hidden string order on 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy super-geometry in the construction of supersymmetric versions of VBS (SVBS) states, and give a pedagogical introduction of SVBS models and their properties [arXiv:0809.4885, 1105.3529, 1210.0299]. As concrete examples, we present detail analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i:e: UOSp(Nj2) and UOSp(Nj4) SVBS states whose mathematics are closely related to fuzzy twoand four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with superconducting property that interpolate various VBS states depending on value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate regardless of the parity of bulk (super)spins. Stability of topological phase with supersymmetry is discussed with emphasis on its relation to particular edge (super)spin states.