Abstract
Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “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 a topological order known as a hidden string order on the 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy supergeometry in the construction of supersymmetric versions of VBS (SVBS) states and give a pedagogical introduction of SVBS models and their properties. As concrete examples, we present detailed analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e., UOSp(N|2) and UOSp(N|4) SVBS states, whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with a superconducting property that interpolates various VBS states, depending on the 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. The stability of the topological phase with supersymmetry is discussed, with emphasis on its relation to particular edge (super)spin states.
Highlights
Correlated systems, such as cuprate superconductors, quantum Hall systems, and quantum anti-ferromagnets (QAFM), have been offering arenas for unexpected emergent phenomena brought about by strong many-body correlation
We reviewed the constructions and basic properties of the fuzzy superspheres and supersymmetric valence bond solid (SVBS) models
valence bond solid (VBS) states were emphasized based on the Schwinger operator formalism
Summary
Correlated systems, such as cuprate superconductors, quantum Hall systems, and quantum anti-ferromagnets (QAFM), have been offering arenas for unexpected emergent phenomena brought about by strong many-body correlation. We have already seen that the Bloch spin-coherent state enables us to relate the Laughlin-Haldane wave functions to the VBS states In correspondence with such QHE, a variety of VBS models have been constructed with the symmetries, such as SO(5), SO(2n + 1) [112,113,114], SU (N + 1) [115,116,117,118], Sp(N ) [119,120] and q-deformed SU (2) [22,23,121,122,123] [see Figure 2]. Since the MPS formalism naturally incorporates edge states, MPS provides a powerful formalism to discuss relations between topological order and edge state Taking this advantage, we can obtain general lessons for SUSY effects in topological phases. Fuzzy four-superspheres with an arbitrary number of SUSY are described in Appendix B
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