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Symmetry 2013, 5(2), 119-214;

Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry

Department of General Education, Kagawa National College of Technology, Takuma, Mitoyo,Kagawa 769-1192, Japan
Yukawa Institute for Theoretical Physics, Kyoto University, Oiwake-cho, Kyoto 606-8502, Japan
Author to whom correspondence should be addressed.
Received: 10 March 2013 / Accepted: 1 April 2013 / Published: 26 April 2013
(This article belongs to the Special Issue Supersymmetry)


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. View Full-Text
Keywords: quantum antiferromagnet; supersymmetry; fuzzy sphere; matrix product state; quantum entanglement; topological order quantum antiferromagnet; supersymmetry; fuzzy sphere; matrix product state; quantum entanglement; topological order

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This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Hasebe, K.; Totsuka, K. Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry. Symmetry 2013, 5, 119-214.

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