Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry

Abstract

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(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 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.

Review article, 1+104 pages, 37 figures, published version

References in corpus (18)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Real time evolution using the density matrix renormalization group
• Matrix product states represent ground states faithfully
• The iTEBD algorithm beyond unitary evolution
• Class of exactly solvable SO(n) symmetric spin chains with matrix product ground states
• On topological phases of spin chains
• Exact results for SU(3) spin chains: trimer states, valence bond solids, and their parent Hamiltonians
• Simplex solid states of SU(N) quantum antiferromagnets
• Supersymmetry and Goldstino-like Mode in Bose-Fermi Mixtures
• Fragility of String Orders
• String order and hidden topological symmetry in the SO(2n+1) symmetric matrix product states
• Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
• Valence bond solid states with symplectic symmetry
• Hyperbolic Supersymmetric Quantum Hall Effect
• Continuous quantum phase transition between two topologically distinct valence bond solid states associated with the same spin value
• Generalized Berezin-Toeplitz quantization of Kaehler supermanifolds
• Matrix models and the gravitational interaction
• A (p,q)-deformed Landau problem in a spherical harmonic well: spectrum and noncommuting coordinates