Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry

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., U OSp(N |2) and U OSp(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.

Tsirelson's bound and supersymmetric entangled states

“…It is worth remarking, however, that supersymmetry also appears in several proposals from condensed matter physics. In particular, the orthosymplectic algebra osp(1|2) plays a key role in [18,19] and superqubit basis states make an appearance in the context of supersymmetric valence-bond solid states in [20,21].…”

Divergence theorems and the supersphere

“…The supersphere occurs naturally in the theory of the supersymmetric quantum hall effect ( [Has08], [HT13]) and underlies certain field theories, see [SW05]. In a series of papers ( [DBS07], [CDS09], [Cou12]), an integration over the supersphere was introduced, first by an extension of Pizzetti's formula to super-polynomials, then by a formula for general superfunctions later expressed in terms of an embedding, while a particular case of this integral was already studied in [Jar88].…”

The structure of superqubit states

“…The use of super-Hilbert spaces implies a non-trivial departure from standard quantum theory; the term supersymmetry in this context should not be confused with its more familiar usage in the field of high energy physics, which operates exclusively in the realm of standard quantum theory. While the infinitesimal symmetries and representations relevant to superqubits appear in a variety of physically motivated contexts [7][8][9][10][11][12][13][14][15][16], it is not clear what relation they have to superqubits themselves, especially given the use of super-Hilbert space.…”