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Bulletin of the American Physical Society

A Linear Dependency Structure Arising from Weyl-Heisenberg Symmetry

Hoan Bui Dang, Marcus Appleby, Ingemar Bengtsson, Kate Blanchfield, Åsa Ericsson, Christopher A. Fuchs, Matthew A. Graydon, Gelo Noel M. Tabia

Published Mar 22, 2011• 0 citations
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Five years ago, Lane Hughston showed that some of the symmetric informationally complete positive operator valued measures (SICs) in dimension 3 coincide with the Hesse configuration (a structure well known to algebraic geometers, which arises from the torsion points of a certain elliptic curve). This connection with elliptic curves is signalled by the presence of linear dependencies among the SIC vectors. Here we look for analogous connections between SICs and algebraic geometry by performing computer searches for linear dependencies in higher dimensional SICs. We prove that linear dependencies will always emerge in Weyl–Heisenberg orbits when the fiducial vector lies in a certain subspace of an order 3 unitary matrix. This includes SICs when the dimension is divisible by 3 or equal to 8 mod 9. We examine the linear dependencies in dimension 6 in detail and show that smaller dimensional SICs are contained within this structure, potentially impacting the SIC existence problem. We extend our results to look for linear dependencies in orbits when the fiducial vector lies in an eigenspace of other elements of the Clifford group that are not order 3. Finally, we align our work with recent studies on representations of the Clifford group.
Beginning from the conventional square-lattice nearest-neighbor antiferromagnetic Heisenberg model, we allow the Jx{J}_{x}Jx​ and Jy{J}_{y}Jy​ couplings to be anisotropic, with their values depending on the bond orientation. The emergence of anisotropic, bond-dependent couplings should be expected to occur naturally in most antiferromagnetic compounds which undergo structural transitions that reduce the point-group symmetry at lower temperature. Using the spin-wave approximation, we study the model in several parameter regimes by diagonalizing the reduced Hamiltonian exactly and computing the edge spectrum and Berry connection vector, which show clear evidence of localized topological charges. We discover phases that exhibit Weyl-type spin-wave dispersion, characterized by pairs of degenerate points and edge states, as well as phases supporting lines of degeneracy. We also identify a parameter regime in which there is an exotic state hosting gapless linear spin-wave dispersions with different longitudinal and transverse spin-wave velocities.