430 research outputs found
Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8
We show that the Leech lattice gives a sphere covering which is locally least
dense among lattice coverings. We show that a similar result is false for the
root lattice E8. For this we construct a less dense covering lattice whose
Delone subdivision has a common refinement with the Delone subdivision of E8.
The new lattice yields a sphere covering which is more than 12% less dense than
the formerly best known given by the lattice A8*. Currently, the Leech lattice
is the first and only known example of a locally optimal lattice covering
having a non-simplicial Delone subdivision. We hereby in particular answer a
question of Dickson posed in 1968. By showing that the Leech lattice is rigid
our answer is even strongest possible in a sense.Comment: 13 pages; (v2) major revision: proof of rigidity corrected, full
discussion of E8-case included, src of (v3) contains MAGMA program, (v4) some
correction
Semiclassical degeneracies and ordering for highly frustrated magnets in a field
We discuss ground state selection by quantum fluctuations in frustrated
magnets in a strong magnetic field. We show that there exist dynamical
symmetries -- one a generalisation of Henley's gauge-like symmetry for
collinear spins, the other the quantum relict of non-collinear weathervane
modes -- which ensure a partial survival of the classical degeneracies. We
illustrate these for the case of the kagome magnet, where we find zero-point
energy differences to be rather small everywhere except near the collinear
`up-up-down` configurations, where there is rotational but not translational
symmetry breaking. In the effective Hamiltonian, we demonstrate the presence of
a term sensitive to a topological `flux'. We discuss the connection of such
problems to gauge theories by casting the frustrated lattices as medial
lattices of appropriately chosen simplex lattices, and in particular we show
how the magnetic field can be used to tune the physical sector of the resulting
gauge theories.Comment: 10 pages, 8 figure
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