488 research outputs found
On well-rounded sublattices of the hexagonal lattice
We produce an explicit parameterization of well-rounded sublattices of the
hexagonal lattice in the plane, splitting them into similarity classes. We use
this parameterization to study the number, the greatest minimal norm, and the
highest signal-to-noise ratio of well-rounded sublattices of the hexagonal
lattice of a fixed index. This investigation parallels earlier work by
Bernstein, Sloane, and Wright where similar questions were addressed on the
space of all sublattices of the hexagonal lattice. Our restriction is motivated
by the importance of well-rounded lattices for discrete optimization problems.
Finally, we also discuss the existence of a natural combinatorial structure on
the set of similarity classes of well-rounded sublattices of the hexagonal
lattice, induced by the action of a certain matrix monoid.Comment: 21 pages (minor correction to the proof of Lemma 2.1); to appear in
Discrete Mathematic
On similarity classes of well-rounded sublattices of
A lattice is called well-rounded if its minimal vectors span the
corresponding Euclidean space. In this paper we study the similarity classes of
well-rounded sublattices of . We relate the set of all such
similarity classes to a subset of primitive Pythagorean triples, and prove that
it has structure of a noncommutative infinitely generated monoid. We discuss
the structure of a given similarity class, and define a zeta function
corresponding to each similarity class. We relate it to Dedekind zeta of
, and investigate the growth of some related Dirichlet series,
which reflect on the distribution of well-rounded lattices. Finally, we
construct a sequence of similarity classes of well-rounded sublattices of
, which gives good circle packing density and converges to the
hexagonal lattice as fast as possible with respect to a natural metric we
define.Comment: 27 pages, 2 figures; added a lemma on Diophantine approximation by
quotients of Pythagorean triples; final version to be published in Journal of
Number Theor
Revisiting the hexagonal lattice: on optimal lattice circle packing
In this note we give a simple proof of the classical fact that the hexagonal
lattice gives the highest density circle packing among all lattices in .
With the benefit of hindsight, we show that the problem can be restricted to
the important class of well-rounded lattices, on which the density function
takes a particularly simple form. Our proof emphasizes the role of well-rounded
lattices for discrete optimization problems.Comment: 8 pages, 1 figure; to appear in Elemente der Mathemati
Well-rounded zeta-function of planar arithmetic lattices
We investigate the properties of the zeta-function of well-rounded
sublattices of a fixed arithmetic lattice in the plane. In particular, we show
that this function has abscissa of convergence at with a real pole of
order 2, improving upon a recent result of S. Kuehnlein. We use this result to
show that the number of well-rounded sublattices of a planar arithmetic lattice
of index less or equal is as . To obtain these
results, we produce a description of integral well-rounded sublattices of a
fixed planar integral well-rounded lattice and investigate convergence
properties of a zeta-function of similarity classes of such lattices, building
on some previous results of the author.Comment: 12 pages; to appear in PAM
Phases and phase transitions of a perturbed Kekul\'e-Kitaev model
We study the quantum spin liquid phase in a variant of the Kitaev model where
the bonds of the honeycomb lattice are distributed in a Kekul\'e pattern. The
system supports gapped and gapless Z_2 quantum spin liquids with interesting
differences from the original Kitaev model, the most notable being a gapped Z_2
spin liquid on a Kagome lattice. Perturbing the exactly solvable model with
antiferromagnetic Heisenberg perturbations, we find a magnetically ordered
phase stabilized by a quantum `order by disorder' mechanism, as well as an
exotic continuous phase transition between the topological spin liquid and this
magnetically ordered phase. Using a combination of field theory and Monte-Carlo
simulations, we find that the transition likely belongs to the 3D-XYxZ_2
universality class.Comment: 15 pages, 11 figure
Ripple state in the frustrated honeycomb-lattice antiferromagnet
We discover a new type of multiple- state, "ripple state", in a frustrated
honeycomb-lattice Heisenberg antiferromagnet under magnetic fields. The ground
state has an infinite ring-like degeneracy in the wavevector space, exhibiting
a cooperative paramagnetic state, "ring-liquid" state. We elucidate that the
system exhibits the ripple state as a new low-temperature thermodynamic phase
via a second-order phase transition from the ring-liquid state, keeping the
ring-like spin structure factor. The spin texture in real space looks like a
"water ripple" and can induce a giant electric polarization vortex. Possible
relationship to the honeycomb-lattice compound, , is discussed.Comment: 6+7 pages, 3+7 figures, revised manuscript accepted in PR
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