399 research outputs found
Crystallization in a model glass: influence of the boundary conditions
Using molecular dynamics calculations and the Voronoi tessellation, we study
the evolution of the local structure of a soft-sphere glass versus temperature
starting from the liquid phase at different quenching rates. This study is done
for different sizes and for two different boundary conditions namely the usual
cubic periodic boundary conditions and the isotropic hyperspherical boundary
conditions for which the particles evolve on the surface of a hypersphere in
four dimensions. Our results show that for small system sizes, crystallization
can indeed be induced by the cubic boundary conditions. On the other hand we
show that finite size effects are more pronounced on the hypersphere and that
crystallization is artificially inhibited even for large system sizes.Comment: 11 pages, 2 figure
Borcherds symmetries in M-theory
It is well known but rather mysterious that root spaces of the Lie
groups appear in the second integral cohomology of regular, complex, compact,
del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms)
of toroidal compactifications of M theory. Their Borel subgroups are actually
subgroups of supergroups of finite dimension over the Grassmann algebra of
differential forms on spacetime that have been shown to preserve the
self-duality equation obeyed by all bosonic form-fields of the theory. We show
here that the corresponding duality superalgebras are nothing but Borcherds
superalgebras truncated by the above choice of Grassmann coefficients. The full
Borcherds' root lattices are the second integral cohomology of the del Pezzo
surfaces. Our choice of simple roots uses the anti-canonical form and its known
orthogonal complement. Another result is the determination of del Pezzo
surfaces associated to other string and field theory models. Dimensional
reduction on corresponds to blow-up of points in general position
with respect to each other. All theories of the Magic triangle that reduce to
the sigma model in three dimensions correspond to singular del Pezzo
surfaces with (normal) singularity at a point. The case of type I and
heterotic theories if one drops their gauge sector corresponds to non-normal
(singular along a curve) del Pezzo's. We comment on previous encounters with
Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real
fermionic simple roots when they would naively aris
Multiple CSLs for the body centered cubic lattice
Ordinary Coincidence Site Lattices (CSLs) are defined as the intersection of
a lattice with a rotated copy of itself. They are useful for
classifying grain boundaries and have been studied extensively since the mid
sixties. Recently the interests turned to so-called multiple CSLs, i.e.
intersections of rotated copies of a given lattice , in particular
in connection with lattice quantizers. Here we consider multiple CSLs for the
3-dimensional body centered cubic lattice. We discuss the spectrum of
coincidence indices and their multiplicity, in particular we show that the
latter is a multiplicative function and give an explicit expression of it for
some special cases.Comment: 4 pages, SSPCM (31 August - 7 September 2005, Myczkowce, Poland
Vison states and confinement transitions of Z2 spin liquids on the kagome lattice
We present a projective symmetry group (PSG) analysis of the spinless
excitations of Z2 spin liquids on the kagome lattice. In the simplest case,
vortices carrying Z2 magnetic flux ('visons') are shown to transform under the
48 element group GL(2, Z3). Alternative exchange couplings can also lead to a
second case with visons transforming under 288 element group GL(2, Z3) \times
D3. We study the quantum phase transition in which visons condense into
confining states with valence bond solid order. The critical field theories and
confining states are classified using the vison PSGs.Comment: 25 pages, 13 figure
Affine Wa(A4), Quaternions, and Decagonal Quasicrystals
We introduce a technique of projection onto the Coxeter plane of an arbitrary
higher dimensional lattice described by the affine Coxeter group. The Coxeter
plane is determined by the simple roots of the Coxeter graph I2 (h) where h is
the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh
of order 2h as a maximal subgroup. As a simple application we demonstrate
projections of the root and weight lattices of A4 onto the Coxeter plane using
the strip (canonical) projection method. We show that the crystal spaces of the
affine Wa(A4) can be decomposed into two orthogonal spaces whose point groups
is the dihedral group D5 which acts in both spaces faithfully. The strip
projections of the root and weight lattices can be taken as models for the
decagonal quasicrystals. The paper also revises the quaternionic descriptions
of the root and weight lattices, described by the affine Coxeter group Wa(A3),
which correspond to the face centered cubic (fcc) lattice and body centered
cubic (bcc) lattice respectively. Extensions of these lattices to higher
dimensions lead to the root and weight lattices of the group Wa(An), n>=4 . We
also note that the projection of the Voronoi cell of the root lattice of Wa(A4)
describes a framework of nested decagram growing with the power of the golden
ratio recently discovered in the Islamic arts.Comment: 26 pages, 17 figure
Quaternionic Root Systems and Subgroups of the
Cayley-Dickson doubling procedure is used to construct the root systems of
some celebrated Lie algebras in terms of the integer elements of the division
algebras of real numbers, complex numbers, quaternions and octonions. Starting
with the roots and weights of SU(2) expressed as the real numbers one can
construct the root systems of the Lie algebras of SO(4),SP(2)=
SO(5),SO(8),SO(9),F_{4} and E_{8} in terms of the discrete elements of the
division algebras. The roots themselves display the group structures besides
the octonionic roots of E_{8} which form a closed octonion algebra. The
automorphism group Aut(F_{4}) of the Dynkin diagram of F_{4} of order 2304, the
largest crystallographic group in 4-dimensional Euclidean space, is realized as
the direct product of two binary octahedral group of quaternions preserving the
quaternionic root system of F_{4}.The Weyl groups of many Lie algebras, such
as, G_{2},SO(7),SO(8),SO(9),SU(3)XSU(3) and SP(3)X SU(2) have been constructed
as the subgroups of Aut(F_{4}). We have also classified the other non-parabolic
subgroups of Aut(F_{4}) which are not Weyl groups. Two subgroups of orders192
with different conjugacy classes occur as maximal subgroups in the finite
subgroups of the Lie group of orders 12096 and 1344 and proves to be
useful in their constructions. The triality of SO(8) manifesting itself as the
cyclic symmetry of the quaternionic imaginary units e_{1},e_{2},e_{3} is used
to show that SO(7) and SO(9) can be embedded triply symmetric way in SO(8) and
F_{4} respectively
Quaterionic Construction of the W(F_4) Polytopes with Their Dual Polytopes and Branching under the Subgroups B(B_4) and W(B_3)*W(A_1)
4-dimensional polytopes and their dual polytopes have been
constructed as the orbits of the Coxeter-Weyl group where the group
elements and the vertices of the polytopes are represented by quaternions.
Branchings of an arbitrary \textbf{} orbit under the Coxeter groups
and have been presented. The role of
group theoretical technique and the use of quaternions have been emphasizedComment: 26 pages, 10 figure
Classification and stability of simple homoclinic cycles in R^5
The paper presents a complete study of simple homoclinic cycles in R^5. We
find all symmetry groups Gamma such that a Gamma-equivariant dynamical system
in R^5 can possess a simple homoclinic cycle. We introduce a classification of
simple homoclinic cycles in R^n based on the action of the system symmetry
group. For systems in R^5, we list all classes of simple homoclinic cycles. For
each class, we derive necessary and sufficient conditions for asymptotic
stability and fragmentary asymptotic stability in terms of eigenvalues of
linearisation near the steady state involved in the cycle. For any action of
the groups Gamma which can give rise to a simple homoclinic cycle, we list
classes to which the respective homoclinic cycles belong, thus determining
conditions for asymptotic stability of these cycles.Comment: 34 pp., 4 tables, 30 references. Submitted to Nonlinearit
Hunting for the New Symmetries in Calabi-Yau Jungles
It was proposed that the Calabi-Yau geometry can be intrinsically connected
with some new symmetries, some new algebras. In order to do this it has been
analyzed the graphs constructed from K3-fibre CY_d (d \geq 3) reflexive
polyhedra. The graphs can be naturally get in the frames of Universal
Calabi-Yau algebra (UCYA) and may be decode by universal way with changing of
some restrictions on the generalized Cartan matrices associated with the Dynkin
diagrams that characterize affine Kac-Moody algebras. We propose that these new
Berger graphs can be directly connected with the generalizations of Lie and
Kac-Moody algebras.Comment: 29 pages, 15 figure
Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)
Snub 24-cell is the unique uniform chiral polytope in four dimensions
consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the
4-dimensional semi-regular polytope snub 24-cell and its symmetry group
{(W(D_{4})\mathord{/{\vphantom {(W(D_{4}) C_{2}}}. \kern-\nulldelimiterspace}
C_{2}}):S_{3} of order 576 are obtained from the quaternionic representation
of the Coxeter-Weyl group \textbf{}The symmetry group is an
extension of the proper subgroup of the Coxeter-Weyl group
\textbf{}by the permutation symmetry of the Coxeter-Dynkin diagram
\textbf{} The 96 vertices of the snub 24-cell are obtained as the
orbit of the group when it acts on the vector \textbf{}or\textbf{}on the vector\textbf{}in the Dynkin basis with\textbf{} The two different sets represent the
mirror images of the snub 24-cell. When two mirror images are combined it leads
to a quasi regular 4D polytope invariant under the Coxeter-Weyl group
\textbf{}Each vertex of the new polytope is shared by one cube and
three truncated octahedra. Dual of the snub 24 cell is also constructed.
Relevance of these structures to the Coxeter groups \textbf{}has been pointed out.Comment: 15 pages, 8 figure
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