3,718 research outputs found
Quantum Phase Transitions in Anti-ferromagnetic Planar Cubic Lattices
Motivated by its relation to an -hard problem, we analyze the
ground state properties of anti-ferromagnetic Ising-spin networks embedded on
planar cubic lattices, under the action of homogeneous transverse and
longitudinal magnetic fields. This model exhibits a quantum phase transition at
critical values of the magnetic field, which can be identified by the
entanglement behavior, as well as by a Majorization analysis. The scaling of
the entanglement in the critical region is in agreement with the area law,
indicating that even simple systems can support large amounts of quantum
correlations. We study the scaling behavior of low-lying energy gaps for a
restricted set of geometries, and find that even in this simplified case, it is
impossible to predict the asymptotic behavior, with the data allowing equally
good fits to exponential and power law decays. We can therefore, draw no
conclusion as to the algorithmic complexity of a quantum adiabatic ground-state
search for the system.Comment: 7 pages, 13 figures, final version (accepted for publication in PRA
Characterizing fully principal congruence representable distributive lattices
Motivated by a recent paper of G. Gr\"atzer, a finite distributive lattice
is said to be fully principal congruence representable if for every subset
of containing , , and the set of nonzero join-irreducible
elements of , there exists a finite lattice and an isomorphism from the
congruence lattice of onto such that corresponds to the set of
principal congruences of under this isomorphism. Based on earlier results
of G. Gr\"atzer, H. Lakser, and the present author, we prove that a finite
distributive lattice is fully principal congruence representable if and
only if it is planar and it has at most one join-reducible coatom. Furthermore,
even the automorphism group of can arbitrarily be stipulated in this case.
Also, we generalize a recent result of G. Gr\"atzer on principal congruence
representable subsets of a distributive lattice whose top element is
join-irreducible by proving that the automorphism group of the lattice we
construct can be arbitrary.Comment: 20 pages, 8 figure
Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices
In continuum mechanics, the non-centrosymmetric micropolar theory is usually
used to capture the chirality inherent in materials. However when reduced to a
two dimensional (2D) isotropic problem, the resulting model becomes non-chiral.
Therefore, influence of the chiral effect cannot be properly characterized by
existing theories for 2D chiral solids. To circumvent this difficulty, based on
reinterpretation of isotropic tensors in a 2D case, we propose a continuum
theory to model the chiral effect for 2D isotropic chiral solids. A single
material parameter related to chirality is introduced to characterize the
coupling between the bulk deformation and the internal rotation which is a
fundamental feature of 2D chiral solids. Coherently, the proposed continuum
theory is also derived for a triangular chiral lattice from a homogenization
procedure, from which the effective material constants of the lattice are
analytically determined. The unique behavior in the chiral lattice is
demonstrated through the analyses of a static tension problem and a plane wave
propagation problem. The results, which cannot be predicted by the non-chiral
model, are validated by the exact solution of the discrete model.Comment: 33 pages, 7 figure
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
K. Adaricheva and M. Bolat have recently proved that if and are
circles in a triangle with vertices , then there exist and such that is included in the convex hull
of . One could say disks instead of
circles. Here we prove the existence of such a and for the more general
case where and are compact sets in the plane such that is
obtained from by a positive homothety or by a translation. Also, we give
a short survey to show how lattice theoretical antecedents, including a series
of papers on planar semimodular lattices by G. Gratzer and E. Knapp, lead to
our result.Comment: 28 pages, 7 figure
Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions
The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771–2796, 2004 and Mareno and Healey in SIAM J. Math. Anal. 38:103–115, 2006). The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell’Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127–162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850–858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255–2259, 2000). In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677–678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev’s space as the most suitable energy space E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations
Characterizing a vertex-transitive graph by a large ball
It is well-known that a complete Riemannian manifold M which is locally
isometric to a symmetric space is covered by a symmetric space. Here we prove
that a discrete version of this property (called local to global rigidity)
holds for a large class of vertex-transitive graphs, including Cayley graphs of
torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free
virtually nilpotent groups. By contrast, we exhibit various examples of Cayley
graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this
property, answering a question of Benjamini, Ellis, and Georgakopoulos.
Answering a question of Cornulier, we also construct a continuum of non
pairwise isometric large-scale simply connected locally finite
vertex-transitive graphs. This question was motivated by the fact that
large-scale simply connected Cayley graphs are precisely Cayley graphs of
finitely presented groups and therefore have countably many isometric classes.Comment: v1: 38 pages. With an Appendix by Jean-Claude Sikorav v2: 48 pages.
Several improvements in the presentation. To appear in Journal of Topolog
Critical and Tricritical Hard Objects on Bicolorable Random Lattices: Exact Solutions
We address the general problem of hard objects on random lattices, and
emphasize the crucial role played by the colorability of the lattices to ensure
the existence of a crystallization transition. We first solve explicitly the
naive (colorless) random-lattice version of the hard-square model and find that
the only matter critical point is the non-unitary Lee-Yang edge singularity. We
then show how to restore the crystallization transition of the hard-square
model by considering the same model on bicolored random lattices. Solving this
model exactly, we show moreover that the crystallization transition point lies
in the universality class of the Ising model coupled to 2D quantum gravity. We
finally extend our analysis to a new two-particle exclusion model, whose
regular lattice version involves hard squares of two different sizes. The exact
solution of this model on bicolorable random lattices displays a phase diagram
with two (continuous and discontinuous) crystallization transition lines
meeting at a higher order critical point, in the universality class of the
tricritical Ising model coupled to 2D quantum gravity.Comment: 48 pages, 13 figures, tex, harvmac, eps
Dimensional crossover of Bose-Einstein condensation phenomena in quantum gases confined within slab geometries
We investigate systems of interacting bosonic particles confined within
slab-like boxes of size L^2 x Z with Z<<L, at their three-dimensional (3D) BEC
transition temperature T_c, and below T_c where they experience a quasi-2D
Berezinskii-Kosterlitz-Thouless transition (at T_BKT < T_c depending on the
thickness Z). The low-temperature phase below T_BKT shows quasi-long-range
order: the planar correlations decay algebraically as predicted by the 2D
spin-wave theory. This dimensional crossover, from a 3D behavior for T > T_c to
a quasi-2D critical behavior for T < T_BKT, can be described by a transverse
finite-size scaling limit in slab geometries. We also extend the discussion to
the off-equilibrium behavior arising from slow time variations of the
temperature across the BEC transition. Numerical evidence of the 3D->2D
dimensional crossover is presented for the Bose-Hubbard model defined in
anisotropic L^2 x Z lattices with Z<<L.Comment: 16 page
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