Discrete Symmetry Group Research Articles

Autoparallel curves and Riemannian geodesics for materially uniform but inhomogeneous bodies

According to the theory of materially uniform but inhomogeneous bodies two geometric structures can be defined on the body manifold when a uniform reference is known. The first is given by the material connection and the second by the intrinsic Riemannian metric. Two important classes of curves related with these structures are the geodesic and the autoparallel curves. The goal of the present contribution is to define and characterize mechanically these kind of curves for materially uniform but inhomogeneous bodies. We propose the use of these curves for constructing the stress-free non-Euclidean material manifold that plays the role of the reference configuration for the dislocated problem. A generic scheme for the construction of this manifold based on geodesics/autoparallel curves is given as well as a discussion related with the field of internal stresses. Attention is then focused on a continuous distribution of edge dislocations. We solve numerically the geodesic equation that corresponds to a solid body with a continuous symmetry group. By using the norm for the dislocation density tensor we conclude that the higher the dislocation density the greater the deviation from the straight line is. For the same distribution of dislocations, but for a solid body with a discrete symmetry group, we solve analytically the autoparallel curves.

An exact quantification of backreaction in relativistic cosmology

An important open question in cosmology is the degree to which the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) solutions of Einstein's equations are able to model the large-scale behavior of the locally inhomogeneous observable universe. We investigate this problem by considering a range of exact n-body solutions of Einstein's constraint equations. These solutions contain discrete masses, and so allow arbitrarily large density contrasts to be modeled. We restrict our study to regularly arranged distributions of masses in topological 3-spheres. This has the benefit of allowing straightforward comparisons to be made with FLRW solutions, as both spacetimes admit a discrete group of symmetries. It also provides a time-symmetric hypersurface at the moment of maximum expansion that allows the constraint equations to be solved exactly. We find that when all the mass in the universe is condensed into a small number of objects ($\ensuremath{\lesssim}10$) then the amount of back-reaction in dust models can be large, with $O(1)$ deviations from the predictions of the corresponding FLRW solutions. When the number of masses is large ($\ensuremath{\gtrsim}100$), however, then our measures of back-reaction become small ($\ensuremath{\lesssim}1%$). This result does not rely on any averaging procedures, which are notoriously hard to define uniquely in general relativity, and so provides (to the best of our knowledge) the first exact and unambiguous demonstration of back-reaction in general relativistic cosmological modelling. Discrete models such as these can therefore be used as laboratories to test ideas about back-reaction that could be applied in more complicated and realistic settings.

Rod groups and their settings as special geometric realisations of line groups

Rod groups (monoperiodic subgroups of the 3-periodic space groups) are considered as a special case of the commensurate line groups (discrete symmetry groups of the three-dimensional objects translationally periodic along a line). Two different factorizations of line groups are considered: (1) The standard L = T(a)F used in crystallography for rod groups; F is a finite system of representatives of line-group decomposition in cosets of 1-periodic translation group T(a); (2) L = ZP used in the theory of line groups; Z is a cyclic generalized translation group and P is a finite point group. For symmorphic line groups (five line-group families of 13 families) the two factorizations are equivalent: the cyclic group Z is a monoperiodic translation group and P is the point group defining the crystal class. For each of the remaining eight families of non-symmorphic line groups the explicit correspondence between rod groups and relevant geometric realisations of the corresponding line groups is established. The settings of rod groups and line groups are taken into account. The results are presented in a table of 75 rod groups listed (in international and factorized notation) by families of the line groups according to the order of the principal axis q (q = 1, 2, 3, 4, 6) of the corresponding isogonal point group.

Zpscalar dark matter from multi-Higgs-doublet models

In many models, stability of dark matter particles is protected by a conserved Z_2 quantum number. However dark matter can be stabilized by other discrete symmetry groups, and examples of such models with custom-tailored field content have been proposed. Here we show that electroweak symmetry breaking models with N Higgs doublets can readily accommodate scalar dark matter candidates stabilized by groups Z_p with any $p \le 2^{N-1}$, leading to a variety of kinds of microscopic dynamics in the dark sector. We give examples in which semi-annihilation or multiple semi-annihilation processes are allowed or forbidden, which can be especially interesting in the case of asymmetric dark matter.

Molecular torus group

Generally speaking, the highest symmetry of Mobius cyclacene molecule possesses the C2 symmetry based on the theory of point group according to the previous works. However, based on the topology principle, the fundamental group of Mobuis strip is an infinite continuous cyclic group and its border is made up of twice of the generator. Of course, the Mobius strip-like molecule is associated with a finite discrete cyclic symmetry group. For the cyclacene isomers constructed by several (n) benzene rings, these isomers include: the common cylinder Huckel cyclacene (HC-[n]) molecules, the Mobius cyclacene (MC-[n]) molecules by twisting the linear precursor one time (180°), and the multi-twisted Mobius strip-like cyclacene (M m C−[n]) molecules by twisting the linear precursor m times (m × 180°). The relevant results suggest that in addition to the point symmetry, there is a new kind of symmetry called the torus screw rotation (denoted as TSR). In this article, we take the M m C−[n] molecules as examples to discuss their TSR group and point group symmetry, and the relative symmetry adaptive atom sets as well as their atomic orbital (AO) sets. Here, the Cartesian coordinates is not quite fit for the investigation of these AOs, so it is replaced by either the torus orthogonal curvilinear coordinates (for M m C−[n] molecule) or the cylinder orthogonal curvilinear coordinates (for HC-[n] molecule). According to the features of cyclic group, the character table of the irreducible representation of the TSR group could be constructed easily. Some other relevant point-group symmetries maybe also belong to the molecule, so its symmetry maybe called as the torus group symmetry. For multi-twisted Mobius strip-like molecule, there are some correlations in topology characteristics.

Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations

Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry protected topological orders exist. In this paper, we present a model in a 2D interacting spin system with nontrivial on-site $Z_2$ symmetry protected topological order. The order is nontrivial because we can prove that the 1D system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the $Z_2$ group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if $Z_2$ symmetry is not broken. We prove the latter by developing the tool of matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and revealing the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with on-site $Z_2$ symmetry protected topological order.

Order by disorder and phase transitions in a highly frustrated spin model on the triangular lattice

Frustration has proved to give rise to an extremely rich phenomenology in both quantum and classical systems. The leading behavior of the system can often be described by an effective model, where only the lowest-energy degrees of freedom are considered. In this paper we study a system corresponding to the strong trimerization limit of the spin 1/2 kagome antiferromagnet in a magnetic field. It has been suggested that this system can be realized experimentally by a gas of spinless fermions in an optical kagome lattice at 2/3 filling. We investigate the low-energy behavior of both the spin 1/2 quantum version and the classical limit of this system by applying various techniques. We study in parallel both signs of the coupling constant J since the two cases display qualitative differences. One of the main peculiarities of the J>0 case is that, at the classical level, there is an exponentially large manifold of lowest-energy configurations. This renders the thermodynamics of the system quite exotic and interesting in this case. For both cases, J>0 and J<0, a finite-temperature phase transition with a breaking of the discrete dihedral symmetry group D_6 of the model is present. For J<0, we find a transition temperature T^<_c/|J| = 1.566 +/- 0.005, i.e., of order unity, as expected. We then analyze the nature of the transition in this case. While we find no evidence for a discontinuous transition, the interpretation as a continuous phase transition yields very unusual critical exponents violating the hyperscaling relation. By contrast, in the case J>0 the transition occurs at an extremely low temperature, T^>_c ~= 0.0125 J. Presumably this low transition temperature is connected with the fact that the low-temperature ordered state of the system is established by an order-by-disorder mechanism in this case.

Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group

On the Engel group a nilpotent sub-Riemannian problem is considered, a 4-dimensional optimal control problem with a 2-dimensional linear control and an integral cost functional. It arises as a nilpotent approximation to nonholonomic systems with 2-dimensional control in a 4-dimensional space (for example, a system describing the navigation of a mobile robot with trailer). A parametrization of extremal trajectories by Jacobi functions is obtained. A discrete symmetry group and its fixed points, which are Maxwell points, are described. An estimate for the cut time (the time of the loss of optimality) on extremal trajectories is derived on this basis.Bibliography: 25 titles.

Calabi-Yau orbifolds and torus coverings

The theory of coverings of the two-dimensional torus is a standard part of algebraic topology and has applications in several topics in string theory, for example, in topological strings. This paper initiates applications of this theory to the counting of orbifolds of toric Calabi-Yau singularities, with particular attention to Abelian orbifolds of $ {\mathbb{C}^D} $ . By doing so, the work introduces a novel analytical method for counting Abelian orbifolds, verifying previous algorithm results. One identifies a p-fold cover of the torus $ {\mathbb{T}^{D - 1}} $ with an Abelian orbifold of the form $ {{{{\mathbb{C}^D}}} \left/ {{{\mathbb{Z}_p}}} \right.} $ , for any dimension D and a prime number p. The counting problem leads to polynomial equations modulo p for a given Abelian subgroup of S D , the group of discrete symmetries of the toric diagram for $ {\mathbb{C}^D} $ . The roots of the polynomial equations correspond to orbifolds of the form $ {{{{\mathbb{C}^D}}} \left/ {{{\mathbb{Z}_p}}} \right.} $ , which are invariant under the corresponding subgroup of S D . In turn, invariance under this subgroup implies a discrete symmetry for the corresponding quiver gauge theory, as is clearly seen by its brane tiling formulation.

From Quantum AN(Calogero) to H4(Rational) Model

A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the Hamiltonian, (ii) a number of polynomial eigenfunctions, (iii) a factorization property for eigenfunctions, and admit (iv) the separation of the radial coordinate and, hence, the existence of the 2nd order integral, (v) an algebraic form in invariants of a discrete symmetry group (in space of orbits).

The golden ratio prediction for the solar angle from a natural model with A 5 flavour symmetry

We formulate a consistent model predicting, in the leading order approximation, maximal atmospheric mixing angle, vanishing reactor angle and tan {\theta}_12 = 1/{\phi} where {\phi} is the Golden Ratio. The model is based on the flavour symmetry A5 \times Z5 \times Z3, spontaneously broken by a set of flavon fields. By minimizing the scalar potential of the theory up to the next-to-leading order in the symmetry breaking parameter, we demonstrate that this mixing pattern is naturally achieved in a finite portion of the parameter space, through the vacuum alignment of the flavon fields. The leading order approximation is stable against higher-order corrections. We also compare our construction to other models based on discrete symmetry groups.

Self-organization and pattern formation in coupled Lorenz oscillators under a discrete symmetric transformation

We present a spatial array of Lorenz oscillators, with each cell lattice in the chaotic regime. This system shows spatial ordering due to self-organization of chaos synchronization after a bifurcation. It is shown that an array of such oscillators transformed under a discrete symmetry group, does not maintain the global dynamics, although each transformed unit cell is locally identical to its precursor. Alternatively, it is shown that in a 1-dimensional lattice, the coupling destroy the chaotic behavior but there are similar global behaviors between both coupled arrays, suggesting that is the local equivariance which controls the dynamics.

Statistical properties of cosmological billiards

Belinski, Khalatnikov, and Lifshitz pioneered the study of the statistical properties of the never-ending oscillatory behavior (among successive Kasner epochs) of the geometry near a spacelike singularity. We show how the use of a ``cosmological billiard'' description allows one to refine and deepen the understanding of these statistical properties. Contrary to previous treatments, we do not quotient the dynamics by its discrete symmetry group (of order 6), thereby uncovering new phenomena, such as correlations between the successive billiard corners in which the oscillations take place. Starting from the general integral invariants of Hamiltonian systems, we show how to construct invariant measures for various projections of the cosmological-billiard dynamics. In particular, we exhibit, for the first time, a (non-normalizable) invariant measure on the ``Kasner circle'' which parametrizes the exponents of successive Kasner epochs. Finally, we discuss the relation between: (i) the unquotiented dynamics of the Bianchi-IX ($a$, $b$, $c$ or mixmaster) model; (ii) its quotienting by the group of permutations of ($a$, $b$, $c$); and (iii) the billiard dynamics that arose in recent studies suggesting the hidden presence of Kac-Moody symmetries in cosmological billiards.

Noncommutative Bloch analysis of Bochner Laplacians with nonvanishing gauge fields

Given an invariant gauge potential and a periodic scalar potential V ̃ on a Riemannian manifold M ̃ with a discrete symmetry group Γ , consider a Γ -periodic quantum Hamiltonian H ̃ = − Δ ̃ B + V ̃ where Δ ̃ B is the Bochner Laplacian. Both the gauge group and the symmetry group Γ can be noncommutative, and the gauge field need not vanish. On the other hand, Γ is supposed to be of type I. With any unitary representation Λ of Γ one associates a Hamiltonian H Λ = − Δ B Λ + V on M = M ̃ / Γ where V is the projection of V ̃ onto M . We describe a construction of the Bloch decomposition of H ̃ into a direct integral whose components are H Λ , with Λ running over the dual space Γ ˆ . The evolution operator and the resolvent decompose correspondingly. Conversely, given Λ ∈ Γ ˆ , one can express the propagator K t Λ ( y 1 , y 2 ) (the kernel of exp ( − i t H Λ ) ) in terms of the propagator K ̃ t ( y 1 , y 2 ) (the kernel of exp ( − i t H ̃ ) ) as a weighted sum over Γ . Such a formula is known in theoretical physics for the case when the gauge field vanishes and M ̃ is a universal covering space of a multiply connected manifold M . We show that these constructions are mutually inverse. Analogous formulas exist for resolvents and their kernels (Green functions) as well.

Running effects on lepton mixing angles in flavour models with type I seesaw

We study renormalization group running effects on neutrino mixing patterns when a (type I) seesaw model is implemented by suitable flavour symmetries. We are particularly interested in mass-independent mixing patterns to which the widely studied tribimaximal mixing pattern belongs. In this class of flavour models, the running contribution from neutrino Yukawa coupling, which is generally dominant at energies above the seesaw threshold, can be absorbed by a small shift on neutrino mass eigenvalues leaving mixing angles unchanged. Consequently, in the whole running energy range, the change in mixing angles is due to the contribution coming from charged lepton sector. Subsequently, we analyze in detail these effects in an explicit flavour model for tribimaximal neutrino mixing based on an A 4 discrete symmetry group. We find that for normally ordered light neutrinos, the tribimaximal prediction is essentially stable under renormalization group evolution. On the other hand, in the case of inverted hierarchy, the deviation of the solar angle from its TB value can be large depending on mass degeneracy, putting strong constraints on the flavour model.

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.

Islamic geometrical patterns indexing and classification using discrete symmetry groups

In this article, we propose a general computational model for the extraction of symmetry features of Islamic geometrical patterns' (IGP) images. We describe IGP images using the discrete symmetry groups theory. Our model contains the three following steps. (1) By noting that these patterns fall into three major categories, we begin our indexation process by classifying every pattern into one of these categories. The first pattern category describes all the patterns generated by translation along one direction. Every pattern of this category can be classified into one of the seven Frieze groups. The second type of pattern contains translational symmetries in two independent directions. Patterns of this category can be classified into one of the seventeen Wallpaper groups. The last type, called rosettes, describes patterns which begin at a central point and grow radially outward. We use rosette symmetry groups to classify patterns of this latter category. (2) For every pattern, we extract the symmetry features, namely, the symmetry group and the fundamental region, which is a representative region in the image from which the whole image can be regenerated. But for rosette groups, we can also compute the number of folds. (3) Finally, we describe the fundamental region by a simple color histogram and build the feature vector which is a combination of the symmetry feature (defined in the second step) and histogram information. Experiments show promising results for either IGP images' classification or indexing. Efforts for the subsequent task of classifying Islamic geometrical patterns' images can be significantly reduced.

The quantum dilogarithm and representations of quantum cluster varieties

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. One of applications is quantization of higher Teichmuller spaces. The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms. We introduce the symplectic/quantum double of cluster varieties and related them to the representations.

Flavon inflation

We propose an entirely new class of particle physics models of inflation based on the phase transition associated with the spontaneous breaking of family symmetry responsible for the generation of the effective quark and lepton Yukawa couplings. We show that the Higgs fields responsible for the breaking of family symmetry, called flavons, are natural candidates for the inflaton field in new inflation, or the waterfall fields in hybrid inflation. This opens up a rich vein of possibilities for inflation, all linked to the physics of flavour, with interesting cosmological and phenomenological implications. Out of these, we discuss two examples which realise flavon inflation: a model of new inflation based on the discrete non-Abelian family symmetry group A4 or Δ27, and a model of hybrid inflation embedded in an existing flavour model with a continuous SU(3) family symmetry. With the inflation scale and family symmetry breaking scale below the Grand Unification Theory (GUT) scale, these classes of models are free of the monopole (and similar) problems which are often associated with the GUT phase transition.

Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries

At special loci in their moduli spaces, Calabi–Yau manifolds are endowed with discrete symmetries. Over the years, such spaces have been intensely studied and have found a variety of important applications. As string compactifications they are phenomenologically favored, and considerably simplify many important calculations. Mathematically, they provided the framework for the first construction of mirror manifolds, and the resulting rational curve counts. Thus, it is of significant interest to investigate such manifolds further. In this paper, we consider several unexplored loci within familiar families of Calabi–Yau hypersurfaces that have large but unexpected discrete symmetry groups. By deriving, correcting, and generalizing a technique similar to that of Candelas, de la Ossa and Rodriguez–Villegas, we find a calculationally tractable means of finding the Picard–Fuchs equations satisfied by the periods of all 3–forms in these families. To provide a modest point of comparison, we then briefly investigate the relation between the size of the symmetry group along these loci and the number of nonzero Yukawa couplings. We include an introductory exposition of the mathematics involved, intended to be accessible to physicists, in order to make the discussion self–contained.