Abelian Symmetry Group Research Articles

Topological response theory of Abelian symmetry-protected topological phases in two dimensions.

It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with Zm symmetry and find the classification of SPT phases depends on the parity of m: for even m there are 2m classes, m out of which is intrinsically fermionic SPT phases and cannot be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.

Classical Topological Order in Abelian and Non-Abelian Generalized Height Models

We present MonteCarlo simulations on a new class of lattice models in which the degrees of freedom are elements of an Abelian or non-Abelian finite symmetry group G, placed on directed edges of a two-dimensional lattice. The plaquette group product is constrained to be the group identity. In contrast to discrete gauge models (but similar to past work on height models), only elements of symmetry-related subsets S∈G are allowed on edges. These models have topological sectors labeled by group products along topologically nontrivial loops. Measurement of relative sector probabilities and the distribution of distance between defect pairs are done to characterize the types of order (topological or quasi-long-range order) exhibited by these models. We present particular models in which fully local non-Abelian constraints lead to global topological liquid properties.

Abelian symmetries of the N-Higgs-doublet model with Yukawa interactions

We investigate finite abelian groups which can represent symmetries of the N-Higgs-doublet models with quarks. We build a general formalism based on the powerful method of the Smith normal form and obtain an analytic upper bound on the order of abelian symmetry groups for any N. We investigate in detail the case N=2 and rederive known results in a more compact and intuitive fashion. We also study the NHDM with the maximal finite abelian symmetry for all small N cases up to N=5, and show that in each case all Yukawa textures compatible with such symmetry originate from a unique basic structure. This work opens the way to a systematic exploration of phenomenology of the NHDM with a desired symmetry, and illustrates the power of the Smith normal form technique.

Abelian symmetries in multi-Higgs-doublet models

N-Higgs doublet models (NHDMs) are a popular framework to construct electroweak symmetry breaking mechanisms beyond the standard model. Usually, one builds an NHDM scalar sector which is invariant under a certain symmetry group. Although several such groups have been used, no general analysis of symmetries possible in the NHDM scalar sector exists. Here, we make the first step toward this goal by classifying the elementary building blocks, namely the Abelian symmetry groups, with a special emphasis on finite groups. We describe a strategy that identifies all Abelian groups which are realizable as symmetry groups of the NHDM Higgs potential. We consider both the groups of Higgs-family transformations only and the groups which also contain generalized CP transformations. We illustrate this strategy with the examples of 3HDM and 4HDM and prove several statements for arbitrary N.

Reduction of exterior differential systems for ordinary variational problems

This paper deals with the reduction of ordinary variational problems with Abelian and non-Abelian symmetry groups. It differs from the existing results on geometric reduction in the approach that is completely Lagrangian and based on exterior differential systems. In this setting, two procedures of reduction come out naturally: the first one is the Lagrangian analog of the Marsden–Weinstein reduction holding in the symplectic case and the other one, especially in the non-Abelian case, comes out naturally in our setting.

Extreme covariant quantum observables in the case of an Abelian symmetry group and a transitive value space

We represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference, and time observables.

Zero textures of the neutrino mass matrix from cyclic family symmetry

We present the symmetry realization of the phenomenologically viable Frampton–Glashow–Marfatia (FGM) two zero texture neutrino mass matrices in the flavor basis within the framework of the type (I+II) seesaw mechanism natural to SO(10) grand unification. A small Abelian cyclic symmetry group Z3 is used to realize these textures except for class C for which the symmetry is enlarged to Z4. The scalar sector is restricted to the Standard Model (SM) Higgs doublet to suppress the flavor changing neutral currents. Other scalar fields used for symmetry realization are at the most two scalar triplets and, in some cases, a complex scalar singlet. Symmetry realization of one zero textures has, also, been presented.

Extreme Covariant Observables for Type I Symmetry Groups

The structure of covariant observables—normalized positive operator measures (POMs)—is studied in the case of a type I symmetry group. Such measures are completely determined by kernels which are measurable fields of positive semidefinite sesquilinear forms. We produce the minimal Kolmogorov decompositions for the kernels and determine those which correspond to the extreme covariant observables. Illustrative examples of the extremals in the case of the Abelian symmetry group are given.

Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation

A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension ( 2 n + 2 ) with an n-dimensional abelian symmetry group. Examples for n = 1 , 2 , 3 are presented to illustrate application of the theory.

The Complexity of Symmetry-Breaking Formulas

Symmetry-breaking formulas for a constraint-satisfaction problem are satisfied by exactly one member (e.g., the lexicographic leader) from each set of “symmetrical points” in the search space. Thus, the incorporation of such formulas can accelerate the search for a solution without sacrificing satisfiability. We study the computational complexity of generating lex-leader formulas. We show, even for abelian symmetry groups, that the number of essential clauses in the “natural” lex-leader formula could be exponential. Furthermore, we show the intractability (NP-hardness) of finding any expression of lex-leadership without reordering the variables, even for elementary abelian groups with orbits of size 3. Nevertheless, using techniques of computational group theory, we describe a reordering relative to which we construct small lex-leader formulas for abelian groups.

A coordinate-free reduction for flows on the volume manifold

In this paper, instead of previous methods such as the Lie group method dependent of coordinate transformation, a coordinate-free and pure geometric reduction procedure for flows preserving volume form on any n-dimensional volume manifold is given by the general differential form theory. That is, a volume form-preserving flow with a r-parameter Abelian volume-preserving symmetry group on any n-dimensional volume manifold can be reduced into a volume-preserving flow on the corresponding ( n-r)-dimensional manifold by the differential geometric method.

Leapfrogging vortex rings: Hamiltonian structure, geometric phases and discrete reduction

We present two interesting features of vortex rings in incompressible, Newtonian fluids that involve their Hamiltonian structure. The first feature is for the Hamiltonian model of dynamically interacting thin-cored, coaxial, circular vortex rings described, for example, in the works of Dyson (Philos. Trans. Roy. Soc. London Ser. A 184 (1893) 1041) and Hicks (Proc. Roy. Soc. London Ser. A 102 (1922) 111). For this model, the symplectic reduced space associated with the translational symmetry is constructed. Using this construction, it is shown that for periodic motions on this reduced space, the reconstructed dynamics on the momentum level set can be split into a dynamic phase and a geometric phase. This splitting is done relative to a cotangent bundle connection defined for abelian isotropy symmetry groups. In this setting, the translational motion of leapfrogging vortex pairs is interpreted as the total phase, which has a dynamic and a geometric component. Second, it is shown that if the rings are modeled as coaxial circular filaments, their dynamics and Hamiltonian structure is derivable from a more general Hamiltonian model for N interacting filament rings of arbitrary shape in R 3 , where the mutual interaction is governed by the Biot–Savart law for filaments and the self-interaction is determined by the local induction approximation. The derivation is done using the fixed point set for the action of the group of rotations about the axis of symmetry using methods of discrete reduction theory.

Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics

In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G= U(1), there exists a generalization, known as p-form electrodynamics, in which ( p−1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p=2 to possibly non-Abelian symmetry groups. An important feature of our model is that it involves both parallel transports along paths and generalized transports along surfaces with a non-trivial interplay of these two types of variables. Our main result is the geometric picture, namely the assignment of non-Abelian quantities to geometrical objects in a coordinate free way. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.

Divergence-free vector-field and reduction

By the Lie symmetry group, the reduction for divergence-free vector-fields (DFVs) is studied, and the following results are found. A n-dimensional DFV can be locally reduced to a ( n − 1)-dimensional DFV if it admits a one-parameter symmetry group that is spatial and divergenceless. More generally, a n-dimensional DFV admitting a r-parameter, spatial, divergenceless Abelian (commutable) symmetry group can be locally reduced to a ( n − r)-dimensional DFV.

Principles of Symmetry, Dynamics and Spectroscopy

A Review of Matrix Algebra and Quantum Mechanics. Basic Theory and Applications of Symmetry Representations (Abelian Symmetry Groups). Basic Theory and Applications of Symmetry Representations (Non-Abelian Symmetry Groups). Theory and Applications of Higher Finite Symmetry and Induced Representations. Representations of Continuous Rotation Groups and Applications. Theory and Applications of Symmetry Representation Products (Finite Groups). Theory and Application of Symmetry Representation Products (Continuous Rotation Groups). Symmetry Analysis for Semiclassical and Quantum Mechanics: Dynamics with High Quanta. Appendices. Index.

Phase structure of two-dimensional spin models and percolation

For a class of classical spin models in 2D satisfying a certain continuity constraint it is proven that some of their correlations do not decay exponentially. The class contains discrete and continuous spin systems with Abelian and non-Abelian symmetry groups. For the discrete models our results imply that they show either long-range order or are in a soft phase characterized by powerlike decay of correlations; for the continuous models only the second possibility exists. The continuous models include a version of the plane rotator [O(2)] model; for this model we rederive, modulo two conjectures, the Frohlich-Spencer result on the existence of the Kosterlitz-Thouless phase in a very simple way. The proof is based on percolation-theoretic and topological arguments.

Galerkin projections and the proper orthogonal decomposition for equivariant equations

In this paper we are interested in Galerkin projections of certain partial differential evolution equations, that retain the symmetries of the original nontruncated equations. In particular, we extend several results concerning the Galerkin projections, based on the proper orthogonal decomposition (which is also known as the Karhunen-Loève expansion) for equations which are equivariant under the action of compact Abelian symmetry groups. We provide sufficient conditions for numerical schemes, based on the linear and the nonlinear Galerkin method, to retain the symmetries of the original system. We perform a rational analysis of methods which use symmetry to enlarge the ensemble size of a data set. Our analysis has the potential for computational savings.

Lagrangian reduction and the double spherical pendulum

This paper studies the stability and bifurcations of the relative equilibrium of the double spherical pendulum, which has the circle as its symmetry group. The example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group.

On the use of spatial symmetry in ab initio calculations. Transformation of the two-electron integrals from atomic orbital to localized molecular orbital basis

Anm5-dependent integral transformation procedure from atomic orbital basis to localized molecular orbitals is described for spatially extended systems with some Abelian symmetry groups. It is shown that exploiting spatial symmetry, the number of non-redundant integrals for normal saturated hydrocarbons can be reduced by a factor of 2.5-3.5, depending on the size of the system and on the basis. Starting from a list of integrals over basis functions in canonical order, the number of multiplications of the four-index transformation is reduced by a factor of 2.8-3.5 as compared to that of Diercksen's algorithm. It is pointed out that even larger reduction can be achieved if negligible integrals over localized molecular orbitals are omitted from the transformation in advance.

Abelian symmetry groups in social choice

Abstract For every finite abelian group G there exists (i) a social choice rule f for which the group of permutations of individuals leaving f unchanged is isomorphic to G ; and (ii) a social choice rule g for which the group of permutations of alternatives leaving g unchanged is isomorphic to G .