Topic

# Rotational symmetry

About: Rotational symmetry is a research topic. Over the lifetime, 4531 publications have been published within this topic receiving 81400 citations.

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TL;DR: In this article, the authors present a quantitative continuum theory of flock behavior, which predicts the existence of an ordered phase of flocks, in which all members of even an arbitrarily large flock move together with the same mean velocity.

Abstract: We present a quantitative continuum theory of ``flocking'': the collective coherent motion of large numbers of self-propelled organisms. In agreement with everyday experience, our model predicts the existence of an ``ordered phase'' of flocks, in which all members of even an arbitrarily large flock move together with the same mean velocity $〈\stackrel{\ensuremath{\rightarrow}}{v}〉\ensuremath{
e}0.$ This coherent motion of the flock is an example of spontaneously broken symmetry: no preferred direction for the motion is picked out a priori in the model; rather, each flock is allowed to, and does, spontaneously pick out some completely arbitrary direction to move in. By analyzing our model we can make detailed, quantitative predictions for the long-distance, long-time behavior of this ``broken symmetry state.'' The ``Goldstone modes'' associated with this ``spontaneously broken rotational symmetry'' are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole. These ``Goldstone modes'' mix with modes associated with conservation of bird number to produce propagating sound modes. These sound modes lead to enormous fluctuations of the density of the flock, far larger, at long wavelengths, than those in, e.g., an equilibrium gas. Our model is similar in many ways to the Navier-Stokes equations for a simple compressible fluid; in other ways, it resembles a relaxational time-dependent Ginsburg-Landau theory for an $n=d$ component isotropic ferromagnet. In spatial dimensions $dg4,$ the long-distance behavior is correctly described by a linearized theory, and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems. For $dl4,$ nonlinear fluctuation effects radically alter the long distance behavior, making it different from that of any known equilibrium model. In particular, we find that in $d=2,$ where we can calculate the scaling exponents exactly, flocks exhibit a true, long-range ordered, spontaneously broken symmetry state, in contrast to equilibrium systems, which cannot spontaneously break a continuous symmetry in $d=2$ (the ``Mermin-Wagner'' theorem). We make detailed predictions for various correlation functions that could be measured either in simulations, or by quantitative imaging of real flocks. We also consider an anisotropic model, in which the birds move preferentially in an ``easy'' (e.g., horizontal) plane, and make analogous, but quantitatively different, predictions for that model as well. For this anisotropic model, we obtain exact scaling exponents for all spatial dimensions, including the physically relevant case $d=3.$

1,365 citations

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TL;DR: In this paper, it was shown that the boundary excitations of SPT phases can be described by a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear σ models.

Abstract: Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H^(1+d)[G,UT(1)], the Borel (1+d)-group-cohomology classes of G over the G module UT(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological θ term in continuous nonlinear σ models to lattice nonlinear σ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear σ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear σ models. As a result, the symmetry G must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use H^(1+d)[U(1)⋊ Z^(T)_(2),U_T(1)] to obtain interacting bosonic topological insulators (protected by time reversal Z2T and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of H^(1+d)[Z^(T)_(2),U_T(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use H1+d[U(1)×Z2T,UT(1)] to construct the SPT phases of integer spin systems with time-reversal and U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H,G_Ψ,H^(1+d)[G_Ψ,U_T(1)]), where G_H is the symmetry group of the Hamiltonian and G_Ψ the symmetry group of the ground states.

1,001 citations

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TL;DR: In this article, the concept of molecular symmetry is extended to molecules such as ethane and hydrazine, which can pass from one conformation to another, and examples are given to illustrate the use of this concept in determining the statistical weights of individual levels and selection rules for electric dipole transitions between them.

Abstract: The concept of molecular symmetry is extended to molecules such as ethane and hydrazine which can pass from one conformation to another. The symmetry group of such a molecule is the set of (i) all feasible permutations of the positions and spins of identical nuclei and (ii) all feasible permutation-inversions, which simultaneously invert the coordinates of all particles in the centre of mass. According to the representations of this group one can classify not only the spin states and states of motions of the nuclei, but even the electronic states of the molecule. Examples are given to illustrate the use of this concept in determining the statistical weights of individual levels and selection rules for electric dipole transitions between them.

963 citations

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TL;DR: In this paper, a particularly symmetric model of neutrino mixings where, with good accuracy, the atmospheric mixing angle �23 is maximal, �13 = 0 and the solar angle satisfies sin 2 �12 = 1 (HPS) matrix is discussed.

612 citations

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TL;DR: In this article, the authors proposed a continuous symmetry measure to quantify the distance of a given (distorted molecular) shape from any chosen element of symmetry, allowing one to compare the symmetry distance of several objects relative to a single symmetry element.

Abstract: We advance the notion that for many realistic issues involving symmetry in chemistry, it is more natural to analyze symmetry properties in terms of a continuous scale rather than in terms of "yes or no". Justification of that approach is dealt with in some detail using examples such as: symmetry distortions due to vibrations; changes in the "allowedness" of electronic transitions due to deviations from an ideal symmetry; continuous changes in environmental symmetry with reference to crystal and ligand field effects; non-ideal symmetry in concerted reactions; symmetry issues of polymers and large random objects. A versatile, simple tool is developed as a continuous symmetry measure. Its main property is the ability to quantify the distance of a given (distorted molecular) shape from any chosen element of symmetry. The generality of this symmetry measure allows one to compare the symmetry distance of several objects relative to a single symmetry element and to compare the symmetry distance of a single object relative to various symmetry elements. The continuous symmetry approach is presented in detail for the case of cyclic molecules, first in a practical way and then with a rigorous mathematical analysis. The versatility of the approah is then further demonstrated with alkane conformations, with a vibrating ABA water-like molecule, and with a three-dimensional analysis of the symmetry of a (2 3 21 reaction in which the double bonds are not ideally aligned.

584 citations