Molien series and low-degree invariants for a natural action of ${\rm SO}(3)\;\wr \;{{{\rm Z}}_{2}}$

Chillingworth, D. R. J.; Lauterbach, Reiner; Turzi, Stefano

doi:10.1088/1751-8113/48/1/015203

Cited by 5 publications

(20 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the symmetry of a phase may be also determined by the "eigenvalues" and the distribution of nonzero entries of O G i [61]. Studies with this regard so far mostly concentrate on the rank-2 D ∞h and D 2h ordering tensors [12,[62][63][64], it would be interesting to consider the ordering of the tensors in Table I in full generality without assumptions on microscopic configurations of a particular model.…”

Section: Determining the Symmetry Of A Phase With A High-rank Order P...mentioning

confidence: 99%

Classification of point-group-symmetric orientational ordering tensors

et al. 2016

View full text Add to dashboard Cite

The concept of symmetry breaking has been a propelling force in understanding phases of matter. While rotational-symmetry breaking is one of the most prevalent examples, the rich landscape of orientational orders breaking the rotational symmetries of isotropic space, i.e., O(3), to a three-dimensional point group remain largely unexplored, apart from simple examples such as ferromagnetic or uniaxial nematic ordering. Here we provide an explicit construction, utilizing a recently introduced gauge-theoretical framework, to address the three-dimensional point-group-symmetric orientational orders on a general footing. This unified approach allows us to enlist order parameter tensors for all three-dimensional point groups. By construction, these tensor order parameters are the minimal set of simplest tensors allowed by the symmetries that uniquely characterize the orientational order. We explicitly give these for the point groups {C n ,D n ,T ,O,I } ⊂ SO(3) andfor n,2n ∈ {1,2,3,4,6,∞}. This central result may be perceived as a road map for identifying exotic orientational orders that may become more and more in reach in view of rapid experimental progress in, e.g., nanocolloidal systems and novel magnets.

show abstract

Section: Determining the Symmetry Of A Phase With A High-rank Order P...mentioning

confidence: 99%

Classification of point-group-symmetric orientational ordering tensors

et al. 2016

View full text Add to dashboard Cite

show abstract

“…Conway and Smith [3] show that this is a two-to-one map on SO (4) where the needed identification is [1,1] = [−1, −1] yielding [l, r] = [−l, −r], which obviously both map a point x to the same image point. The authors have used this characterization to classify the closed subgroups of SO(4) in terms of the biquaternionic notation.…”

Section: Resultsmentioning

confidence: 99%

“…It is a subtle yet very important observation that this map is -when taking the identification into account -a bijection but it is not a group homomorphism. Following Chillingworth, Lauterbach, and Turzi [1] we define a similar map via [l, r] : x → lxr.…”

Section: Resultsmentioning

confidence: 99%

Equivariant bifurcations in four-dimensional fixed point spaces

Lauterbach

Schwenker

2016

Dynamical Systems

View full text Add to dashboard Cite

In honor of Marty Golubitsky on the occasion of his seventieth birthday. AbstractIn this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in the previous papers by Lauterbach [14] and Lauterbach and Matthews [15] we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behavior is different from what we have seen in the known examples.

show abstract

“…In other words, the group of 3D rotations. 9. Similarly, O(2) is the orthogonal group in two dimensions, and SO(2) is the special orthogonal group in two dimensions 10.…”

Section: The Scalar Product Between Two Second-rank Tensors T L Is Dmentioning

confidence: 99%

“…The definitions we provide here best fit the group-theoretic analysis put forward in the rest of the paper, and allow taking a non-standard view on this topic, by describing the ordering tensor in terms of a linear map in the space of symmetric, traceless second rank tensors; an analogous approach is found in Refs. [7,8,9,10]. In Secs.…”

Section: Introductionmentioning

confidence: 99%

Determination of the symmetry classes of orientational ordering tensors

Turzi

Bisi

2017

Nonlinearity

View full text Add to dashboard Cite

The orientational order of nematic liquid crystals is traditionally studied by means of the secondrank ordering tensor . When this is calculated through experiments or simulations, the symmetry group of the phase is not known a-priori, but needs to be deduced from the numerical realisation of , which is affected by numerical errors. There is no generally accepted procedure to perform this analysis. Here, we provide a new algorithm suited to identifying the symmetry group of the phase. As a by product, we prove that there are only five phase-symmetry classes of the second-rank ordering tensor and give a canonical representation of for each class. The nearest tensor of the assigned symmetry is determined by group-projection. In order to test our procedure, we generate uniaxial and biaxial phases in a system of interacting particles, endowed with D ∞h or D 2h symmetry, which mimic the outcome of Monte-Carlo simulations. The actual symmetry of the phases is correctly identified, along with the optimal choice of laboratory frame.

show abstract

Molien series and low-degree invariants for a natural action of ${\rm SO}(3)\;\wr \;{{{\rm Z}}_{2}}$

Cited by 5 publications

References 22 publications

Classification of point-group-symmetric orientational ordering tensors

Classification of point-group-symmetric orientational ordering tensors

Equivariant bifurcations in four-dimensional fixed point spaces

Determination of the symmetry classes of orientational ordering tensors

Contact Info

Product

Resources

About