Showing papers on "Symmetry (geometry) published in 2013"

Classifying fractionalization: Symmetry classification of gapped Z 2 spin liquids in two dimensions

Abstract: We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with ${\mathbb{Z}}_{2}$ topological order, that is, on gapped ${\mathbb{Z}}_{2}$ spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group ${H}^{2}(G,{\mathbb{Z}}_{2})$. This result leads us to a symmetry classification of gapped ${\mathbb{Z}}_{2}$ spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and $\mathrm{SO}(3)$ spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric ${\mathbb{Z}}_{2}$ spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.

The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations

Abstract: If a system undergoes symmetric dynamics, then the final state of the system can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state's asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be understood in terms of information-theoretic concepts, for instance in terms of the state's ability to encode information about an element of the symmetry group. We show that the asymmetry properties of a pure state ψ relative to the symmetry group G are completely specified by the characteristic function of the state, defined as χψ(g) ≡ 〈ψ|U(g)|ψ〉 where g∈G and U is the unitary representation of interest. For a symmetry described by a compact Lie group G, we show that two pure states can be reversibly interconverted one to the other by symmetric operations if and only if their characteristic functions are equal up to a one-dimensional representation of the group. Characteristic functions also allow us to easily identify the conditions for one pure state to be converted to another by symmetric operations (in general irreversibly) for the various paradigms of single-copy transformations: deterministic, state-to-ensemble, stochastic and catalyzed.

Braiding statistics approach to symmetry-protected topological phases

Abstract: We construct a 2D quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a \"symmetry-protected topological phase.\" We describe a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z_2 gauge field and then show that the \\pi-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.

Gravity from rational curves in twistor space.

Abstract: This Letter presents a new formula which is conjectured to yield all tree amplitudes in $\mathcal{N}=8$ supergravity. The amplitudes are described in terms of higher degree rational maps to twistor space. The resulting expression has manifest $\mathcal{N}=8$ supersymmetry and is manifestly permutation symmetric in all external states. It depends monomially on the infinity twistor that explicitly breaks conformal symmetry to Poincar\'e symmetry. We have carried out various nontrivial analytic and numerical checks of the formula for up to eight external states with arbitrary helicities.

Symmetry Relationships between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry

Abstract: 1. Introduction PART I: CRYSTALLOGRAPHIC FOUNDATIONS 2. Basics of crystallography, part 1 3. Mappings 4. Basics of crystallography, part 2 5. Group theory 6. Basics of crystallography, part 3 7. Subgroups and supergroups of point and space groups 8. Conjugate subgroups, normalizers and equivalent descriptions of crystal structures 9. How to handle space groups PART II: SYMMETRY RELATIONS BETWEEN SPACE GROUPS AS A TOOL TO DISCLOSE CONNECTIONS BETWEEN CRYSTAL STRUCTURES 10. The group-theoretical presentation of crystal-chemical relationships 11. Symmetry relations between between related crystal structures 12. Pitfalls when setting up group-subgroup relations 13. Derivation of crystal structures from closest packings of spheres 14. Crystal structures of molecular compounds 15. Symmetry relations at phase transitions 16. Topotactic reactions 17. Group-subgroup relations as an aid for structure determination 18. Prediction of possible structure types 19. Historical remarks Appendix A: Isomorphic subgroups Appendix B: On the theory of phase transitions Appendix C: Symmetry species Appendix D: Solutions to the exercises References Glossary Index

ψ-epistemic theories: The role of symmetry

Abstract: Formalizing an old desire of Einstein, "psi-epistemic theories" try to reproduce the predictions of quantum mechanics, while viewing quantum states as ordinary probability distributions over underlying objects called "ontic states." Regardless of one's philosophical views about such theories, the question arises of whether one can cleanly rule them out, by proving no-go theorems analogous to the Bell Inequality. In the 1960s, Kochen and Specker (who first studied these theories) constructed an elegant psi-epistemic theory for Hilbert space dimension d=2, but also showed that any deterministic psi-epistemic theory must be "measurement contextual" in dimensions 3 and higher. Last year, the topic attracted renewed attention, when Pusey, Barrett, and Rudolph (PBR) showed that any psi-epistemic theory must "behave badly under tensor product." In this paper, we prove that even without the Kochen-Specker or PBR assumptions, there are no psi-epistemic theories in dimensions d>=3 that satisfy two reasonable conditions: (1) symmetry under unitary transformations, and (2) "maximum nontriviality" (meaning that the probability distributions corresponding to any two non-orthogonal states overlap). This no-go theorem holds if the ontic space is either the set of quantum states or the set of unitaries. The proof of this result, in the general case, uses some measure theory and differential geometry. On the other hand, we also show the surprising result that without the symmetry restriction, one can construct maximally-nontrivial psi-epistemic theories in every finite dimension d.

Calibrating Photometric Stereo by Holistic Reflectance Symmetry Analysis

Abstract: Under unknown directional lighting, the uncalibrated Lambertian photometric stereo algorithm recovers the shape of a smooth surface up to the generalized bas-relief (GBR) ambiguity. We resolve this ambiguity from the half vector symmetry, which is observed in many isotropic materials. Under this symmetry, a 2D BRDF slice with low-rank structure can be obtained from an image, if the surface normals and light directions are correctly recovered. In general, this structure is destroyed by the GBR ambiguity. As a result, we can resolve the ambiguity by restoring this structure. We develop a simple algorithm of auto-calibration from separable homogeneous specular reflection of real images. Compared with previous methods, this method takes a holistic approach to exploiting reflectance symmetry and produces superior results.

Symmetry-protected topological phases and orbifolds: Generalized Laughlin's argument

Abstract: We consider nonchiral symmetry-protected topological phases of matter in two spatial dimensions protected by a discrete symmetry such as ${\mathbb{Z}}_{K}$ or ${\mathbb{Z}}_{K}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{K}$ symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons $K$-matrix theories and fermionic phases relevant to topological superconductors as an example, we demonstrate explicitly that when the modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge state.

Symmetries in the representation of grain boundary-plane distributions

Abstract: Symmetries play a crucial role in the theoretical analysis and visualization of the five macroscopic grain boundary parameters, including the misorientation (three parameters) and the orientation of the boundary-plane (two parameters). The symmetry aspects of the misorientation spaces are very well documented and in this article all possible boundary-plane symmetries are enumerated for the 32 crystallographic point groups. It is observed that the boundary-plane spaces exhibit a wide variety of point group symmetries, which depend both on the crystallographic point group and on the corresponding misorientation (i.e. location in the fundamental zone). The list of symmetries presented here should serve as a guide for graphical representations of not only the distributions of boundary-plane orientations but also for the representation of boundary-plane related properties such as energy, mobility etc.

Classification of Lie point symmetries for quadratic Liénard type equation ẍ+f(x)ẋ2+g(x)=0

Abstract: In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Lienard type equation, x+f(x)x2+g(x)=0, where f(x) and g(x) are arbitrary functions of x. The symmetry analysis gets divided into two cases, (i) the maximal (eight parameter) symmetry group and (ii) non-maximal (three, two, and one parameter) symmetry groups. We identify the most general form of the quadratic Lienard equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable as well as isochronic. We present specific examples of physical interest. For the non-maximal cases, the identified equations are all integrable and include several physically interesting examples such as the Mathews-Lakshmanan oscillator, particle on a rotating parabolic well, etc. We also analyse the underlying equivalence transformations.

Classification of Plane Symmetric Static Space-Times According to Their Noether Symmetries

Abstract: In this paper we give a classification of plane symmetric static space-times using symmetry method. For this purpose we consider the Lagrangian corresponding to the general plane symmetric static metric in the Noether symmetry equation. This provides a system of determining equations. Solutions of this system give us classification of the plane symmetric static space-times according to their Noether symmetries. During this classification we recover all the results listed in Feroze et al. (J. Math. Phys. 42:4947, 2001) and Bashir and Ehsan (Il Nuovo Cimento B 123:1, 2008).

The Small Step toward Asymmetry: Aesthetic Judgment of Broken Symmetries:

Abstract: Symmetry and complexity both affect the aesthetic judgment of abstract patterns. However, although beauty tends to be associated with symmetry, there are indications that small asymmetries can also be beautiful. We investigated the influence of small deviations from symmetry on people's aesthetic liking for abstract patterns. Breaking symmetry not only decreased patterns' symmetry but also increased their complexity. While an increase of complexity normally results in a higher liking, we found that even a small decrease of symmetry has a strong effect, such that patterns with slightly broken symmetries were significantly less liked than fully symmetric ones.

The small step toward asymmetry: Aesthetic judgment of broken symmetries

Abstract: Symmetry and complexity both affect the aesthetic judgment of abstract patterns. However, although beauty tends to be associated with symmetry, there are indications that small asymmetries can also be beautiful. We investigated the influence of small deviations from symmetry on people's aesthetic liking for abstract patterns. Breaking symmetry not only decreased patterns' symmetry but also increased their complexity. While an increase of complexity normally results in a higher liking, we found that even a small decrease of symmetry has a strong effect, such that patterns with slightly broken symmetries were significantly less liked than fully symmetric ones.

A K3 sigma model with Z_2^8:M_{20} symmetry

Abstract: The K3 sigma model based on the Z_2-orbifold of the D_4-torus theory is studied. It is shown that it has an equivalent description in terms of twelve free Majorana fermions, or as a rational conformal field theory based on the affine algebra su(2)^6. By combining these different viewpoints we show that the N=(4,4) preserving symmetries of this theory are described by the discrete symmetry group Z_2^8:M_{20}. This model therefore accounts for one of the largest maximal symmetry groups of K3 sigma models. The symmetry group involves also generators that, from the orbifold point of view, map untwisted and twisted sector states into one another.

About the x-y symmetry of the F_g algebraic invariants

Abstract: We complete the proof of the x-y symmetry of symplectic invariants of [EO]. We recall the main steps of the proof of [EO2], and we include the integration constants absent in [EO2].

Reflections on the four facets of symmetry: how physics exemplifies rational thinking

Abstract: In contemporary theoretical physics, the powerful notion of symmetry stands for a web of intricate meanings among which I identify four clusters associated with the notion of transformation, comprehension, invariance and projection. While their interrelations are examined closely these four facets of symmetry are scrutinised one after the other in great detail. This decomposition allows us to carefully examine the multiple different roles symmetry plays in many places in physics. Furthermore, some connections with other disciplines like neurobiology, epistemology, cognitive sciences and, not least, philosophy are proposed in an attempt to show that symmetry can be an organising principle also in these fields.

On the Hopf conjecture with symmetry

Abstract: The Hopf conjecture states that an even-dimensional manifold with positive curvature has positive Euler characteristic. We show that this is true under the assumption that a torus of sufficiently large dimension acts by isometries. This improves previous results by replacing linear bounds by a logarithmic bound. The new method that is introduced is the use of Steenrod squares combined with geometric arguments of a similar type to what was done before.

Noetherianity up to symmetry

Abstract: These lecture notes for the 2013 CIME/CIRM summer school Combinatorial Algebraic Geometry deal with manifestly infinite-dimensional algebraic varieties with large symmetry groups. So large, in fact, that subvarieties stable under those symmetry groups are defined by finitely many orbits of equations---whence the title Noetherianity up to symmetry. It is not the purpose of these notes to give a systematic, exhaustive treatment of such varieties, but rather to discuss a few "personal favourites": exciting examples drawn from applications in algebraic statistics and multilinear algebra. My hope is that these notes will attract other mathematicians to this vibrant area at the crossroads of combinatorics, commutative algebra, algebraic geometry, statistics, and other applications.

Inferring mirror symmetric 3D shapes from sketches

Abstract: We describe a system for taking a 2D sketch of a mirror-symmetric 3D shape and lifting the curves to 3D, inferring the symmetry relationship from the original hand-drawn curves. The system takes as input a hand-drawn sketch and generates a set of 3D curves such that their orthogonal projection matches the input sketch. The main contribution is a method which is able to identify the symmetry relationship among the hand-drawn curves even in the presence of ambiguity in the sketch.

Symmetry Enriched Phases via Pseudo Anyon Condensation

Abstract: We show that a large class of symmetry enriched (topological) phases of matter in 2+1 dimensions can be embedded in "larger" topological phases- phases describable by larger hidden Hopf symmetries. Such an embedding is analogous to anyon condensation, although no physical condensation actually occurs. This generalizes the Landau-Ginzburg paradigm of symmetry breaking from continuous groups to quantum groups- in fact algebras- and offers a potential classification of the symmetry enriched (topological) phases thus obtained, including symmetry protected trivial phases as well, in a unified framework.

On the Symmetry Classification of Energy Bands in Graphene

Abstract: In the previous publications (E. Kogan and V. U. Nazarov, Phys. Rev. B {\bf 85}, 115418 (2012) and E. Kogan, Graphene {\bf 2}, 74 (2013)) we presented point group symmetry classification of energy bands in graphene. In the present note we generalize classification by considering the space group symmetry. Also in the framework of group theory we describe destruction of the Dirac points in graphene by a supercell potential.

Dark Matter in a twisted bottle

Abstract: The real projective plane is a compact, non-orientable orbifold of Euler charac- teristic 1 without boundaries, which can be described as a twisted Klein bottle. We shortly review the motivations for choosing such a geometry among all possible two-dimensional orbifolds, while the main part of the study will be devoted to dark matter study and limits in Universal Extra Dimensional (UED) models based on this peculiar geometry. In the following we consider such a UED construction based on the direct product of the real projective plane with the standard four-dimensional Minkowski space-time and discuss its relevance as a model of a weakly interacting Dark Matter candidate. One important difference with other typical UED models is the origin of the symmetry leading to the stability of the dark matter particle. This symmetry in our case is a remnant of the six-dimensional Minkowski space-time symmetry partially broken by the compactifi- cation. Another important difference is the very small mass splitting between the particles of a given Kaluza-Klein tier, which gives a very important role to co-annihilation effects. Finally the role of higher Kaluza-Klein tiers is also important and is discussed together with a detailed numerical description of the influence of the resonances.

Trinity symmetry and kaleidoscopic regular maps

Abstract: A cellular embedding of a connected graph (also known as a map) on an orientable surface has trinity symmetry if it is isomorphic to both its dual and its Petrie dual. A map is regular if for any two incident vertex-edge pairs there is an automorphism of the map sending the first pair onto the second. Given a map M with all vertices of the same degree d, for any e relatively prime to d the power map M e is formed from M by replacing the cyclic rotation of edges at each vertex on the surface with the e th power of the rotation. A map is kaleidoscopic if all of its power maps are pairwise isomorphic. In this paper, we present a covering construction that gives infinite families of kaleidoscopic regular maps with trinity symmetry.

Long time solutions for wave maps with large data

Abstract: For (2 + 1)-dimensional wave maps with 𝕊2 as the target, we show that for all positive numbers T0 > 0 and E0 > 0, there exist Cauchy initial data with energy at least E0, so that the solution's life-span is at least [0, T0]. We assume neither symmetry nor closeness to harmonic maps.