Showing papers on "Symmetry (geometry) published in 2011"

Classifying quantum phases using matrix product states and projected entangled pair states

Abstract: We give a classification of gapped quantum phases of one-dimensional systems in the framework of matrix product states (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states and in both the absence and the presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground-state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, that is, the equivalence classes of its projective representations, a result first derived by Chen, Gu, and Wen [Phys. Rev. B 83, 035107 (2011)]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labeled by both the permutation action of the former and the cohomology class of the latter. Using projected entangled pair states (PEPS), we subsequently extend our framework to the classification of two-dimensional phases in the neighborhood of a number of important cases, in particular, systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS, which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus we can focus on to the classification of isometric forms.

Torus knots and mirror symmetry

Abstract: We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar-Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.

The theory of manipulations of pure state asymmetry I: basic tools and equivalence classes of states under symmetric operations

Abstract: the nal state 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 relative to the symmetry groupG are completely specied by the characteristic function of the state, dened as (g) h jU(g)j i where g 2 G and U is the unitary representation of interest. Among other results, we show that for a symmetry described by a compact Lie group G, two pure states can be reversibly interconverted one to the other by symmetric dynamics if and only if their characteristic functions are equal up to a 1-dimensional representation of the group. PACS numbers:

Rotation-reversal symmetries in crystals and handed structures

Abstract: The symmetries of crystals are an important factor in the understanding of their properties. The discovery of a new symmetry type, rotation-reversal symmetry, may lead to the discovery of new rotation-based phenomena, for example in multiferroic materials.

Splitting of operations, Manin products and Rota-Baxter operators

Abstract: This paper provides a general operadic definition for the notion of splitting the operations of algebraic structures. This construction is proved to be equivalent to some Manin products of operads and it is shown to be closely related to Rota-Baxter operators. Hence, it gives a new effective way to compute Manin black products. The present construction is shown to have symmetry properties. Finally, this allows us to describe the algebraic structure of square matrices with coefficients in algebras of certain types. Many examples illustrate this text, including the case of Jordan algebras.

Entanglement and symmetry in permutation-symmetric states

Abstract: We investigate the relationship between multipartite entanglement and symmetry, focusing on permutation symmetric states. We give a highly intuitive geometric interpretation to entanglement via the Majorana representation, where these states correspond to points on a unit sphere. We use this to show how various entanglement properties are determined by the symmetry properties of the states. The geometric measure of entanglement is thus phrased entirely as a geometric optimization and a condition for the equivalence of entanglement measures written in terms of point symmetries. Finally, we see that different symmetries of the states correspond to different types of entanglement with respect to interconvertibility under stochastic local operations and classical communication.

Bonus Yangian Symmetry for the Planar S-Matrix of N=4 Super Yang-Mills

Abstract: Recent developments in the determination of the planar S-matrix of N = 4 Super Yang-Mills are closely related to its Yangian symmetry. Here we provide evidence for a yet unobserved additional symmetry: the Yangian level-one helicity operator.

1D symmetry for solutions of semilinear and quasilinear elliptic equations

Abstract: Several new $ 1$D results for solutions of possibly singular or degenerate elliptic equations, inspired by a conjecture of De Giorgi, are provided. In particular, $ 1$D symmetry is proven under the assumption that either the profiles at infinity are $ 2$D, or that one level set is a complete graph, or that the solution is minimal or, more generally, $ Q$-minimal.

A biologically plausible model of human shape symmetry perception

Abstract: Symmetry is usually computationally expensive to encode reliably, and yet it is relatively effortless to perceive. Here, we extend F. J. A. M. Poirier and H. R. Wilson's (2006) model for shape perception to account for H. R. Wilson and F. Wilkinson's (2002) data on shape symmetry. Because the model already accounts for shape perception, only minimal neural circuitry is required to enable it to encode shape symmetry as well. The model is composed of three main parts: (1) recovery of object position using large-scale non-Fourier V4-like concentric units that respond at the center of concentric contour segments across orientations, (2) around that recovered object center, curvature mechanisms combine multiplicatively the responses of oriented filters to encode object-centric local shape information, with a preference for convexities, and (3) object-centric symmetry mechanisms. Model and human performances are comparable for symmetry perception of shapes. Moreover, with some improvement of edge recovery, the model can encode symmetry axes in natural images such as faces.

Quantum [FORMULA]-symmetry in AdS3

Abstract: It has recently been argued that, classically, massless higher spin theories in AdS 3 have an enlarged [FORMULA]-symmetry as the algebra of asymptotic isometries. In this note we provide evidence that this symmetry is realised (perturbatively) in the quantum theory. We perform a one loop computation of the fluctuations for a massless spin s field around a thermal AdS 3 background. The resulting determinants are evaluated using the heat kernel techniques of arXiv:0911.5085. The answer factorises holomorphically, and the contributions from the various spin s fields organise themselves into vacuum characters of the [FORMULA] symmetry. For the case of the hs(1, 1) theory consisting of an infinite tower of massless higher spin particles, the resulting answer can be simply expressed in terms of (two copies of) the MacMahon function.

Interactive Visualization of Rotational Symmetry Fields on Surfaces

Abstract: Rotational symmetries (RoSys) have found uses in several computer graphics applications, such as global surface parameterization, geometry remeshing, texture and geometry synthesis, and nonphotorealistic visualization of surfaces. The visualization of N-way rotational symmetry (N-RoSy) fields is a challenging problem due to the ambiguities in the N directions represented by an N-way symmetry. We provide an algorithm that allows faithful and interactive representation of N-RoSy fields in the plane and on surfaces, by adapting the well-known line integral convolution (LIC) technique from vector and second-order tensor fields. Our algorithm captures N directions associated with each point in a given field by decomposing the field into multiple different vector fields, generating LIC images of these fields, and then blending the results. To address the loss of contrast caused by the blending of images, we observe that the pixel values in LIC images closely approximate normally distributed random variables. This allows us to use concepts from probability theory to correct the loss of contrast without the need to perform any image analysis at each frame.

Translation symmetry detection in a fronto-parallel view

Abstract: In this paper, we present a method of detecting translation symmetries from a fronto-parallel image. The proposed method automatically detects unknown multiple repetitive patterns of arbitrary shapes, which are characterized by translation symmetries on a plane. The central idea of our approach is to take advantage of the interesting properties of translation symmetries in both image space and the space of transformation group. We first detect feature points in input image as sampling points. Then for each sampling point, we search for the most probable corresponding lattice structures in the image and transform spaces using scale-space similarity maps. Finally, using a MRF formulation, we optimally partition the graph of all sampling points associated with the estimated lattices into subgraphs of sampling points and lattices belonging to the same symmetry pattern. Our method is robust because of the joint analysis in image and transform spaces, and the MRF optimization. We demonstrate the robustness and effectiveness of our method on a large variety of images.

Relating crystal structures by group–subgroup relations

Abstract: The relations between crystal structures that are related by symmetry can be set forth in a concise manner with a tree of group–subgroup relations of their space groups (called a Bar­nig­hausen tree). At its top, the tree starts from the space-group symbol of an aristotype, i.e. a simple, highly symmetrical crystal structure. Arrows pointing downwards depict symmetry reductions that result from structural distortions or partial substitutions of atoms; each arrow represents the relation from a space group to a maximal subgroup. In the middle of each arrow the kind of the subgroup is marked by a t (translationengleiche), k (klassengleiche) or i (isomorphic) followed by the index of the symmetry reduction. In addition, changes of the basis vectors and origin shifts are marked. Each step of the symmetry reduction may involve moderate changes of the atomic coordinates that have to be monitored carefully. An aristotype can be at the head of a large family of structures. From the kinds of subgroups it can be deduced what and how many kinds of domains can result at a phase transition or topotactic reaction involving a symmetry reduction.

Moments of wishart-laguerre and jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities

Abstract: We collect explicit and user-friendly expressions for one-point densities of the real eigenvalues $\{\lambda_i\}$ of $N\times N$ Wishart-Laguerre and Jacobi random matrices with orthogonal, unitary and symplectic symmetry. Using these formulae, we compute integer moments $\tau_n= $ for all symmetry classes without any large $N$ approximation. In particular, our results provide exact expressions for moments of transmission eigenvalues in chaotic cavities with time-reversal or spin-flip symmetry and supporting a finite and arbitrary number of electronic channels in the two incoming leads.

Dense 3D reconstruction of symmetric scenes from a single image

Abstract: A system is presented that takes a single image as an input (e.g. showing the interior of St.Peter's Basilica) and automatically detects an arbitrarily oriented symmetry plane in 3D space. Given this symmetry plane a second camera is hallucinated that serves as a virtual second image for dense 3D reconstruction, where the point of view for reconstruction can be chosen on the symmetry plane. This naturally creates a symmetry in the matching costs for dense stereo. Alternatively, we also show how to enforce the 3D symmetry in dense depth estimation for the original image. The two representations are qualitatively compared on several real world images, that also validate our fully automatic approach for dense single image reconstruction.

Design of a spherical focal surface using close-packed relay optics

Abstract: This paper presents a design strategy for close-packing circular finite-conjugate optics to create a spherical focal surface. Efficient packing of circles on a sphere is commonly referred to as the Tammes problem and various methods for packing optimization have been investigated, such as iterative point-repulsion simulations. The method for generating the circle distributions proposed here is based on a distorted icosahedral geodesic. This has the advantages of high degrees of symmetry, minimized variations in circle separations, and computationally inexpensive generation of configurations with N circles, where N is the number of vertices on the geodesic. These properties are especially beneficial for making a continuous focal surface and results show that circle packing densities near steady-state maximum values found with other methods can be achieved.

Is there a connection between face symmetry and face recognition

Abstract: Recent research in the area of automatic machine recognition of human faces has shown that there may be an advantage in utilizing face symmetry to improve recognition accuracy. While promising, this work has led to several open questions. What is a good feature description or score of the symmetry of the face? Is there a statistical significance between face symmetry and face recognition? We present new symmetry scores of the face and use the scores to compare the symmetry in several subgroups of a face database. A 3D face database is used to remove the effects of illumination which should improve the reliability of the symmetry score. We find a significant difference in face symmetry between the men and women subjects in the database. The database is then partitioned into most symmetric and least symmetric subjects based on the symmetry scores. The average-half-face is utilized in our face recognition experiments to take into account the symmetry of the face. Face recognition with eigenfaces using the average-half-face is significantly higher than using the full face in all subgroups regardless of symmetry score. However, face recognition using the full face does depend on the symmetry score and generally favors the least symmetric subjects.

Quantum \mathcal{W} -symmetry in AdS3

Abstract: It has recently been argued that, classically, massless higher spin theories in AdS3 have an enlarged $ {\mathcal{W}_N} $ -symmetry as the algebra of asymptotic isometries. In this note we provide evidence that this symmetry is realised (perturbatively) in the quantum theory. We perform a one loop computation of the fluctuations for a massless spin s field around a thermal AdS3 background. The resulting determinants are evaluated using the heat kernel techniques of arXiv:0911.5085. The answer factorises holomorphically, and the contributions from the various spin s fields organise themselves into vacuum characters of the $ {\mathcal{W}_N} $ symmetry. For the case of the hs(1, 1) theory consisting of an infinite tower of massless higher spin particles, the resulting answer can be simply expressed in terms of (two copies of) the MacMahon function.

Geometry of the generalized Bloch sphere for qutrits

Abstract: The geometry of the generalized Bloch sphere $\Omega_3$, the state space of a qutrit, is studied Closed form expressions for $\Omega_3$, its boundary $\partial \Omega_3$, and the set of extremals $\Omega_3^{\rm ext}$ are obtained by use of an elementary observation These expressions and analytic methods are used to classify the 28 two-sections and the 56 three-sections of $\Omega_3$ into unitary equivalence classes, completing the works of earlier authors It is shown, in particular, that there are families of two-sections and of three-sections which are equivalent geometrically but not unitarily, a feature that does not appear to have been appreciated earlier A family of three-sections of obese-tetrahedral shape whose symmetry corresponds to the 24-element tetrahedral point group $T_d$ is examined in detail This symmetry is traced to the natural reduction of the adjoint representation of $SU(3)$, the symmetry underlying $\Omega_3$, into direct sum of the two-dimensional and the two (inequivalent) three-dimensional irreducible representations of $T_d$

Teaching Point-Group Symmetry with Three-Dimensional Models.

Abstract: Three tools for teaching symmetry in the context of an upper-level undergraduate or introductory graduate course on the chemical applications of group theory are presented. The first is a collection of objects that have the symmetries of all the low-symmetry and high-symmetry point groups and the point groups with rotational symmetries from 2-fold to 6-fold. Students can sort the objects by point group and can also see similarities and differences between point groups. The second is a magnet-backed mirror onto which are placed molecular models that have been modified to rest on the mirror and represent atoms and bonds that are on the mirror plane. The third is a frame to show the perpendicular C2 axes that are a feature of all the Dnh, Dnd, and Dn point groups.

Small Orbits

Abstract: We study both the "large" and "small" U-duality charge orbits of extremal black holes appearing in D = 5 and D = 4 Maxwell-Einstein supergravity theories with symmetric scalar manifolds We exploit a formalism based on cubic Jordan algebras and their associated Freudenthal triple systems, in order to derive the minimal charge representatives, their stabilizers and the associated "moduli spaces" After recalling N = 8 maximal supergravity, we consider N = 2 and N = 4 theories coupled to an arbitrary number of vector multiplets, as well as N = 2 magic, STU, ST^2 and T^3 models While the STU model may be considered as part of the general N = 2 sequence, albeit with an additional triality symmetry, the ST^2 and T^3 models demand a separate treatment, since their representative Jordan algebras are Euclidean or only admit non-zero elements of rank 3, respectively Finally, we also consider minimally coupled N = 2, matter coupled N = 3, and "pure" N = 5 theories