Showing papers on "Symmetry (geometry) published in 2017"

Reflection-Symmetric Second-Order Topological Insulators and Superconductors

Abstract: Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but with topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five of the ten Altland-Zirnbauer symmetry classes allow for the existence of such second-order topological insulators in two and three dimensions. We show that reflection symmetry can be employed to systematically generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken. A three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of e^{2}/h.

On finite symmetries and their gauging in two dimensions

Abstract: It is well-known that if we gauge a $\mathbb{Z}_n$ symmetry in two dimensions, a dual $\mathbb{Z}_n$ symmetry appears, such that re-gauging this dual $\mathbb{Z}_n$ symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.

Carroll symmetry of plane gravitational waves

Abstract: The well-known 5-parameter isometry group of plane gravitational waves in $4$ dimensions is identified as Levy-Leblond's Carroll group in $2+1$ dimensions with no rotations. Our clue is that plane waves are Bargmann spaces into which Carroll manifolds can be embedded. We also comment on the scattering of light by a gravitational wave and calculate its electric permittivity considered as an impedance-matched metamaterial.

Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory

Abstract: Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this work, we revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. We construct very general fixed point SPT wavefunctions for interacting fermion systems. We naturally reproduce the partial classifications given by special group super-cohomology theory, and we show that with an additional $\tilde{B}H^2(G_b, \mathbb Z_2)$ (the so-called obstruction free subgroup of $H^2(G_b, \mathbb Z_2)$) structure, a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group $G_f=G_b\times \mathbb Z_2^f$ can be obtained for unitary symmetry group $G_b$. We also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions.

Composite Weyl nodes stabilized by screw symmetry with and without time-reversal invariance

Abstract: The authors demonstrate that in nonmagnetic crystals with threefold or sixfold symmetry, such as trigonal Te or hexagonal NbSi${}_{2}$, three linear band crossings (Weyl nodes) of the same chirality can merge at a high-symmetry point on the symmetry axis, forming a triple Weyl node with a cubic off-axis dispersion. If time reversal invariance is broken, the triple node in NbSi${}_{2}$ is displaced from the symmetry point along the axis, while its off-axis dispersion becomes quadratic. In Te, however, it splits into three linear Weyl nodes, since threefold symmetry is unable to stabilize triple Weyl nodes in the absence of time reversal symmetry.

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases.

Abstract: We define and compute many-body topological invariants of interacting fermionic symmetry-protected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a nonlocal operation, "partial" reflection or transpose As an application of our scheme, we study the Z_{8} and Z_{16} classification of topological superconductors in one and three dimensions

On gauging finite subgroups

Abstract: We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $\Gamma$-symmetric theory. Depending on how anomalous $\Gamma$ is, we find that the symmetry of the gauged theory can be i) a direct product of $G=\Gamma/A$ and a higher-form symmetry $\hat A$ with a mixed anomaly, where $\hat A$ is the Pontryagin dual of $A$; ii) an extension of the ordinary symmetry group $G$ by the higher-form symmetry $\hat A$; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the $H^3(G,\hat A)$ symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

Brane brick models in the mirror

Abstract: Brane brick models are Type IIA brane configurations that encode the 2d $$ \mathcal{N}=\left(0,2\right) $$ gauge theories on the worldvolume of D1-branes probing toric Calabi-Yau 4-folds. We use mirror symmetry to improve our understanding of this correspondence and to provide a systematic approach for constructing brane brick models starting from geometry. The mirror configuration consists of D5-branes wrapping 4-spheres and the gauge theory is determined by how they intersect. We also explain how 2d (0, 2) triality is realized in terms of geometric transitions in the mirror geometry. Mirror symmetry leads to a geometric unification of dualities in different dimensions, where the order of duality is n − 1 for a Calabi-Yau n-fold. This makes us conjecture the existence of a quadrality symmetry in 0d. Finally, we comment on how the M-theory lift of brane brick models connects to the classification of 2d (0, 2) theories in terms of 4-manifolds.

Freeness versus maximal global Tjurina number for plane curves

Abstract: We give a characterization of nearly free plane curves in terms of their global Tjurina numbers, similar to the characterization of free curves as curves with a maximal Tjurina number, due to A. A. du Plessis and C.T.C. Wall. It is also shown that an irreducible plane curve having a 1-dimensional symmetry is nearly free. A new numerical characterization of free curves and a simple characterization of nearly free curves in terms of their syzygies conclude this note.

Orientation and Symmetries of Alexandrov Spaces with Applications in Positive Curvature

Abstract: We develop two new tools for use in Alexandrov geometry: a theory of ramified orientable double covers and a particularly useful version of the Slice Theorem for actions of compact Lie groups. These tools are applied to the classification of compact, positively curved Alexandrov spaces with maximal symmetry rank.

Exactly marginal deformations from exceptional generalised geometry

Abstract: We apply exceptional generalised geometry to the study of exactly marginal deformations of $$ \mathcal{N} $$ = 1 SCFTs that are dual to generic AdS5 flux backgrounds in type IIB or eleven-dimensional supergravity. In the gauge theory, marginal deformations are parametrised by the space of chiral primary operators of conformal dimension three, while exactly marginal deformations correspond to quotienting this space by the complexified global symmetry group. We show how the supergravity analysis gives a geometric interpretation of the gauge theory results. The marginal deformations arise from deformations of generalised structures that solve moment maps for the generalised diffeomorphism group and have the correct charge under the generalised Reeb vector, generating the R-symmetry. If this is the only symmetry of the background, all marginal deformations are exactly marginal. If the background possesses extra isometries, there are obstructions that come from fixed points of the moment maps. The exactly marginal deformations are then given by a further quotient by these extra isometries. Our analysis holds for any $$ \mathcal{N} $$ = 2 AdS5 flux background. Focussing on the particular case of type IIB Sasaki-Einstein backgrounds we recover the result that marginal deformations correspond to perturbing the solution by three-form flux at first order. In various explicit examples, we show that our expression for the three-form flux matches those in the literature and the obstruction conditions match the one-loop beta functions of the dual SCFT.

Geometry of special Galileons

Abstract: A theory known as special Galileon has recently attracted considerable interest due to its peculiar properties. It has been shown that it represents an extremal member of the set of effective field theories with an enhanced soft limit. This property makes its tree-level $S$-matrix fully on-shell reconstructible and representable by means of the Cachazo-He-Yuan representation. The enhanced soft limit is a consequence of new hidden symmetry of the special Galileon action; however, until now, the origin of this peculiar symmetry has remained unclear. In this paper we interpret this symmetry as a special transformation of the coset space $\mathrm{GAL}(D,1)/SO(1,D\ensuremath{-}1)$ and show that there exists a three-parametric family of invariant Galileon actions. The latter family is closed under the duality which appears as a natural generalization of the above mentioned symmetry. We also present a geometric construction of the special Galileon action using a $D$-dimensional brane propagating in $2D$-dimensional flat pseudo-Riemannian space. Within such a framework, the special Galileon symmetry emerges as a $U(1,D\ensuremath{-}1)$ symmetry of the target space, which can be treated as a $D$-dimensional K\"ahler manifold. Such a treatment allows for classification of the higher order invariant Lagrangians needed as counterterms on the quantum level. We also briefly comment on the relation between such higher order Lagrangians and the Lagrangians that are invariant with respect to the polynomial shift symmetry.

Single Nodal Loop of Accidental Degeneracies in Minimal Symmetry: Triclinic CaAs 3

Abstract: The existence of closed loops of degeneracies in crystals has been intimately connected with associated crystal symmetries, raising the following question: What is the minimum symmetry required for topological character, and can one find an example? Triclinic ${\mathrm{CaAs}}_{3}$, in the space group $P\overline{1}$ with only a center of inversion, has been found to display, without need for tuning, a nodal loop of accidental degeneracies with topological character, centered on one face of the Brillouin zone that is otherwise fully gapped. The small loop is very flat in energy, yet is cut four times by the Fermi energy, a condition that results in an intricate repeated touching of inversion related pairs of Fermi surfaces at Weyl points. Spin-orbit coupling lifts the fourfold degeneracy along the loop, leaving trivial Kramers pairs. With its single nodal loop that emerges without protection from any point group symmetry, ${\mathrm{CaAs}}_{3}$ represents the primal ``hydrogen atom'' of nodal loop systems.

2017 ICCV Challenge: Detecting Symmetry in the Wild

Abstract: Motivated by various new applications of computational symmetry in computer vision and in an effort to advance machine perception of symmetry in the wild, we organize the third international symmetry detection challenge at ICCV 2017, after the CVPR 2011/2013 symmetry detection competitions. Our goal is to gauge the progress in computational symmetry with continuous benchmarking of both new algorithms and datasets, as well as more polished validation methodology. Different from previous years, this time we expand our training/testing data sets to include 3D data, and establish the most comprehensive and largest annotated datasets for symmetry detection to date; we also expand the types of symmetries to include densely-distributed and medial-axis-like symmetries; furthermore, we establish a challenge-and-paper dual track mechanism where both algorithms and articles on symmetry-related research are solicited. In this report, we provide a detailed summary of our evaluation methodology for each type of symmetry detection algorithm validated. We demonstrate and analyze quantified detection results in terms of precision-recall curves and F-measures for all algorithms evaluated. We also offer a short survey of the paper-track submissions accepted for our 2017 symmetry challenge.

Measuring space-group symmetry fractionalization in Z 2 spin liquids

Abstract: The interplay of symmetry and topological order leads to a variety of distinct phases of matter, the symmetry enriched topological (SET) phases. Here we discuss physical observables that distinguish different SETs in the context of ${\mathbb{Z}}_{2}$ quantum spin liquids with SU(2) spin rotation invariance. We focus on the cylinder geometry, and show that ground-state quantum numbers for different topological sectors are robust invariants which can be used to identify the SET phase. More generally, these invariants are related to 1D symmetry protected topological phases when viewing the cylinder geometry as a 1D spin chain. In particular, we show that the kagome spin liquid SET can be determined by measurements on one ground state, by wrapping the kagome in a few different ways on the cylinder. In addition to guiding numerical studies, this approach provides a transparent way to connect bosonic and fermionic mean-field theories of spin liquids. When fusing quasiparticles, it correctly predicts nontrivial phase factors for combining their space group quantum numbers.

Exploiting Symmetry and/or Manhattan Properties for 3D Object Structure Estimation from Single and Multiple Images

Abstract: Many man-made objects have intrinsic symmetries and Manhattan structure. By assuming an orthographic projection model, this paper addresses the estimation of 3D structures and camera projection using symmetry and/or Manhattan structure cues, which occur when the input is single-or multiple-image from the same category, e.g., multiple different cars. Specifically, analysis on the single image case implies that Manhattan alone is sufficient to recover the camera projection, and then the 3D structure can be reconstructed uniquely exploiting symmetry. However, Manhattan structure can be difficult to observe from a single image due to occlusion. To this end, we extend to the multiple-image case which can also exploit symmetry but does not require Manhattan axes. We propose a novel rigid structure from motion method, exploiting symmetry and using multiple images from the same category as input. Experimental results on the Pascal3D+ dataset show that our method significantly outperforms baseline methods.

Dimensional reduction and topological invariants of symmetry-protected topological phases

Abstract: We review the dimensional reduction procedure in the group cohomology classification of bosonic SPT phases with finite Abelian unitary symmetry group. We then extend this to include general reductions of arbitrary dimensions and also extend the procedure to fermionic SPT phases described by the Gu-Wen supercohomology model. We then show that we can define topological invariants as partition functions on certain closed orientable/spin manifolds equipped with a flat connection. The invariants are able to distinguish all phases described within the respective models. Finally, we establish a connection to invariants obtained from braiding statistics of the corresponding gauged theories.

The neural basis of mirror symmetry detection: a review

Abstract: The human visual system is extremely sensitive to the presence of bilateral (mirror) symmetry. In this review, I summarise the results of recent work investigating the neural basis of mirror symmetry detection, focusing in particular on brain stimulation evidence. Overall, available findings converge in pointing to the lateral occipital (LO) complex, especially in the right hemisphere, as a key region causally involved in symmetry detection. Interestingly, they also suggest that another region in the right extrastriate visual cortex, the occipital face area (OFA), is causally implied in symmetry detection, posing an interesting connection at the neural level between visual cortex responses to faces and to symmetry. Finally, this review also considers evidence on haptic symmetry detection in sighted and early blind individuals that points to LO as a multi-modal symmetry-sensitive region, and suggests that symmetry is a salient perceptual feature mediated by LO even when any visual experience is mis...

The dynamical Manin-Mumford problem for plane polynomial automorphisms

Abstract: Let f be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve C. We conjecture that this hap- pens if and only if f admits a time-reversal symmetry; in particular the Jacobian Jac(f) must be a root of unity. As a step towards this conjecture, we prove that its Jacobian, together with all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed Jac(f) is a root of unity. We use these results to show in various cases that any two automor- phisms sharing an infinite set of periodic points must have a common it- erate, in the spirit of recent results by Baker-DeMarco and Yuan-Zhang.

Equivariant Topological Quantum Field Theory and Symmetry Protected Topological Phases

Abstract: Short-Range Entangled topological phases of matter are closely related to Topological Quantum Field Theory. We use this connection to classify Symmetry Protected Topological phases in low dimensions, including the case when the symmetry involves time-reversal. To accomplish this, we generalize Turaev’s description of equivariant TQFT to the unoriented case. We show that invertible unoriented equivariant TQFTs in one or fewer spatial dimensions are classified by twisted group cohomology, in agreement with the proposal of Chen, Gu, Liu and Wen. We also show that invertible oriented equivariant TQFTs in spatial dimension two or fewer are classified by ordinary group cohomology.

Qudit quantum computation on matrix product states with global symmetry

Abstract: Resource states that contain nontrivial symmetry-protected topological order are identified for universal single-qudit measurement-based quantum computation. Our resource states fall into two classes: one as the qudit generalizations of the one-dimensional qubit cluster state, and the other as the higher-symmetry generalizations of the spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) state, namely, with unitary, orthogonal, or symplectic symmetry. The symmetry in cluster states protects information propagation (identity gate), while the higher symmetry in AKLT-type states enables nontrivial gate computation. This work demonstrates a close connection between measurement-based quantum computation and symmetry-protected topological order.

Symmetry in abstract elementary classes with amalgamation

Abstract: This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-elementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolated by the first author) in AECs with amalgamation that satisfy a local definition of superstability. The key results are a downward transfer of symmetry and a deduction of symmetry from failure of the order property. These results are then used to prove several structural properties in categorical AECs, improving classical results of Shelah who focused on the special case of categoricity in a successor cardinal. We also study the interaction of symmetry with tameness, a locality property for Galois (orbital) types. We show that superstability and tameness together imply symmetry. This sharpens previous work of Boney and the second author.

Exactly Solvable Models for Symmetry-Enriched Topological Phases

Abstract: We construct fixed-point wave functions and exactly solvable commuting-projector Hamiltonians for a large class of bosonic symmetry-enriched topological (SET) phases, based on the concept of equivalent classes of symmetric local unitary transformations. We argue that for onsite unitary symmetries, our construction realizes all SETs free of anomaly, as long as the underlying topological order itself can be realized with a commuting-projector Hamiltonian. We further extend the construction to anti-unitary symmetries (e.g. time-reversal symmetry), mirror-reflection symmetries, and to anomalous SETs on the surface of three-dimensional symmetry-protected topological phases. Mathematically, our construction naturally leads to a generalization of group extensions of unitary fusion categories to anti-unitary symmetries.

Symmetry-Enforced Line Nodes in Unconventional Superconductors

Abstract: We classify line nodes in superconductors with strong spin-orbit interactions and time-reversal symmetry, where the latter may include nonprimitive translations in the magnetic Brillouin zone to account for coexistence with antiferromagnetic order. We find four possible combinations of irreducible representations of the order parameter on high-symmetry planes, two of which allow for line nodes in pseudospin-triplet pairs and two that exclude conventional fully gapped pseudospin-singlet pairs. We show that the former can only be realized in the presence of band-sticking degeneracies, and we verify their topological stability using arguments based on Clifford algebra extensions. Our classification exhausts all possible symmetry protected line nodes in the presence of spin-orbit coupling and a (generalized) time-reversal symmetry. Implications for existing nonsymmorphic and antiferromagnetic superconductors are discussed.