The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles

We study four-dimensional gauge theories with arbitrary simple gauge group with 1-form global center symmetry and 0-form parity or discrete chiral symmetry. We canonically quantize on 𝕋3, in a fixed background field gauging the 1-form symmetry. We show that the mixed 0-form/1-form ’t Hooft anomaly results in a central extension of the global-symmetry operator algebra. We determine this algebra in each case and show that the anomaly implies degeneracies in the spectrum of the Hamiltonian at any finite- size torus. We discuss the consistency of these constraints with both older and recent semiclassical calculations in SU(N) theories, with or without adjoint fermions, as well as with their conjectured infrared phases.

Quantum Phases of Time Order in Many-Body Ground States

Understanding phases of matter is of both fundamental and practical importance. Prior to the widespread appreciation and acceptance of topological order, the paradigm of spontaneous symmetry breaking, formulated along the Landau-Ginzburg-Wilson (LGW) dogma, is central to understanding phases associated with order parameters of distinct symmetries and transitions between phases. This work proposes to identify ground state phases of quantum many-body system in terms of time order, which is operationally defined by the appearance of nontrivial temporal structure in the two-time auto-correlation function of a symmetry operator (order parameter). As a special case, the (symmetry protected) time crystalline order phase detects continuous time crystal (CTC). Time order phase diagrams for spin-1 atomic Bose-Einstein condensate (BEC) and quantum Rabi model are fully worked out. Besides time crystalline order, the intriguing phase of time functional order is discussed in two non-Hermitian interacting spin models.

The first-order symmetry operator on gravitational perturbations in the 5D Myers–Perry spacetime with equal angular momenta

It has been revealed that the first-order symmetry operator for the linearized Einstein equation on a vacuum spacetime can be constructed from a Killing–Yano 3-form. This might be used to construct all or part of the solutions to the field equation. In this paper, we perform a mode decomposition of a metric perturbation on the Schwarzschild spacetime and the Myers–Perry spacetime with equal angular momenta in 5 dimensions, and investigate the action of the symmetry operator on specific modes concretely. We show that, on such spacetimes, there is no transition between the modes of a metric perturbation by the action of the symmetry operator, and it ends up being the linear combination of the infinitesimal transformations of isometry.

Reflection Symmetry Axes Localization Framework for Object Intelligent Perception based on Salient Symmetry Feature

This paper presents an optimized feature-centered reflection symmetry axis detection and localization framework for object perception. The proposed framework is formed to obtain an improved reflection symmetry axis based on the salient symmetry feature. It starts with a refined Multi-scale Saliency Symmetry Model (MSSM), which is realized by applying isotropic symmetry operator on salient points in scale-space rather than all pixels. In each scale, salient points are initially extracted as local extremal from an image, and they are further refined by a multi-scale implementation for generating salient symmetry feature maps. A Symmetric Transformation Matrix is then computed using the optimal feature matching pairs, which can be explicitly used as an abstract representation of the constraint regions of symmetry objects in an image to optimize the performance of the potential symmetry axis detection. The framework has been investigated experimentally both on the classical dataset from a symmetry detection challenge and the latest dataset. It has shown that the framework can get a better or comparative result and also can be further adapted into terminated human--computer equipment for reflection symmetry object perception and tracking.

Emergence of form in embryogenesis

The development of form in an embryo is the result of a series of topological and informational symmetry breakings. We introduce the vector–reaction–diffusion–drift (VRDD) system where the limit cycle of spatial dynamics is morphogen concentrations with Dirac delta-type distributions. This is fundamentally different from the Turing reaction–diffusion system, as VRDD generates system-wide broken symmetry. We developed ‘fundamental forms’ from spherical blastula with a single organizing axis (rotational symmetry), double axis (mirror symmetry) and triple axis (no symmetry operator in three dimensions). We then introduced dynamics for cell differentiation, where genetic regulatory states are modelled as a finite-state machine (FSM). The state switching of an FSM is based on local morphogen concentrations as epigenetic information that changes dynamically. We grow complicated forms hierarchically in spatial subdomains using the FSM model coupled with the VRDD system. Using our integrated simulation model with four layers (topological, physical, chemical and regulatory), we generated life-like forms such as hydra. Genotype–phenotype mapping was investigated with continuous and jump mutations. Our study can have applications in morphogenetic engineering, soft robotics and biomimetic design.

Crystal structures of 5,12-dimethyl-1,4,8,11-tetraazacyclotetradecane cobalt(III) mono-phenylacetylide and bis-phenylacetylide

Reported in this contribution are the synthesis and crystal structures of new mono- and bis-phenylacetylides based on CoIII(DMC) (DMC is 5,12-dimethyl-1,4,8,11-tetraazacyclotetradecane). Chlorido(5,12-dimethyl-1,4,8,11-tetraazacyclotetradecane)(phenylethynyl)cobalt(III) chloride–acetonitrile–methanol (1/1/1), [Co(C8H5)Cl(C12H28N4)]Cl·CH3CN·CH3OH, 1, and (5,12-dimethyl-1,4,8,11-tetraazacyclotetradecane)bis(phenylethynyl)cobalt(III) trifluoromethanesulfonate–dichloromethane (2/1), [Co(C8H5)2(C12H28N4)]2(CF3SO3)2·CH2Cl2, 2, were prepared under weak-base conditions in satisfactory yields. Single-crystal X-ray diffraction studies revealed that both 1 and 2 adopt a pseudo-octahedral symmetry in which the Cl—Co—C angles of 1 and C—Co—C of 2 range from 177.7 (2) to 178.0 (2)° and from 177.67 (9) to 179.67 (9)°, respectively. In both structures, the CoIII metal center is coordinated in the equatorial plane by four N atoms, in which the N—Co—N angles range from 85.6 (3) to 94.4 (3)°. The structure of 1 features two crystallographically independent molecules in its triclinic cell (Z′ = 2), which are related to each other by pseudo-monoclinic symmetry. The crystal investigated was twinned by a symmetry operator of the approximate double-volume C-centered cell (180° rotation around [201] of the actual triclinic cell), with a refined twin ratio of 0.798 (3) to 0.202 (3). Both methanol solvent molecules in 1 are disordered, the major occupancy rates refined to 0.643 (16) and 0.357 (16). Compound 2 also contains two molecules in the asymmetric unit, together with two trifluoromethanesulfonate anions [of which one is disordered; occupancy values of 0.503 (16) and 0.497 (16)] and a disordered dichloromethane [occupancy values of 0.545 (12) and 0.455 (12)].

A Note on the Symmetry Properties of Löwdin's Orthogonalization Schemes

We point out that the well-known symmetry properties of the symmetrically and canonically orthogonalized vectors hold only under certain conditions on the overlapping vectors. In particular, the matrix of the transformation induced by the symmetry operator must be unitary. This requirement is not fulfilled if Cartesian d or f functions are used in the basis set. If such functions are present, canonically orthogonalized orbitals do not transform according to representations of the molecular point group; nor do Löwdin orthogonalized vectors preserve symmetry relation of the original vectors.