Projective Symmetry Research Articles

A Precise Algorithm for Known Spherical Marker Position Retrieval Using an Ideal Pinhole Camera in a Diffuse Light Scene

The exact marker image was computed for various positions in the workspace and for each image was computed back the approximation of the marker position based on it. The position computation used the symmetry of the scene and the particular characteristics of the ellipse marker projection. Was established that the center of symmetry of the marker projection is different than the projection of the marker center itself and the the relation between two was found using the Dandeline Spheres. For a given marker radius the ratio between the area of the right cone base that contains the marker center projection and the area of the marker projection was expressed as a law based only on the angle of the line passing through the focal point and the symmetry center of the marker projection and the view axis. Based on all these intermediary results was obtained an algorithm with a precision under 0.04 mm in sagital plane, and under 0.1 mm on the depth view axis with a modest 1920x1680 camera resolution.

Photon spheres and sonic horizons in black holes from supergravity and other theories

We study closed photon orbits in spherically-symmetric static solutions of supergravity theories, a Horndeski theory, and a theory of quintessence. These orbits lie in what we shall call a photon sphere (anti-photon sphere) if the orbit is unstable (stable). We show that in all the asymptotically flat solutions we examine that admit a regular event horizon, and whose energy-momentum tensor satisfies the strong energy condition, there is one and only one photon sphere outside the event horizon. We give an example of a Horndeski theory black hole (whose energy-momentum tensor violates the strong energy condition) whose metric admits both a photon sphere and an anti-photon sphere. The uniqueness and non-existence also holds for asymptotically anti-de Sitter solutions in gauged supergravity. The latter also exhibit the projective symmetry that was first discovered for the Schwarzschild-de Sitter metrics: the unparameterised null geodesics are the same as when the cosmological or gauge coupling constant vanishes. We also study the closely related problem of accretion flows by perfect fluids in these metrics. For a radiation fluid, Bondi's sonic horizon coincides with the photon sphere. For a general polytropic equation of state this is not the case. Finally we exhibit counterexamples to a conjecture of Hod's.

On the Anosov character of the Pappus–Schwartz representations

In the paper Pappus's Theorem and The Modular Group (1993) [4], R.E. Schwartz observed that the classical Pappus theorem gives rise to an action of the modular group on the space of marked boxes. He inferred from this a 2-dimensional family of faithful representations of the modular group into the group of projective symmetries. These representations have a dynamical behavior very similar to the one of Anosov representations, even if they are never Anosov themselves. In this note, we announce the main result of V. Pardini Valério (2016) [3], which elucidates this Anosov character of the Schwartz representations by proving that their restrictions to the index-2 subgroup are limits of Anosov representations.

Spin structure factors of chiral quantum spin liquids on the kagome lattice

We calculate dynamical spin structure factors for gapped chiral spin liquid states in the spin-1/2 Heisenberg antiferromagnet on the kagome lattice using Schwinger-boson mean-field theory. In contrast to static (equal-time) structure factors, the dynamical structure factor shows clear signatures of time-reversal symmetry breaking for chiral spin liquid states. In particular, momentum inversion $\bm{k} \to -\bm{k}$ symmetry as well as the six-fold rotation symmetry around the $\Gamma$-point are lost. We highlight other interesting features, such as a relatively flat onset of the two-spinon continuum for the \emph{cuboc1} state. Our work is based on the projective symmetry group classification of time-reversal symmetry breaking Schwinger-boson mean-field states by Messio, Lhuillier, and Misguich.

Eisenhart lifts and symmetries of time-dependent systems

Certain dissipative systems, such as Caldirola and Kannai’s damped simple harmonic oscillator, may be modelled by time-dependent Lagrangian and hence time dependent Hamiltonian systems with n degrees of freedom. In this paper we treat these systems, their projective and conformal symmetries as well as their quantisation from the point of view of the Eisenhart lift to a Bargmann spacetime in n+2 dimensions, equipped with its covariantly constant null Killing vector field. Reparametrisation of the time variable corresponds to conformal rescalings of the Bargmann metric. We show how the Arnold map lifts to Bargmann spacetime. We contrast the greater generality of the Caldirola–Kannai approach with that of Arnold and Bateman. At the level of quantum mechanics, we are able to show how the relevant Schrödinger equation emerges naturally using the techniques of quantum field theory in curved spacetimes, since a covariantly constant null Killing vector field gives rise to well defined one particle Hilbert space.Time-dependent Lagrangians arise naturally also in cosmology and give rise to the phenomenon of Hubble friction. We provide an account of this for Friedmann–Lemaître and Bianchi cosmologies and how it fits in with our previous discussion in the non-relativistic limit.

Schwinger boson spin-liquid states on square lattice

We study possible spin liquids on square lattice that respect all lattice symmetries and time-reversal symmetry within the framework of Schwinger boson (mean-field) theory. Such spin liquids have spin gap and emergent Z_2 gauge field excitations. We classify them by the projective symmetry group method, and find six spin liquid states that are potentially relevant to the J_1-J_2 Heisenberg model. The properties of these states are studied under mean-field approximation. Interestingly we find a spin liquid state that can go through continuous phase transitions to either the N\'eel magnetic order or magnetic orders of the wavevector at Brillouin zone edge center. We also discuss the connection between our results and the Abrikosov fermion spin liquids.

Low-Energy Spin Dynamics of the Honeycomb Spin Liquid Beyond the Kitaev Limit

We investigate the generic features of the low energy dynamical spin structure factor of the Kitaev honeycomb quantum spin liquid perturbed away from its exact soluble limit by generic symmetry-allowed exchange couplings. We find that the spin gap persists in the Kitaev-Heisenberg model, but generally vanishes provided more generic symmetry-allowed interactions exist. We formulate the generic expansion of the spin operator in terms of fractionalized Majorana fermion operators according to the symmetry enriched topological order of the Kitaev spin liquid, described by its projective symmetry group. The dynamical spin structure factor displays power-law scaling bounded by Dirac cones in the vicinity of the Γ, K, and K^{'} points of the Brillouin zone, rather than the spin gap found for the exactly soluble point.

Conformal and projective symmetries in Newtonian cosmology

Definitions of non-relativistic conformal transformations are considered both in the Newton-Cartan and in the Kaluza-Klein-type Eisenhart/Bargmann geometrical frameworks. The symmetry groups that come into play are exemplified by the cosmological, and also the Newton-Hooke solutions of Newton's gravitational field equations. It is shown, in particular, that the maximal symmetry group of the standard cosmological model is isomorphic to the 13-dimensional conformal-Newton-Cartan group whose conformal-Bargmann extension is explicitly worked out. Attention is drawn to the appearance of independent space and time dilations, in contrast with the Schr{\"o}dinger group or the Conformal Galilei Algebra.

Confinement transition to density wave order in metallic doped spin liquids

Insulating quantum spin liquids can undergo a confinement transition to a valence bond solid via the condensation of topological excitations of the associated gauge theory. We extend the theory of such transitions to fractionalized Fermi liquids (FL*): These are metallic doped spin liquids in which the Fermi surfaces only have gauge neutral quasiparticles. Using insights from a duality transform on a doped quantum dimer model for the U(1)-FL* state, we show that projective symmetry group of the theory of the topological excitations remains unmodified, but the Fermi surfaces can lead to additional frustrating interactions. We propose a theory for the confinement transition of ${\mathbb{Z}}_{2}$-FL* states via the condensation of visons. A variety of confining, incommensurate density wave states are possible, including some that are similar to the incommensurate $d$-form factor density wave order observed in several recent experiments on the cuprate superconductors.

Projective symmetry group classification of chiral spin liquids

We present a general review of the projective symmetry group classification of fermionic quantum spin liquids for lattice models of spin $S=1/2$. We then introduce a systematic generalization of the approach for symmetric $\mathbb{Z}_2$ quantum spin liquids to the one of chiral phases (i.e., singlet states that break time reversal and lattice reflection, but conserve their product). We apply this framework to classify and discuss possible chiral spin liquids on triangular and kagome lattices. We give a detailed prescription on how to construct quadratic spinon Hamiltonians and microscopic wave functions for each representation class on these lattices. Among the chiral $\mathbb{Z}_2$ states, we study the subset of U(1) phases variationally in the antiferromagnetic $J_1$-$J_2$-$J_d$ Heisenberg model on the kagome lattice. We discuss static spin structure factors and symmetry constraints on the bulk spectra of these phases.

Submaximally symmetric c-projective structures

[Formula: see text]-projective structures are analogues of projective structures in the almost complex setting. The maximal dimension of the Lie algebra of [Formula: see text]-projective symmetries of a complex connection on an almost complex manifold of [Formula: see text]-dimension [Formula: see text] is classically known to be [Formula: see text]. We prove that the submaximal dimension is equal to [Formula: see text]. If the complex connection is minimal (encoded as a normal parabolic geometry), the harmonic curvature of the [Formula: see text]-projective structure has three components and we specify the submaximal symmetry dimensions and the corresponding geometric models for each of these three pure curvature types. If the connection is non-minimal, we introduce a modified normalization condition on the parabolic geometry and use this to resolve the symmetry gap problem. We prove that the submaximal symmetry dimension in the class of Levi-Civita connections for pseudo-Kähler metrics is [Formula: see text], and specializing to the Kähler case, we obtain [Formula: see text]. This resolves the symmetry gap problem for metrizable [Formula: see text]-projective structures.

Proper projective symmetry in the most general non-static spherically symmetric four-dimensional Lorentzian manifolds

A study of proper projective symmetry in the most general form of non-static spherically symmetric space-time is given using direct integration and algebraic techniques. In this study, we show that when the above space-time admits proper projective symmetry it becomes a very special class of static spherically symmetric space-times.

Approximate singly excited states from a two-component Hartree-Fock reference.

For many molecules, relaxing the spin symmetry constraint on the wave function results in the lowest energy mean-field solution. The two-component Hartree-Fock (2cHF) method relaxes all spin symmetry constraints, and the wave function is no longer an eigenfunction of the total spin, spin projection, or time-reversal symmetry operators. For ground state energies, 2cHF is a superior mean-field method for describing spin-frustrated molecules. For excited states, the utility of 2cHF is uncertain. Here, we implement the 2cHF extensions of two single-reference excited state methods, the two-component configuration interaction singles and time-dependent Hartree-Fock. We compare the results to the analogous methods based off of the unrestricted Hartree-Fock approximation, as well as the full configuration interaction for three small molecules with distinct 2cHF solutions, and discuss the nature of the 2cHF excited state solutions.

Projective symmetry of partons in the Kitaev honeycomb model

Low-energy states of quantum spin liquids are thought to involve partons living in a gauge-field background. We study the spectrum of Majorana fermions of Kitaev's honeycomb model on spherical clusters. The gauge field endows the partons with half-integer orbital angular momenta. As a consequence, the multiplicities reflect not the point-group symmetries of the cluster, but rather its projective symmetries, operations combining physical and gauge transformations. The projective symmetry group of the ground state is the double cover of the point group.

Scanning of two-dimensional space groups.

Tables of the scanning of two-dimensional space groups are presented to determine the frieze-group symmetry of lines that transect two-dimensional crystals. It is shown how these tables can be used to predict the (001) projection symmetries of migration-related segments of coincidence site lattice tilt boundaries with [001] tilt axis.

C-Arm Assisted Zygoma Fracture Repair: A Critical Analysis of the First 20 Cases

Currently used open reduction and internal fixation techniques of zygoma fracture repair are not optimal. Surgical exposure of those sites needed to allow for accurate reduction and for rigid fixation has a high possibility of negative consequences. The objective of the present study was to present a single-incision, single-fixation site zygoma fracture repair technique using a single zygoma c-arm view to quantitatively determine its accuracy, complication rate, and practical aspects in a clinical series. In a prospective study, consecutive patients with isolated, unilateral, displaced zygoma fractures not requiring orbital floor exploration treated using a c-arm-assisted repair technique at the author's institution from 2009 to 2011 were included. Objective outcomes assessed included accuracy of zygoma realignment (on postoperative computed tomogram), ocular globe projection symmetry (using a Naugle exophthalmometer), complication rate, and operative duration. Statistical analysis was performed using the Student t test. Twenty patients were included. Differences in zygoma projection, width, and height between the uninjured and repaired sides of the face were clinically noteworthy (>3mm) in the first patient only. Average differences of these parameters for all 20 patients were clinically and statistically insignificant. Differences in ocular globe projection between the uninjured and repaired sides of the face for each patient were no greater than 2mm. The average difference in globe projection for all 20 patients was also clinically and statistically insignificant. No major complications occurred, and the average operative duration was 76minutes. The present study shows that the c-arm-assisted zygoma fracture repair technique is accurate, has a low complication rate, can be performed quickly, and has a relatively low level of difficulty.

Classification of spin liquids on the square lattice with strong spin-orbit coupling

Spin liquids represent exotic types of quantum matter that evade conventional symmetry-breaking order even at zero temperature. Exhaustive classifications of spin liquids have been carried out in several systems, particularly in the presence of full SU(2) spin-rotation symmetry. Real magnetic compounds, however, generically break SU(2) spin symmetry as a result of spin-orbit coupling - which in many materials provides an `order one' effect. We generalize previous works by using the projective symmetry group method to classify $Z_2$ spin liquids on the square lattice when SU(2) spin symmetry is maximally lifted. We find that, counterintuitively, the lifting of spin symmetry actually results in vastly more spin liquid phases compared to SU(2)-invariant systems. A generic feature of the SU(2)-broken case is that the spinons naturally undergo $p+ip$ pairing; consequently, many of these $Z_2$ spin liquids feature a topologically nontrivial spinon band structure supporting gapless Majorana edge states. We study in detail several spin-liquid phases with varying numbers of gapless edge states and discuss their topological protection. The edge states are often protected by a combination of time reversal and lattice symmetries and hence resemble recently proposed topological crystalline superconductors.

A note on proper projective symmetry in cylindrical symmetric non-static space-times

In this paper a study of the most general form of cylindrical symmetric non-static space-times is given using direct integration and algebraic techniques. In this study we deal with the cases in which all eigenvalues of the Riemann tensor are different. From this study we show that when the cylindrical symmetric non-static space-times admit proper projective symmetry, it turns out to be a special class of the space-like flat Friedmann-Robertson-Walker model.

Nipple-areola complex reconstruction.

Nipple areolar reconstruction (NAC) was introduced since 1940s and evolved as parallel with breast reconstruction since era of breast cancer treatment. It consists of nipple and areolar reconstruction. Ideal reconstruction of the NAC requires symmetry in position, size, shape, texture, and pigmentation and permanent projection. There are many innovative ways to create a nipple and each method has its unique characteristics that apply to certain breast types. NAC reconstruction techniques comprises of composite nipple grafts, local flap, flaps with autologous graft augmentation, flaps with alloplastic augmentation and flaps with allograft augmentation. Areolar reconstruction by using skin grafting and tattooing are the easiest and most common techniques. With the evolution of techniques and technology, perhaps the newer methods of NAC reconstruction can produce promising long-lasting aesthetically acceptable result with minimal morbidity.

Quantum spin liquids in the absence of spin-rotation symmetry: Application to herbertsmithite

It has been suggested that the nearest-neighbour (NN) antiferromagnetic Heisenberg (HAF) model on the Kagome lattice may be a good starting point to understand the quantum spin-liquid (QSL) behaviour discovered in Herbertsmithite. We investigate possible QSL phases in the presence of experimentally relevant spin-rotation symmetry-breaking perturbations such as Dzyaloshinskii-Moriya and Ising interactions, as well as second-neighbour antiferromagnetic Heisenberg interactions. We use the projective symmetry group analysis within the slave-fermion framework of QSL phases and systematically classify possible QSLs in the presence of these perturbations. The dynamical spin-structure factor for relevant QSLs is computed and their effects are studied. Our calculations reveal dispersive features in the spin structure factor embedded in a generally diffuse background due to the existence of fractionalized S=1/2 excitations called spinons. For two of the previously proposed $Z_2$ states, the dispersive features are almost absent, and diffuse scattering dominates over a large energy window throughout the Brillouin zone. This resembles the structure factor observed in recent inelastic neutron scattering experiments on singlet crystals of Herbertsmithite. Furthermore, one of the $Z_2$ states with the spin structure factor with mostly diffuse scattering is gapped, and it may be adiabatically connected to the gapped QSL state observed in recent DMRG calculations for the NN HAF. The above perturbations are found to enhance the diffuse nature of the spin structure factor and reduce the momentum dependencies of the spin gap. We also calculate the electron spin resonance absorption spectra that further characterize the role of spin-rotation symmetry breaking perturbations, and can shed more light into the nature of the ground state in Herbertsmithite.