Trivial Defect Research Articles

State sum models with defects based on spherical fusion categories

We define a Turaev-Viro-Barrett-Westbury state sum model of triangulated 3-manifolds with surface, line and point defects. Surface defects are oriented embedded 2d PL submanifolds and are labelled with bimodule categories over spherical fusion categories with bimodule traces. Line and point defects form directed graphs on these surfaces and labelled with bimodule functors and bimodule natural transformations. The state sum is based on generalised 6j symbols that encode the coherence isomorphisms of the defect data. We prove the triangulation independence of the state sum and show that it can be computed in terms of polygon diagrams that satisfy the cutting and gluing identities for polygon presentations of oriented surfaces. By computing state sums with defect surfaces, we show that they detect the genus of a defect surface and are sensitive to its embedding. We show that defect lines on defect surfaces with trivial defect data define ribbon invariants for the centre of the underlying spherical fusion category.

Nonadiabatic Shifting of a Topological Interface in an Electroacoustic Su-Schrieffer-Heeger Lattice

We investigate numerical and experimental nonadiabatic relocation of a topological interface. We construct a reconfigurable electroacoustic Su-Schreiffer-Heeger lattice using negative-capacitance (NC) shunted piezoelectric patches attached to a host beam. Tuning the NC circuits allows us to create a topological-interface state by varying the effective elastic modulus along the beam. We then use a microcontroller to switch relays placed in the NC shunt circuits to move the interface by one unit cell. We present the results for continuous and pulse inputs and show the shift of energy localization in the experiment. Further, to distinguish topological protection, we simulate the nonadiabatic shifting of a trivial defect in a discrete lattice. Numerical simulations indicate higher mean energy at the interface after shifting a topological state as compared to a defect state. Topological interfaces as shown in our experiment may have interesting applications in acoustic communication, such as source switching and signal encoding.

Interplay of Leakage Radiation and Protection in Topological Photonic Crystal Cavities

AbstractThe introduction of topological concepts to the design of photonic crystal cavities holds great promise for applications in integrated photonics due to the prospect of topological protection. This study examines the signatures of topological light confinement in the leakage radiation of 2D topological photonic crystal cavities. The cavities are implemented in an all‐dielectric platform that features the photonic quantum spin Hall effect at telecom wavelengths and supports helical edge states that are weakly coupled to the radiation continuum. The modes of resonators scaling down to single point defects in the surrounding bulk lattice are characterized via spectral position and multipolar nature of the eigenstates. The mode profiles in real and momentum space are mapped using far‐field imaging and Fourier‐spectropolarimetry, revealing how certain properties of the cavity modes reflect on their origin in the topological bandstructure. This includes band‐inversion‐induced confinement and inverted scaling of mode spectra for trivial and topological defect cavities. Furthermore, hallmarks of topological protection in the loss rates are demonstrated, which are largely unaffected by cavity shape and size. The results constitute an important step toward the use of radiative topological cavities for on‐chip confinement of light, control of emitted wave fronts, and enhancement of light–matter interactions.

Localized magnetic field in the O(N) model

We consider the critical O(N) model in the presence of an external magnetic field localized in space. This setup can potentially be realized in quantum simulators and in some liquid mixtures. The external field can be understood as a relevant perturbation of the trivial line defect, and thus triggers a defect Renormalization Group (RG) flow. In agreement with the g-theorem, the external localized field leads at long distances to a stable nontrivial defect CFT (DCFT) with g < 1. We obtain several predictions for the corresponding DCFT data in the epsilon expansion and in the large N limit. The analysis of the large N limit involves a new saddle point and, remarkably, the study of fluctuations around it is enabled by recent progress in AdS loop diagrams. Our results are compatible with results from Monte Carlo simulations and we make several predictions that can be tested in the future.

Topologically valley-polarized edge states in elastic phononic plates yielded by lattice defects

Lattice defects (including disclinations and dislocations) which commonly exist in various materials, have shown a fantastic ability to produce excellent mechanical and physical properties of matters. In this paper, lattice defects are introduced into valley-polarized elastic phononic plates. Deformations of lattices yield interfaces expressing as topologically protected waveguides, due to the valley-polarized phase transition of phononic crystals (PnCs) across interfaces. The topological interface states immunize against the finite sizes and the moderate disturbances of plates, essentially differing from the trivial lattice defect states. The discovery of topologically valley-polarized edge states yielded by lattice defects unveils a new horizon in topological mechanics and physics. It provides a novel platform to implement elastic devices with robust topological waveguides.

Comparative study of Hermitian and non-Hermitian topological dielectric photonic crystals

The effects of gain and loss on the band structures of a bulk topological dielectric photonic crystal (PC) with ${C}_{6v}$ symmetry and the PC-air-PC interface are studied based on first-principle calculation. To illustrate the importance of parity-time (PT) symmetry, three systems are considered, namely the PT-symmetric, PT-asymmetric, and lossy systems. We find that the system with gain and loss distributed in a PT symmetric manner exhibits a phase transition from a PT exact phase to a PT broken phase as the strength of the gain and loss increases, while for the PT-asymmetric and lossy systems, no such phase transition occurs. Furthermore, based on the Wilson loop calculation, the topology of the PT-symmetric system in the PT exact phase is demonstrated to keep unchanged as the Hermitian system. At last, different kinds of edge states in Hermitian systems under the influences of gain and loss are studied and we find that while the eigenfrequencies of nontrivial edge states become complex conjugate pairs, they keep real for the trivial defect states.

Topological bound modes in optical waveguide arrays with alternating positive and negative couplings

We investigate the topological bound modes in a binary waveguide array with alternating positive and negative couplings. The negative coupling is realized by incorporating additional waveguides into the bare array and suitably tuning the refractive index. We show the band structures close and Dirac point emerges at the Brillouin zone center when the absolute values of real and negative couplings are equal, indicating topological phase transition. The topological bound modes emerge at the edge of topological non-trivial arrays or at the interface of two arrays with different topologies. Moreover, we show the topological bound mode can be isolated from the trivial defect mode by tuning the coupling strength, which is reflected from wave propagation. The study provides a promising approach to realizing robust light transport.

On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras

In 1941, Brauer-Nesbitt established a characterization of a block with trivial defect group as a block $B$ with $k(B) = 1$ where $k(B)$ is the number of irreducible ordinary characters of $B$. In 1982, Brandt established a characterization of a block with defect group of order two as a block $B$ with $k(B) = 2$. These correspond to the cases when the block is Morita equivalent to the one-dimensional algebra and to the non-semisimple two-dimensional algebra, respectively. In this paper, we redefine $k(A)$ to be the codimension of the commutator subspace $K(A)$ of a finite-dimensional algebra $A$ and prove analogous statements for arbitrary (not necessarily symmetric) finite-dimensional algebras. This is achieved by extending the Okuyama refinement of the Brandt result to this setting. To this end, we study the codimension of the sum of the commutator subspace $K(A)$ and $n$th Jacobson radical $\operatorname{Rad}^n(A)$. We prove that this is Morita invariant and give an upper bound for the codimension as well.

Simultaneous topological Bragg and locally resonant edge modes of shear horizontal guided wave in one-dimensional structure

In this paper, we realize the topological edge modes of a shear horizontal (SH) guided wave in the one-dimensional (1D) composite structure that exist at the phononic band gaps opened at the center of the Brillouin zone (BZ), or at the zone boundary, or both. Furthermore, we investigate the topological edge modes in periodic acoustic systems result from both Bragg scattering and local resonances. In particular, we find and demonstrate that, in our system where topological and trivial defect states coexist, these results provide a new paradigm for manipulating the existence of the interface states in a set of prescribed band gaps that can have potential applications in the design of novel acoustic devices.

Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States.

One-dimensional models with topological band structures represent a simple and versatile platform to demonstrate novel topological concepts. Here we experimentally study topologically protected states in silicon at the interface between two dimer chains with different Zak phases. Furthermore, we propose and demonstrate that, in a system where topological and trivial defect modes coexist, we can probe them independently. Tuning the configuration of the interface, we observe the transition between a single topological defect and a compound trivial defect state. These results provide a new paradigm for topologically protected waveguiding in a complementary metal-oxide-semiconductor compatible platform and highlight the novel concept of isolating topological and trivial defect modes in the same system that can have important implications in topological physics.

A Geometric Approach to Stationary Defect Solutions in One Space Dimension

In this manuscript, we consider the impact of a small jump-type spatial heterogeneity on the existence of stationary localized patterns in a system of partial differential equations in one spatial dimension, i.e., defined on $\mathbb{R}$. This problem corresponds to analyzing a discontinuous and non-autonomous $n$-dimensional system, $\scriptsize\dot{u}=\left\{ \begin{array}{ll} f(u),& t\leq0,\\ f(u)+\varepsilon g(u),& t>0, \end{array}\right.$ under the assumption that the unperturbed system, i.e., the $\varepsilon \to 0$ limit system, possesses a heteroclinic orbit $\Gamma$ that connects two hyperbolic equilibrium points (plus several additional nondegeneracy conditions). The unperturbed orbit $\Gamma$ represents a localized structure in the PDE setting. We define the (pinned) defect solution $\Gamma_\varepsilon$ as a heteroclinic solution to the perturbed system such that $\lim_{\varepsilon \to 0} \Gamma_\varepsilon = \Gamma$ (as graphs). We distinguish between three types of defect solutions: trivial, local, and global defect solutions. The main goal of this manuscript is to develop a comprehensive and asymptotically explicit theory of the existence of local defect solutions. We find that both the dimension of the problem as well as the nature of the linearized system near the endpoints of the heteroclinic orbit $\Gamma$ have a remarkably rich impact on the existence of these local defect solutions. We first introduce the various concepts in the setting of planar systems $(n=2)$ and---for reasons of transparency of presentation---consider the three-dimensional problem in full detail. Then, we generalize our results to the $n$-dimensional problem, with special interest for the additional phenomena introduced by having $n \geq 4$. We complement the general approach by working out two explicit examples in full detail: (i) the existence of pinned local defect kink solutions in a heterogeneous Fisher--Kolmogorov equation ($n=4$) and (ii) the existence of pinned local defect front and pulse solutions in a heterogeneous generalized FitzHugh--Nagumo system $(n=6)$.

Should Plastic Surgeons Operate on Patients Diagnosed with Body Dysmorphic Disorders?

Sir: We read the Viewpoint entitled “Should Plastic Surgeons Operate on Patients Diagnosed with Body Dysmorphic Disorders?” (Plast Reconstr Surg. 2012;129:406e–407e). The authors pose four important questions. Body dysmorphic disorder, originally dysmorphophobia, is a somatoform disorder marked by a compulsive thought with an imagined or trivial defect in appearance that causes clinically significant distress or impairment in social life. We defined the more comprehensive term “dysmorphopathy” to refer to all conditions of psychological distress related to body image. A classification of dysmorphopathies was created for plastic surgery patients to help surgeons deliver proper treatment (Table 1).Table 1: Classification of the Psychosomatic DysmorphopathiesFirst, the authors ask whether we should operate on patients diagnosed with body dysmorphic disorder. The answer is yes, it is possible to operate on body dysmorphic disorder patients only after a correct assignment of the patient to a dysmorphopathy subtype. For example, in case of a severe subjective dysmorphopathy, the patient must not be operated on and will need only psychotherapy. Other patients need psychological support or psychotherapy before the operation. Second, can patients with mild body dysmorphic disorder benefit from plastic surgery? Plastic surgery may play the role of a dual treatment in selected cases in which treating the external morphologic appearance cures the internal well-being. We define this process eumorphic plastic surgery.1 Third, is there any way to predict the patient’s satisfaction with the results of plastic surgery? In our practice, we adopted methods to predict the patient’s expectations, such as the pgm A questionnaire on motivations and expectations.2 Our questionnaire is easy to complete for the patient and through a quick glance gives the surgeon an objective idea on the severity of the patient’s body image. The assessment of the expectations is the main moment for the inclusion or exclusion of the patient as a candidate for surgery. Fourth, the authors wonder whether it is ethical to operate on body dysmorphic disorder patients. It is not justified to operate on a patient without making a diagnosis of dysmorphopathy subtype. For example, it is not ethical to operate on a dysmorphophobic who corresponds to the severe subjective dysmorphopathy. On the contrary, if a full assessment of the dysmorphopathy subtype is obtained and surgery appears to be appropriate, not only is the operation ethical, but it is also likely to have a psychotherapeutic effect. We can achieve eumorphic plastic surgery3 when the deep needs of the patient are identified and the deformity is related to an internal psychological disturbance such as the Minotaur syndrome,4 a psychological distress resulting from the appearance of the patient’s face. The decision to operate or not is the plastic surgeon’s responsibility.5 They are supposed to identify the deep motivation or the psychological disorder1–3 and, as medical doctors, are not entitled to take refuge in an alleged role of exclusively surgical technicians performing procedures. DISCLOSURE The authors have no financial interest to declare in relation to the content of this communication. No outside funding was received. Paolo Giovanni Morselli, M.D., Ph.D. Filippo Boriani, M.D. Bologna University Bologna, Italy

Attractor flows from defect lines

Deforming a two-dimensional conformal field theory (CFT) on one side of a trivial defect line gives rise to a defect separating the original theory from its deformation. The Casimir force between these defects and other defect lines or boundaries is used to construct flows on bulk moduli spaces of CFTs. It turns out, that these flows are constant reparametrizations of gradient flows of the logarithm of the g-functions of the chosen defect or boundary condition. The special flows associated with supersymmetric boundary conditions in N = (2, 2) superconformal field theories agree with the attractor flows studied in the context of black holes in N = 2 supergravity.

Zone-Based Inspection and Defect Classification for LCD Manufacturing: Trivial Defect Free Procedure for TFT Glass Inspection

Inspection of thin film transistor (TFT)-patterned glass is one of the important processes for TFT-LCD panel manufacturing. Minute pattern defects and particles coexist on the glass and a few defects can cause serious quality problems. If there was a way to classify the defects by its potential fatality, it would be useful to control the product quality and loss of inspection time. This article introduces a rule-based approach of zone-based inspection and defect classification. For the zone-based processing, a defect classifier and its decision boundary were also proposed to differentiate non-fatal defects. An experimental result showed that the defect classifier was useful to select non-fatal defects and over 70% of the trivial defects could be selected with little risk of quality problems. This article also presents an application result to utilize the method in a plant. Since prejudged trivial defects have little influence on panel quality, they could be eliminated without interfering with the factor effecting overall final quality. This enables to save post-inspection time. The application results are sufficiently promising to save over 30% in inspection time and, furthermore, only 0.02% of fatal defects are not detectable. The proposed method also brings about an additional return of yield improvement because operator-error is found to be decreased as the defect candidates decrease because of the application of the proposed method.

Modular Representation Theory of Blocks with Trivial Intersection Defect Groups

We show that Uno's refinement of the projective conjecture of Dade holds every block whose defect groups intersect trivially modulo the maximal normal p-subgroup. This corresponds to the block having p-local rank one as defined by Jianbei An and Eaton. An immediate consequence is that Dade's projective conjecture, Alperin-McKay conjecture and Puig's nilpotent block conjecture hold for all trivial intersection blocks.

Right axillary incision: A cosmetically superior approach to repair a wide range of congenital cardiac defects

We sought to evaluate the safety of a right axillary incision, a cosmetically superior approach than anterolateral thoracotomy, to repair various congenital heart defects. All the patients who were approached with this incision between March 2001 and October 2004 were included in the study. There were 80 patients (median age, 4 years) with atrial septal defect closure (38 patients), repair of partial abnormal pulmonary venous return (14 patients), partial atrioventricular canal (16 patients), and perimembranous ventricular septal defect (12 patients). The surgical technique involved peripheral and central cannulation for institution of cardiopulmonary bypass. Electrically induced ventricular fibrillation was used for defects located in front of the atrioventricular valves, and cardioplegic arrest was used for those located at the level or behind these valves. The repair was possible without need for conversion to another approach. One patient sustained a transient neurologic deficit. The patients were all in excellent condition after a mean follow-up of 14 months. The cardiac defect was repaired with no residual defect in 75 patients and with trivial residual defect in 5 patients (3 with mitral valve regurgitation, 1 with atrial septal defect, and 1 with ventricular septal defect). The incision healed properly in all, and the thorax showed no deformity. The right axillary incision provides a quality of repair for various congenital defects similar to that obtained by using standard surgical approaches. Because it lies more laterally and is hidden by the resting arm, it provides superior cosmetic results compared with conventional incisions, including the anterolateral thoracotomy. Finally, the incision is unlikely to interfere with subsequent development of the breast.

Correction of the bilateral cleft using the sliding sulcus technique.

The morphology of the bilateral cleft lip can be understood in terms of process theory and field theory. The former holds that an initial event sets in motion processes of division, deficiency, displacement, and distortion; these act on the cleft site to convert a trivial defect into the final result seen at birth. Field theory describes embryonic units that are in disarray by virtue of the cleft. A subperiosteal technique is described that combines these concepts to achieve simultaneous closure of the lip and alveolus. Primary separation of the nasal tip from the philtral-columellar unit is carried out along embryonic fusion lines.

The green correspondents of simple group modules

We shall show that these correspondents are not arbitrary modules, but have a strong property with respect to specializations (see Corollary 2.8) which ensures that they belong to a family of modules that are theoretically classifiable using only local information about the group (see Corollary 3.12 and the remarks following it). This extends to arbitrary simple modular group modules the similar properties proven in [2] for such modules lying in blocks with trivial intersection defect groups. Indeed, we lean heavily on [2], and follow its notation as much as possible. As an addendum we indicate in Section 4 how our results can be extended from Green correspondents to sources of simple group modules.

TREATMENT OF DYSPHAGIA FROM HERNIA THROUGH ESOPHAGEAL HIATUS IN DIAPHRAGM

Herniation of an abdominal organ through the diaphragm into the thorax occurs frequently, and unless physicians are aware of the significance of symptoms produced by this abnormality, the nature of the disability may not be detected. Herniation through the diaphragm into the thorax may result from congenital maldevelopment or from rupture of a normal diaphragm by external trauma. In many instances a trivial congenital defect in the diaphragm that does not cause symptoms may be increased by the thoracic action of suction and positive intraabdominal pressure so that later in life significant discomfort is produced. The purpose of this paper is not to discuss all types of diaphragmatic hernia, but to direct attention to two types in which dysphagia is often the predominating symptom. In patients with hernia of one of these types, the esophageal opening in the diaphragm is larger than normal, and a portion of the stomach, usually