We describe a microscopic model, an extended Hubbard model on the diamond lattice that, within a mean field treatment, leads to this phase. The order parameter supports a number of topological defects. In particular, a vortex like line defect occurs, but with a Z2 charge. This line defect in the STI is found to be associated with a pair of gapless fermionic excitations that travel along its length. These modes are topologically stable against moderate perturbations such as impurities and interactions as long as time reversal symmetry is intact. This is the main result of the chapter – an analytical derivation is provided which relies on the properties of the Dirac equation on a two-dimensional curved surface. We now contrast our results with other recent work. In Chapter 2 we have shown similar exotic behavior also occurs in TIs, along crystal defects such as dislocations. Gapless fermionic excitations emerge there when a Z2 parameter formed by the product of the dislocation Burgers vector and three WTI indices is nonzero – which in principle can occur inboth the weak and strong TI. In contrast, in the present chapter, the fermionic modes along the line defect are solely determined by the more elusive strong index. They are absent in the case of the WTI. Thus far, the characterization of the TI phase has relied on the surface behavior. This result provides a route to identifying the strong TMI via a bulk property. Similar modes have been identified propagating along a solenoid of π flux, inserted into a STI. Here, the 2π rotation of the electron spin around the line defect leads to a Berry’s phase, plastic potting pots providing a physical realization of the π flux. Analogous phenomena occur in the context of line defects in superfluid He3-B.
In most solids where electron-electron interactions are important tend to have some degree of SOIs – which will confine the defects. Hence, we propose realizations of this physics in optical lattices of ultracold atoms, utilizing molecules with multipole moments to obtain the proposed extended Hubbard models. Realizing the three-dimensional case is more challenging, however molecules with electric quadrupole moments confined in optical lattice can realize some of the key ingredients required. This chapter is organized as the following: In Section 3.2, we will present the order parameter manifold and the line modes’ Z2 dependence on the winding number; in Section 3.3, we will justify our claim with numerical and analytical results; another texture Shankar monopole will be discussed in Section 3.4; in Section 3.5, we will establish our model Hamiltonian on a diamond lattice and show the mean field stability of TMI phases; we give two possible experimental realizations in cold atom systems in Section 3.6; we conclude the main result of this chapter in the Section 3.7. Hereafter we use σ and τ for the spin and sublattice degree of freedom, respectively. This chapter incorporates materials previously published in Ref. .An experimental realization of the TMI phase must contend with two challenges. First, the system should have weak intrinsic spin-orbit coupling, but strong interactions. Next, the further neighbour repulsion should be substantial compared to the nearest neighbor interactions. We believe these difficulties can be overcome in cold atom system, where intrinsic spin orbit couplings are irrelevant, if particles with electric multipole moments are confined to optical lattice sites. We first discuss a two-dimensional example involving electric dipoles, for which a fairly definite experimental setup can be constructed.
Although the phase realized here is two-dimensional and does not break SRS completely spin rotation remains unbroken, it illustrates how the necessary ingredients can be assembled.Subsequently we discuss ideas for realizing the three-dimensional TMI, the main subject of this chapter, using electric quadrupole moments. Two-dimensional Case: Electric Dipoles on a diamond lattice layer Dipole-dipole interactions between hetero-nuclear polar molecules, such as Rb87 and K40 have already been shown to be stron. Consider a fermionic spin 1/2 molecule, with an electric dipole moment confined to the sites of an optical lattice. We note here that the diamond lattice has a special property that if the dipole-moment is along the directions, then the nearest neighbor interaction V1 vanishes. Thus, the second nearest neighbor interaction V2 becomes dominant. However, the difficulty is that within the twelve second nearest neighbors, only interactions between neighbors within a plane perpendicular to the dipole moment are repulsive. This problem can be solved if we restrict the molecules within a two-dimensional layer of the diamond lattice , as the sites circled in Figure 3.4. Then if the dipole moment is perpendicular to the plane all possible nearest neighbor interactions are repulsive. We solve for the mean field phase diagram of this model, as was done previously for the three-dimensional case. Note, since the lattice is essentially the honeycomb lattice, this is essentially the model studied in Ref. . There exists a two-dimensional TMI phase at the center of the U − V2 phase diagram . Note this phase diagram differs from the same model in Ref. which has an extended two-dimensional TMI phase. This is because we also allow for the second nearest neighbor CDW that the authors neglected. Though frustrated, this order will dominate at large V2.One of the many advantages of Drosophila flies as model organisms for life science research has long been their benign relationship to our own species, allowing strains and transgenic stocks to be widely shared without the fear of jeopardizing either human endeavours or the natural environment.
The potential for conflict with humans was highlighted, however, following the 2008 identification of Drosphila suzukii in California. In sharp contrast to the vast majority of Drosophila species, which feed on rotting fruit and other decaying vegetation, D. suzukii, a species that is native to east Asia and had not previously been identified on the US mainland, is capable of puncturing the skin of intact, ripening fruit to lay its eggs. Over the past 5 years, D. suzukii has spread widely across North America and Europe, causing extensive agricultural damage. Today, it ranks with the lionfish infestation of the western Atlantic as one of the more severe ongoing biological invasions of the Western Hemisphere. While there has been a proliferation of recent studies on the ecology and pest management of D. suzukii , this work has often been divorced from the broader context of Drosophila evolution. The ability of D. suzukii to lay its eggs in ripening fruit has been attributed to the unusual appearance of its ovipositor, but little research has been carried out on either the morphology or evolutionary origin of this structure. The evolutionary context, however, is critical from the perspective of both basic and applied science. From the vantage point of evolutionary theory, the derived ovipositor is an example of a putative key innovation, conferring an adaptive advantage by allowing D. suzukii to exploit a new ecological niche: young, undamaged fruit that is inaccessible to the larvae of other Drosophila species. From the applied science perspective, it is critical to know the extent to which other relatives of D. suzukii could behave as pests in a similar manner, raspberry container growing in the hope of preventing their spread before they are established. Indeed, popular guides have referred to D. suzukii as ‘spotted wing Drosophila’ but this description applies to a number of flies in this species group and it is not clear how many of them are potentially harmful to agriculture. We carried out a comparative study of fruit susceptibility to D. suzukii and three of its closest relatives, and combined this work with a morphological analysis of their ovipositors. In our experiments, only D. suzukii and D. subpulchrella , the two species with ovipositors that carry enlarged bristles, punctured the intact skin of raspberries and cherries. However, while the number and morphology of enlarged bristles does not differ between these species, only D. suzukii punctured the tough skin of grapes. The shape of the D. suzukii ovipositor differs from the three other species, suggesting that changes in ovipositor shape evolved after the evolution of enlarged bristles. Our results show that D. subpulchrella could be a significant threat to the raspberry and cherry industries, while suggesting that other closely related species, including one with a spotted wing , are unlikely to be harmful.The susceptibility of four varieties of fruit to flies of four species was assayed. All flies were cultured on standard laboratory media. Bottles of flies containing pupae ready to eclose were emptied of all adults. Five to 7 days later, any adults that had emerged from the pupae were transferred to separate bottles and were aged for another 6–7 days. This process ensured that all flies were between 6 and 14 days old prior to the start of the experiment. For each experiment, three female flies of each of the four species were placed in separate plastic bottles with foam plugs, with each bottle containing one raspberry, cherry, red grape or Thompson grape.
Only fresh fruits were used. After 24 h of exposure to the fruit, flies were removed and each fruit was analysed under dissecting microscopes by two raters. In some cases, particularly for D. mimetica, we observed that flies had died during the 24 h period. Eggs in the exposed region of the fruit were counted separately from eggs found in the intact region of the skin, inserted through punctures generated by the fly . The identification of eggs was facilitated by the presence of protruding filaments . Punctures without eggs were counted in a separate tally. In cases where the raters failed to reach a consensus, the mean value of the two counts was used. Ten to 11 simultaneous replicates of each experiment were carried out. Only D. suzukii flies laid eggs in the intact region of Thompson grapes , and these were very rare. Therefore, in order to have enough punctures with eggs to make meaningful comparisons with the punctures without eggs , we carried out a separate experiment where we placed 6–10 D. suzukii female flies per bottle. We measured the area of a total of 18 randomly selected punctures with egg filaments and compared the results to 14 punctures without filaments from the same experiment.Two strains of each of the four species were used for the morphological analysis . Ten ovipositor plates from each strain, each from a separate fly, were analysed, and the total number of bristles on each plate was determined . Some of the ovipositor bristles on D. subpulchrella and D. suzukii are modified, being enlarged and heavily pigmented. These modified bristles were counted and the location of each bristle was recorded . Outlines of the ovipositor plates were generated manually from photographs. In a manner analogous to a study of the posterior lobe of the male genitalia, a horizontalline was drawn at the base of the ovipositor plate where the pigmentation fades and the structure merges with the abdomen . The area and length to width ratio of each plate outline were calculated using the program IMAGEJ. The same program was also used to calculate puncture area and wing area . Although the flies we studied are all closely related, the ovipositors do not contain easily identifiable landmarks that are invariant across species, making it difficult to employ standard landmark-based morphometric techniques. We decided, therefore, to use elliptical Fourier analysis, which does not require the identification of landmarks. The technique uses a series of contours, described by Fourier harmonics, to approximate a shape. Each harmonic is specified by four Fourier coefficients. Following the example of previous studies, we decided to use 25 harmonics. We conducted the EFA on the distal half of the ovipositorplates , because this is the portion that comes into contact with the fruit. As it is difficult to compare outlines on a large number of Fourier coefficients , principal component analysis is typically used to reduce the data to an orthogonal set of variables ordered according to the proportion of variation explained. As the interpretation of the principal components , however, is not immediately clear, we reconstructed the outlines explained by each PC using the inverse Fourier transform . The software package SHAPE was used for the EFA and PCA.All statistical analyses were carried out using the programming language R. The fruit experiments were designed specifically to compare the susceptibility of the exposed and intact region of each type of fruit across the four species. We therefore tested the following model: number of eggs ¼ f, considering each fruit and skin condition separately.