Nodal Knots Research Articles

Symmetry‐Enforced 2D Hourglass Phononic Nodal Net

Topological nodal‐line semimetals have received extensive attention in recent years, and they exhibit a variety of shapes under various crystallographic symmetry protections, including nodal rings, nodal chains, and nodal knots. By using symmetry analysis and an effective force constant model, it is proposed that a class of 2D hourglass nodal nets protected by nonsymmorphic symmetry can exist in the space group , where the glide operations play a crucial role in the formation of the nets. Further effective model analysis shows that the connect points of the nets are Dirac points that are forced by the symmetries. Based on first‐principle calculations, it is predicted that this nodal net can be realized in the phonon dispersion of 3D Sn. The Berry phase and nontrivial surface states further support its nontrivial topological characteristics. Herein, the studies expand the nodal structure in the study of nodal line in phononic systems and offer a solid foundation for subsequent experimental investigation on this challenging topological state.

Tidal surface states as fingerprints of non-Hermitian nodal knot metals

Non-Hermitian nodal knot metals (NKMs) contain intricate complex-valued energy bands which give rise to knotted exceptional loops and new topological surface states. We introduce a formalism that connects the algebraic, geometric, and topological aspects of these surface states with their parent knots. We also provide an optimized constructive ansatz for tight-binding models for non-Hermitian NKMs of arbitrary knot complexity and minimal hybridization range. Specifically, various representative non-Hermitian torus knots Hamiltonians are constructed in real-space, and their nodal topologies studied via winding numbers that avoid the explicit construction of generalized Brillouin zones. In particular, we identify the surface state boundaries as “tidal” intersections of the complex band structure in a marine landscape analogy. Beyond topological quantities based on Berry phases, we further find these tidal surface states to be intimately connected to the band vorticity and the layer structure of their dual Seifert surface, and as such provide a fingerprint for non-Hermitian NKMs.

Imaging nodal knots in momentum space through topolectrical circuits

Knots are intricate structures that cannot be unambiguously distinguished with any single topological invariant. Momentum space knots, in particular, have been elusive due to their requisite finely tuned long-ranged hoppings. Even if constructed, probing their intricate linkages and topological "drumhead” surface states will be challenging due to the high precision needed. In this work, we overcome these practical and technical challenges with RLC circuits, transcending existing theoretical constructions which necessarily break reciprocity, by pairing nodal knots with their mirror image partners in a fully reciprocal setting. Our nodal knot circuits can be characterized with impedance measurements that resolve their drumhead states and image their 3D nodal structure. Doing so allows for reconstruction of the Seifert surface and hence knot topological invariants like the Alexander polynomial. We illustrate our approach with large-scale simulations of various nodal knots and an experiment which maps out the topological drumhead region of a Hopf-link.

Categories of Phononic Topological Weyl Open Nodal Lines and a Potential Material Candidate: Rb2Sn2O3.

The phononic topological Weyl closed nodal lines, including Weyl nodal rings, nodal chains, nodal nets, nodal links, and nodal knots, have been widely studied. The phononic topological Weyl open nodal lines (PTWONLs), however, have not been well investigated so far. By analyzing the coexistence of parity inversion and time-reversal symmetries, we found that the PTWONLs can be divided into three categories, with surface states hosting different shapes and positions in the Brillouin zone (BZ). Specifically, semiconducting Rb2Sn2O3 was found to exhibit perfect PTWONLs in its phonon spectrum, which fills up one of the categories. Numerical calculations showed that the drumhead-like surface states exist on the (010) surface and six PTWONLs appear in the first BZ due to the C3 rotation symmetry in the crystal structure. Their topological nontrivial nature was confirmed by calculating the Berry phase and by the linear phononic bands around the Weyl points. These theoretical findings provide a deep understanding into the phononic Weyl-open-nodal-line physics, and a promising candidate for experimental verification.

Observation of a nodal chain with Dirac surface states in TiB2

Topological nodal-line semimetals (TNLSMs) are characterized by symmetry-protected band crossings extending along one-dimensional lines in momentum space. The nodal lines exhibit a variety of possible configurations, such as nodal ring, nodal link, nodal chain, and nodal knot. Here, using angle-resolved photoemission spectroscopy, we observe nodal rings in the orthogonal kz = 0 and kx = 0 planes of the Brillouin zone in TiB2. The nodal rings connect with each other on the intersecting line {\Gamma}-K of the orthogonal planes, forming a remarkable nodal-chain structure. Furthermore, we observe surface states (SSs) on the (001) cleaved surface, which are consistent with the calculated SSs considering the contribution from both Ti and B terminations. The calculated SSs have novel Dirac-cone-like band structures, which are distinct from the usual drumhead SSs with a single flat band proposed in other TNLSMs.

Experimental discovery of nodal chains

Three-dimensional (3D) topological nodal points, such as Weyl and Dirac nodes have attracted wide-spread interest across multiple disciplines and diverse material systems. Unlike nodal points that contain little structural variations, nodal lines can have numerous topological configurations in the momentum space, forming nodal rings, nodal chains and potentially nodal links and nodal knots. However, nodal lines have much less development for the lack of ideal material platforms. In condensed matter for example, nodal lines are often fragile to spin-orbit-coupling, locating off the Fermi level, coexisting with energy-degenerate trivial bands and dispersing strongly in energy of the line degeneracy. Here, overcoming all above difficulties, we theoretically predict and experimentally observe nodal chains in a metallic-mesh photonic crystal having frequency-isolated linear bandtouching rings chained across the entire Brillouin zone (BZ). These nodal chains are protected by mirror symmetries and have a frequency variation less than 1%. We used angle-resolved transmission (ART) to probe the projected bulk dispersions and performed Fourier-transformed field scan (FTFS) to map out the surface dispersions, which is a quadratic touching between two drumhead surface bands. Our results established an ideal nodal-line material for further studies of topological line-degeneracies with nontrivial connectivities, as well as the consequent wave dynamics richer than 2D Dirac and 3D Weyl materials.

Nodal-knot semimetals

Topological nodal-line semimetals are characterized by one-dimensional lines of band crossing in the Brillouin zone. In contrast to nodal points, nodal lines can be in topologically nontrivial configurations. In this paper, we study the simplest topologically nontrivial forms of nodal line, namely, a single nodal line taking the shape of a knot in the Brillouin zone. We introduce a generic construction for various "nodal-knot semimetals", which yields the simplest trefoil nodal knot and other more complicated nodal knots in the Brillouin zone. The knotted-unknotted transitions by nodal-line reconnections are also studied. Our work brings the knot theory to the subject of topological semimetals.

A New Fungal Parasite of Mosses

Collections of the moss Leptodictyum riparium (Hedw.) Warnst. taken over several years from near Bonito Lake, New Mexico, were heavily parasitized by the fungus Stemphyllium botryosum Walr. Masses of hyphae were imbedded in the central strand of the host stem with the numerous conidia erupting only from the axil of the leaves. This is the first report of a moss parasitized by this species of fungus. Reports of moss parasitism by fungi are fairly common but in most cases the infection has not been described as specifically limited to the gametophyte, the sporophyte, or portions of these structures. A few authors have reported localized sites of hyphal growth or spore production on the moss plant: Stanley (1940) cited Eocronartium musicola solely on the gamretophyte of Campylium chrysophyllum but described no particular host response. Boros (1926) reported the genera Bryum, Buxbaumia, Gyroweissia, Tortula, and Dicranella as hosts of Cladosporium herbarum but gave no specific location of infection or host response. Seymour (1929) listed a considerable number of mosses that host parasitic fungi. Reports of localized infections on the moss sporophyte have included a new species of fungus, Pleospora pottiae, from the capsules of Pottia heimii (Boros 1926) and another new species, Sphaerulina 1 Department of Biology, Texas Technological College, Lubbock, Texas. This content downloaded from 157.55.39.59 on Mon, 17 Oct 2016 05:02:39 UTC All use subject to http://about.jstor.org/terms 244 THE BRYOLOGIST [Volume 69 muscorum, on the capsules of Mnium carolinianum (Wolf 1954). Several genera of mosses from tropical Asia were found to have the fungus Jola javansis infecting the seta and capsule (Giiumann 1922). All these articles merely touched upon the host response by mentioning a shriveled or stunted condition. Among moss collections made in 1964 from Bonito Lake near Alta, New Mexico, Leptodictyum riparium (Hedw.) Warnst. showed a large amnount of a fungus on the gametophytes. Since the locality had been visited several times previously, it was also possible to check moss collections made in 1958, 1960, and 1963. In all the collections of L. riparium some evidence of the parasitic fungus could be found. However, none of the other species of mosses in an extensive collection from this locality showed any evidence of infection with this fungus. With the help of Dr. Arthur Elliot, I identified the fungus as Stemphyllium botryosum Wallr. The perfect stage of this fungus is Pleospora herbarum. Neergaard (1945) described S. botryosum as a facultative parasite on many flowering plants 57 species have been reported as hosts for it but no mosses are included in this list. Seymour (1929) did not give Leptodictyum riparium as a host for any fungi nor was this moss discussed as a host by Nicolas (1932). The present paper is the first report of Stemphyllium botryosum (or its perfect stage) as a parasite of mosses and of Leptodictyum riparium as a host for fungi. Worthy of additional attention is the manner in which both the fungus and the moss express the host and parasite relationship. Figure 1 shows a typical stem segment of the gametophyte of L. riparium in which the hyphal strands of S. botryosum grow and from which the conidia arise. All the infected moss gametophytes are stunted. Fungal hyphae can be seen clearly following the central strand of the moss stem from node to node almost to the tip of the stem. At each node there is a denser accumulation of hyphae forming a knot or ball large enough to make the stem bulge (Fig. 2). From the nodal knot of hyphae a portion extends up the interior of the stem to the next higher node, a second portion branches out into the leaf in the costa region and a conidiophore erupts from the surface of the stem immediately above the point of leaf attachment. The conidiophores never emerge from the stem at any other place. After emergence the conidiophore may remain simple or may branch with each new branch bearing a conidium. This growth pattern is repeated at each node. The growth of the infected moss is different from that of healthy plants in several respects. Even though the plants are generally stunted, the intemodes are longer than those of a noninfected stem. Leaves where conidiophores arise are as much as one-half smaller than healthy leaves, both in length and width. The normal leaves of L. riparium are costate while those with conidiophores in the axil are ecostate. The fact that other species of mosses from the collecThis content downloaded from 157.55.39.59 on Mon, 17 Oct 2016 05:02:39 UTC All use subject to http://about.jstor.org/terms 1966] PRIOR: NEW FUNGAL PARASITE 245