Non-Hermitian nodal knot metals (NKMs) contains intricate complex-valued energy bands gives 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, and 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.
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 re…
We extend the list of known band structure topologies to include a large family of hyperbolic nodal links and knots, occurring both in conventional Hermitian systems where their stability relies on discrete symmetries, and in the dissipative non-Hermitian realm where the knotted nodal lines are generic and thus stable towards any small perturbation. We show that these nodal structures, taking the forms of Turk's head knots, appear in both continuum- and lattice models with relatively short-ranged hopping that is within experimental reach. To determine the topology of the nodal structures, we d…
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 partne…
Topological nodal line semimetals host stable chained, linked, or knotted line degeneracies in momentum space protected by symmetries. In this paper, we use the Jones polynomial as a general topological invariant to capture the global knot topology of the nodal lines. We show that every possible change in Jones polynomial is attributed to the local evolutions around every point where two nodal lines touch. As an application of our theory, we show that nodal chain semimetals with four touching points can evolve to a Hopf-link. We extend our theory to 3D non-Hermitian multi-band exceptional line…
With its boundary tracing out a link or knot in 3D, the Seifert surface is a 2D surface of core importance to topological classification. We propose the first-ever experimentally realistic setup where Seifert surfaces emerge as the boundary states of 4D topological matter. Unlike ordinary real space knots that exist in polymers, biomolecules and everyday life, our knots and their Seifert surfaces exist as momentum space nodal structures, where topological linkages have profound effects on optical and transport phenomena. Realized with 4D circuit lattices, our nodal Seifert systems are freed fr…
Nodal line semimetals have lately aroused much experimental and theoretical interest, with their gap closing along unconventional trajectories in the 3D Brillouin zone. These trajectories or nodal lines can close into loops and trace out intricate knotted or linked configurations with complicated topologies. In this work, we investigate the semi-classical optical response of two nodal loops in linked, unlinked and touching configurations, focusing particularly on the interplay of response anisotropy and non-linearity. We provide a geometric picture that unifies these aspects and sheds light on…
A nodal link is a special form of a line degeneracy (a nodal line) between adjacent bands in the momentum space of a three-dimensional topological crystal. Unlike nodal chains or knots, a nodal link consists of two or more mutually-linked rings that do not touch each other. Recent studies on non-Abelian band topology revealed that the topological charges of the nodal links can have the properties of quaternions. However, a photonic crystal that has a nodal link with non-Abelian charges has not been reported. Here, we propose dielectric photonic crystals in forms of double diamond structures wh…
Three-dimensional (3D) Weyl and Dirac semimetals garner considerable attention in condensed matter physics due to the exploration of entirely new topological phases and related unconventional surface states. Nodal line and ring semimetals on the other hand can facilitate 3D band crossings characterized by nontrivial links such as coupled chains and knots that are protected by the underlying crystal symmetry. Experimental complexities, detrimental effects of the spin-orbit interaction, along with the merging of the underlying surface states into the bulk pose great challenges for growing advanc…
Nodal lines are symmetry-protected one-dimensional band degeneracies in momentum space, which can appear in numerous topological configurations such as nodal rings, chains, links, and knots. Very recently, non-Abelian topological physics has been proposed in space-time inversion (PT) symmetric systems, and attract widespread attention. One of the most special configurations in non-Abelian system is the earring nodal link, composing of a nodal chain linking with an isolated nodal line, is signature of non-Abelian topology and cannot be elucidated using Abelian topological classifications. Howev…
In periodic systems, nodal lines are loops in the three-dimensional momentum space where two bands are degenerate with each other. Nodal lines exhibit rich topological features as they can take various configurations such as rings, links, chains and knots. These line nodes are usually protected by mirror or PT symmetry. Here we propose and demonstrate a novel type of photonic straight nodal lines in a D2d meta-crystal which are protected by roto-inversion time (roto-PT) symmetry. The nodal lines are located at the central axis and hinges of the Brillouin zone. They appear as quadrupole sources…
We experimentally simulate in a photonic setting non-Hermitian (NH) metals characterized by the topological properties of their nodal band structures. Implementing nonunitary time evolution in reciprocal space followed by interferometric measurements, we probe the complex eigenenergies of the corresponding NH Bloch Hamiltonians, and study in detail the topology of their exceptional lines (ELs), the NH counterpart of nodal lines in Hermitian systems. We focus on two distinct types of NH metals: two-dimensional systems with symmetry-protected ELs, and three-dimensional systems possessing symmetr…
Using spin-resolved and angle-resolved photoemission spectroscopy and first-principles calculations, we have identified bulk band inversion and spin polarized surface state evolved from a weak topological insulator (TI) phase in van der Waals materials Nb3XTe6 (X = Si, Ge). The fingerprints of weak TI homologically emerge with hourglass fermions, as multi nodal chains composed by the same pair of valence and conduction bands gapped by spin orbit coupling. The novel topological state, with a pair of valence and conduction bands encoding both weak TI and hourglass semimetal nature, is essential …
Nodal non-centrosymmetric superconductors (NCS) have recently been shown to be topologically non-trivial. An important consequence is the existence of topologically protected flat zero-energy surface bands, which are related to the topological characteristics of the line nodes of the bulk gap via a bulk-boundary correspondence [Schnyder and Ryu, arXiv:1011.1438]. In this paper we examine these zero-energy surface bands using a quasiclassical theory. We determine their spectrum and derive a general condition for their existence in terms of the sign change of the gap functions. A key experimenta…
