Abstract
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 devise an efficient algorithm for computing the Alexander polynomial, linking numbers and higher order Milnor invariants based on an approximate and well controlled parameterisation of the knot.
Highlights
The band structure of such a model hosting an exceptional figure-eight knot and Borromean rings are displayed in Fig. 1 (a) and (c) respectively, while their corresponding Fermi surfaces are shown in Fig. 1 (e) and (g)
We have introduced Hermitian and non-Hermitian band structures hosting hyperbolic nodal structures including the figure-eight knot and the Borromean rings
While the Hermitian models rely on discrete symmetries, such as PT symmetry, the exceptional structures in the non-Hermitian realm are generic, making them stable towards any small perturbation
Summary
In conventional Hermitian systems, the very appearance of nodal lines relies on symmetries, as band touchings there are generically stable only at isolated points in reciprocal space [12]. A PT-symmetric Hamiltonian attains the form, HPT = d x (k)σ x + dz (k)σz and the nodal structure is given by d x2 + dz2 = 0. These systems are stable in the sense that sufficiently small symmetry preserving perturbations do not dissolve the nodal lines into points.
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