Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems

Abstract

Topological insulators constitute one of the most intriguing phenomena in modern condensed matter theory. The unique and exotic properties of topological states of matter allow for unidirectional gapless electron transport and extremely accurate measurements of the Hall conductivity. Recently, new topological effects occurring at Dirac/Weyl points have been better understood and demonstrated using artificial materials such as photonic and phononic crystals, metamaterials and electrical circuits. In comparison, the topological properties of nodal lines, which are one-dimensional degeneracies in momentum space, remain less explored. Here, we explain the theoretical concept of topological nodal lines and review recent and ongoing progress using artificial materials. The review includes recent demonstrations of non-Abelian topological charges of nodal lines in momentum space and examples of nodal lines realized in photonic and other systems. Finally, we will address the challenges involved in both experimental demonstration and theoretical understanding of topological nodal lines.

Multi-gap topological conversion of Euler class via band-node braiding: minimal models, $PT$-linked nodal rings, and chiral heirs

“…computed over base loops encircling the nodal rings (which provides an effective Z 4 counting in each gap, see details below), and of the Z 2 monopole charge [49,76] when more than two bands must be considered (corresponding to the reduction of the Euler class to the Z 2 second Stiefel-Whitney class). This work thus fills these gaps by providing concrete minimal tight-binding models that can be readily used as a guide for the design of acoustic metamaterials [62,64], photonic crystals [63], electronic circuits [66], and optical traps for cold atoms [51,52].…”

“…Such Euler class models are increasingly becoming of importance and have for example been proposed to induce monopole-antimonople generation in quench setups [51], while the observation of this physical observable has just been reported in trapped-ion experiments [52]. In addition, the braiding and emerging of such non-Abelian charges and its relation to Euler class have also been inspiring pursuits in other experimental contexts that range from from phononic systems [53][54][55][56] and electronic systems [38,44,[57][58][59][60][61] to acoustic, photonic and electric circuit metamaterials [62][63][64][65][66].…”

Triple nodal points characterized by their nodal-line structure in all magnetic space groups

“…These emerge when energy bands become degenerate in the vicinity of the Fermi energy, giving rise to so-called band nodes, and they can feature non-trivial topological invariants, boundary signatures, and transport properties. Since the originally proposed Weyl and Dirac point degeneracies, various other types of band nodes have been proposed and studied: from nodal lines (NLs) [10][11][12][13][14] forming intricate linked, knotted, and intersecting structures [15][16][17][18][19], over nodal surfaces [20][21][22][23][24], to nodal points with degeneracies different from two (Weyl) and four (Dirac) [25][26][27][28]. In particular, three-fold degenerate points [29][30][31][32][33][34][35][36][37][38][39][40] [also called triply-degenerate nodal points, triple nodal points, or (for brevity) just triple points (TPs)] have been widely investigated, as they constitute a special intermediate between Weyl and Dirac semimetals.…”

Experimental observation of meronic topological acoustic Euler insulators

“…Similarly, Euler class and non-trivial braiding was predicted in phononic systems [35][36][37][38] and electronic systems that are strained [25,39], that undergo a structural phase transition [40,41], or are submitted to an external magnetic field [42]. The most immediate playground for these uncharted topological phases of matter is however the context of metamaterials [43][44][45][46][47][48]. The non-Abelian topological charges were recently detected in one-dimensional (1D) electrical circuit metamaterials [43].…”