Type-II nodal loops: Theory and material realization

Abstract

Nodal loop appears when two bands, typically one electron-like and one hole-like, are crossing each other linearly along a one-dimensional manifold in the reciprocal space. Here we propose a new type of nodal loop which emerges from crossing between two bands which are both electron-like (or hole-like) along certain direction. Close to any point on such loop (dubbed as a type-II nodal loop), the linear spectrum is strongly tilted and tipped over along one transverse direction, leading to marked differences in magnetic, optical, and transport responses compared with the conventional (type-I) nodal loops. We show that the compound K4P3 is an example that hosts a pair of type-II nodal loops close to the Fermi level. Each loop traverses the whole Brillouin zone, hence can only be annihilated in pair when symmetry is preserved. The symmetry and topological protections of the loops as well as the associated surface states are discussed.Topological metals and semimetals have become a focus of current physics research [1,2]. These materials feature nontrivial band-crossings in their low-energy band structures, around which the quasiparticles behave drastically different from the usual Schrödinger-type fermions. Depending on its dimensionality, the crossing manifold may take zero-dimensional (nodal point), onedimensional (nodal loop), or two-dimensional (nodal surface) form [3]. There has already been extensive studies on nodal points, especially on so-called Weyl and Dirac semimetal materials [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Recently, nodal loops begin to attract considerable interest: several nodal-loop materials have been proposed, with interesting physical consequences revealed [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33].Consider the generic case of a nodal loop formed by the linear crossing between two bands in a three-dimensional system. Close to any point P on the loop, the dispersion is linear along the two transverse directions of the loop, and is at least quadratic along the tangential direction. The low-energy effective model near P can be expressed as (set = 1)up to first order in the wave-vector q measured from P . Here q i 's (i = 1, 2) are the components of q along two orthogonal transverse directions [see Fig. 1(a)], v i 's are

Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric

“…In this sense, our approach offers a direct method by which Floquet nodal lines and nodal surfaces can be detected through quantum-metric measurements. Besides direct applications in quantum-engineered systems, our work suggests several possible extensions, such as the implementation of tilted (type-II) nodal lines in three dimensions [63] and topological nodal lines and nodal spheres in higher dimensions. We also emphasize that, in contrast with solid-state realizations of nodal lines, the quasi-energy spectrum of Floquet systems can give rise to anomalous nodal spheres located at the boundary of the Floquet-Brillouin zone [ Fig.…”

Hourglass Weyl loops in two dimensions: Theory and material realization in monolayer GaTeI family

“…Moving forward, it was realized that crystalline solids may host more types of emergent fermions beyond the Weyl/Dirac paradigm [15][16][17][18][19]. For example, in a three-dimensional (3D) material, besides 0D nodal point, band crossings may also take the form of 1D nodal loops [20][21][22][23][24][25][26][27][28][29] or even 2D nodal surfaces [30][31][32][33][34].…”

Emergence of Type-I and Type-II Dirac line nodes in penta-octa-graphene

“…It was proposed to realize the DLNs in the Cu 3 PdN [16], CaAgX(X=P, As) [23], and ZrSiS family materials [24]. As analogue to the classification of type-I and type-II Dirac/Weyl cones according to their titled degree of cones, type-II DLNs semimetals have been theoretically proposed [25,26]. Compared with the type-I nodal point, the cone of the type-II nodal point is completely tipped over so that there coexists electron and hole pockets at the certain energy level [27].…”