Five-dimensional generalization of the topological Weyl semimetal

Abstract

We generalize the concept of three-dimensional topological Weyl semimetal to a class of five dimensional (5D) gapless solids, where Weyl points are generalized to Weyl surfaces which are twodimensional closed manifolds in the momentum space. Each Weyl surface is characterized by a U(1) second Chern number C2 defined on a four-dimensional manifold enclosing the Weyl surface, which is equal to its topological linking number with other Weyl surfaces in 5D. In analogy to the Weyl semimetals, the surface states of the 5D metal take the form of topologically protected Weyl fermion arcs, which connect the projections of the bulk Weyl surfaces. The further generalization of topological metal in 2n + 1 dimensions carrying the n-th Chern number Cn is also discussed. PACS numbers: 71.10.-w 73.20.At 03.65.VfThe study of topological states of matter has generated enormous interest in condensed matter physics in the past decade. Prominent examples of topological states include quantum Hall states, topological insulators and topological superconductors, which all have a bulk gap and possess protected gapless surface states [1][2][3][4]. Recently, the three-dimensional (3D) Weyl semimetal has been studied extensively, which is a gapless topological state that realizes Weyl fermions and the chiral anomaly in a condensed matter system [5][6][7][8][9][10][11][12][13][14][15][16][17]. The conduction and valence bands of a Weyl semimetal are connected via a number of Weyl points, each of which carries a topological monopole charge given by the surrounding first Chern number C 1 in the momentum space [18]. Consequentially, the surface states of a Weyl semimetal take the form of topological Fermi arcs, which connect the projections of oppositely charged Weyl points [7]. The novel features of Weyl semimetal have motivated the study of other gapless topological phases such as the nodal superconductor [19,20], and their classifications in general [21][22][23][24][25][26][27].As an important aspect of study, the generalization of topological states to higher dimensions often reveals profound understandings of topological physics [28][29][30][31]. In this work, we present a novel generalization of the Weyl semimetal to five spatial dimensions (5D), which goes beyond the current ten-fold-way classification scheme for gapless topological states [21][22][23]. The Weyl points are generalized to two-dimensional (2D) closed Weyl surfaces, while the topological charge is generalized to the second Chern number C 2 of the Berry connection surrounding each Weyl surface in the momentum space. We prove that C 2 of a Weyl surface is exactly its topological linking number with other Weyl surfaces in the 5D momentum space. Similar to the 3D Weyl semimetal, this 5D metal hosts topologically protected arcs of Weyl fermions (Weyl fermion arcs) on its four dimensional (4D) boundaries, and the arcs connect between the projections of Weyl surfaces on the boundaries. In cold atom systems and photonic crystals, synthetic spatial dimensions can be created to reali...

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