In a semimetal, the map isn’t continuous since the occupied states in the vicinity of a gapless point change discontinuously.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
Intuitively, we may imagine a complete Fermi surface being present when we begin with a thin sample of Weyl semimetal.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
The surface states are obtained by mapping three dimensional Weyl semimetals to Chern insulators in two dimensions as we now explain.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
This observation explains why there are surface states in the Weyl semimetal.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
Condensed matter derivation of the anomaly–Consider a Weyl semimetal with cross-sectional area A and length Lz in an electric and a magnetic ﬁeld applied along z .
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
This describes a 2D system with an edge, corresponding to the top surface of the original semimetal.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
Instead, consider the anomalous Hall effect associated with this simple Weyl semimetal with two Weyl points.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
Thus the Weyl semimetal is an intermediate state between a trivial insulator and a layered Chern insulator.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
As for topological insulators, topological semimetals can exist in various numbers of dimensions and with different symmetries, and they all have surface states.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
Is Graphene a Topological Semimetal?: Graphene has two point nodes in its 2D bulk band structure.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
Therefore we conclude that real graphene is a non-topological semimetal.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
The deﬁnition of a topological semimetal as we formulated it generalizes the “topological protection” of Weyl points by their magnetic charge to other systems.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
Indeed, the well known zero bias peaks in d-wave cuprate superconductors are also an example of edge states of a topological semimetal (the d-wave singlet superconductor)26 .
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
Do quantized responses, which we deﬁne as ones that depend only on the locations and topological properties of nodes, generalize as well? So far there is only one example, the Hall effect in a Weyl semimetal.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
Intermediate between these is the Weyl semimetal phase.
Beyond Band Insulators: Topology of Semi-metals and Interacting Phases
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