nLab semi-metal




In condensed matter physics, a semi-metal is a crystalline material whose electronic band structure is such that there is a sizeable gap between the valence band and the conduction band (as for an insulator) except over a submanifold of “nodal loci” inside the Brillouin torus which is of positive codimension 2\geq 2.

This terminology, which was established in the early 2000s, is meant to follow the older terminology semiconductor, which refers to the case where there is globally a gap between valence and conduction band, but just a small one. Notice that a global and large gap corresponds to an insulator, while the absence of a gap, hence the broad overlap of valence and conductions band, corresponds to a metal. Therefore a semi-metal behaves like an insulator for the bulk of its excitation modes, like a semiconductor for excitations close to the nodal loci, and almost like a metal for excitations right at the nodal loci; whence the terminology.

(graphics from SS 22)

If the gap closure happens over an isolated point, one speaks of a nodal point proper, or more specifically of a Dirac point or Weyl point, indicating the “degeneracy” (multiplicity) of the energy eigenstates at the point (2 for Weyl, 4 for Dirac, see below). If the gap closes over a 1-dimensional manifold (a curve or a circle) one speaks of a nodal line.

Due to the gap on (a neighbourhood retract of) the complement of the nodal points/lines inside the Brillouin torus, the Berry connection is well-defined away from the nodal loci, and it tends to be flat in suitable hyperplanes there. It’s holonomy around codimension=2 nodal loci (ie. around nodal lines in 3d materials and nodal points in effectively 2d materials such as graphene, see also at defect brane) – known as a Berry phase – is then a property of the nodal locus alone, and to be interpreted as a measure for the “topological protection” of the gap over the nodal loci (e.g. FWDF 16, II.B, Vanderbilt 18, 5.5.2): Under adiabatic deformations of the material that do not excite modes above the gap, these nodal points may move and merge or split, but their total topological charge cannot change.

Therefore semi-metals with such non-trivial Berry phases are examples of topological phases of matter, and are sometimes called topological semi-metals, for emphasis. In fact, if the topological charges of the nodal loci in a semi-metal do happen to be trivial, then it is (in the deformation class of) a fully gapped insulator and may then be a topological insulator (if the twisted equivariant topological K-theory class of its valence bundle is non-trivial).


Phase decay via mass terms

Near nodal points in the Brillouin torus of a semi-metal, the dispersion relation kE(k)k \mapsto E( k) exhibited by the energy bands is thought to be approximated by that of a massless Dirac equation/Weyl equation, whence the terminology Dirac point or Weyl point.

When the material’s parameters can be and are adiabatically tuned such that this dispersion relation turns into a massive Dirac equation, then the band gap at the former nodal crossing will “open up” in proportion to the coefficient mm of the effective mass term mγ 0m \gamma_0 in the Dirac equation. If this happens to all nodal points (while keeping the band gap open everywhere else), one expects that the topological semi-metal phase decays into a topological insulator-phase.

A necessary condition for such a mass term to exist at all is that a further Clifford generator γ 0\gamma_0 is represented on the Bloch-Hilbert space of electrons such that it skew-commutes with all Clifford momenta k/k\!\!\!/ (e.g. Schnyder 18, Sec. II.A, Schnyder 20, Sec. 2.1, see also Freed & Hopkins 21, around Lem. 9.55).

(graphics from SS22)

By Karoubi/Atiyah-Singer-type theorems, such gap-opening mass terms are expected to again be classified by topological K-theory groups (Chiu & Schnyder 14, Sec. A.2, reviewed in Schnyder 18, Sec. II.A, Schnyder 20, Sec. 2.1, following Morimoto & Furusaki 13, Sec. V, CTSR 15, Sec. III.C).


  1. This means that the topological semi-metal-phase must be classified in a group modulo the group of mass terms (cf. Freed & Hopkins 21, Thm. 9.63). The K-group of mass terms itself at best classifies aspects of the topological insulator-phase that the semi-metal phase may decay to.

  2. The argument about single mass terms in a K-theory group of a point ignores the global topology of the valence bundle.

    This is manifest already in the canonical example of the Haldane model, where for constant mass terms (as usually understood) the total topological charge of all nodal points combined is constrained to vanish.


The archetypical and original example of a semi-metal is graphene, where the gap closes over two Dirac points. (Or almost: a tiny spin-orbit coupling in graphene actually produces a tiny gap even at the would-be Dirac points, which however tends to be too small to be visible in practice. But if one will or would analyze graphene with an accuracy that does resolve this tiny gap-opening, then graphene will or would appear as a topological insulator instead of a semi-metal.)

After the synthesis of graphene, which is an effectively 2-dimensional material (an atomic mono-layer) much attention has shifted to the synthesis and understanding of 3-dimensional semi-metals.



General monographs:

Review with emphasis on mass terms

For references on graphene see also there.

Discussion focusing on 3-dimensional semi-metals:

Discussion of the /2\mathbb{Z}/2-valued Berry phases around codimension=2 nodal loci in PIP I-symmetric (time-reversal + inversion invariant) semi-metals:

More mathematical discussion of the case of Chern semi-metals:

Discussion of (classification of) Dirac/Weyl mass terms:



Related discussion of Dirac mass terms is in:

Lists of actual semi-metals:

  • Md. Rakibul Karim Akand. Catalog of magnetic topological semimetals, : AIP Advances 10 095222 (2020) [doi:10.1063/5.0020096]

Simulation of Dirac semimetals with lattice field theory:

and other numerical calculuations:

  • The study of magnetic topological semimetals by first principles calculations, npj Computational Materials 5 96 (2019) [doi:10.1038/s41524-019-0237-5]

  • Topological semimetals from the perspective of first-principles calculations, Journal of Applied Physics 128 (2020) 191101 [doi:10.1063/5.0025396]

Holographic description

On holographic descriptions of topological semimetals via the AdS-CMT correspondence:

Some of the above material is taken from

Anyons in momentum-space

On non-trivial braiding of nodal points in the Brillouin torus of topological semi-metals (“braiding in momentum space”):

“a new type non-Abelian ‘braiding’ of nodal-line rings inside the momentum space”

“Here we report that Weyl points in three-dimensional (3D) systems with 𝒞 2𝒯\mathcal{C}_2\mathcal{T} symmetry carry non-Abelian topological charges. These charges are transformed via non-trivial phase factors that arise upon braiding the nodes inside the reciprocal momentum space.”

Braiding of Dirac points in twisted bilayer graphene:

Here, we consider an exotic type of topological phases beyond the above paradigms that, instead, depend on topological charge conversion processes when band nodes are braided with respect to each other in momentum space or recombined over the Brillouin zone. The braiding of band nodes is in some sense the reciprocal space analog of the non-Abelian braiding of particles in real space.


we experimentally observe non-Abelian topological semimetals and their evolutions using acoustic Bloch bands in kagome acoustic metamaterials. By tuning the geometry of the metamaterials, we experimentally confirm the creation, annihilation, moving, merging and splitting of the topological band nodes in multiple bandgaps and the associated non-Abelian topological phase transitions

new opportunities for exploring non-Abelian braiding of band crossing points (nodes) in reciprocal space, providing an alternative to the real space braiding exploited by other strategies.

Real space braiding is practically constrained to boundary states, which has made experimental observation and manipulation difficult; instead, reciprocal space braiding occurs in the bulk states of the band structures and we demonstrate in this work that this provides a straightforward platform for non-Abelian braiding.

See also:

Incidentally, references indicating that the required toroidal (or yet higher genus) geometry for anyonic topological order in position space is dubious (as opposed to the evident toroidal geometry of the momentum-space Brillouin torus): Lan 19, p. 1, ….

Knotted nodal lines in 3d semimetals

Beware that various authors consider braids/knots formed by nodal lines in 3d semimetals, i.e. knotted nodal lines in 3 spatial dimensions, as opposed to worldlines (in 2+1 spacetime dimensions) of nodal points in effectively 2d semimetals needed for the anyon-braiding considered above.

An argument that these nodal lines in 3d space, nevertheless, may be controlled by Chern-Simons theory:

Last revised on January 10, 2024 at 00:24:05. See the history of this page for a list of all contributions to it.