# nLab semi-metal

Contents

### Context

#### Topological physics

Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.

General theory:

# Contents

## Idea

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 $\geq 2$.

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).

## Properties

### Phase decay via mass terms

Near nodal points in the Brillouin torus of a semi-metal, the dispersion relation $k \mapsto E( k)$ 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 $m$ of the effective mass term $m \gamma_0$ 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 $\gamma_0$ is represented on the Bloch-Hilbert space of electrons such that it skew-commutes with all Clifford momenta $k\!\!\!/$ (e.g. Schnyder 18, Sec. II.A, Schnyder 20, Sec. 2.1, see also Freed & Hopkins 21, around Lem. 9.55).

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).

Beware:

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.

## Examples

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.

## References

### General

General textbook account:

• David Vanderbilt, Section 5 of: Berry Phases in Electronic Structure Theory – Electric Polarization, Orbital Magnetization and Topological Insulators, Cambridge University Press (2018) (doi:10.1017/9781316662205)

Review with emphasis on mass terms

• Andreas P. Schnyder, Accidental and symmetry-enforced band crossings in topological semimetals, lecture notes (2018) $[$pdf, pdf$]$

• Andreas P. Schnyder, Topological semimetals, lecture notes (2020) $[$pdf, pdf$]$

Discussion focusing on 3-dimensional semi-metals:

• A. A. Burkov, M. D. Hook, Leon Balents, Topological nodal semimetals, Phys. Rev. B 84 (2011) 235126 (arXiv:1110.1089, doi:10.1103/PhysRevB.84.235126)

• Ari M. Turner, Ashvin Vishwanath, Part I of: Beyond Band Insulators: Topology of Semi-metals and Interacting Phases, in: Topological Insulators, Contemporary Concepts of Condensed Matter Science 6 (2013) 293-324 $[$arXiv:1301.0330, ISBN:978-0-444-63314-9$]$

• Bohm-Jung Yang, Naoto Nagaosa, Classification of stable three-dimensional Dirac semimetals with nontrivial topology, Nature Communications 5 (2014) 4898 (doi:10.1038/ncomms5898)

• N. P. Armitage, Eugene Mele, Ashvin Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90 015001 (2018) $[$doi:10.1103/RevModPhys.90.015001$]$

• Jiaheng Li, Zetao Zhang, Chong Wang, Huaqing Huang, Bing-Lin Gu, Wenhui Duan, Topological semimetals from the perspective of first-principles calculations, Journal of Applied Physics 128, 191101 (2020) (doi:10.1063/5.0025396)

Discussion of the $\mathbb{Z}/2$-valued Berry phases around codimension=2 nodal loci in $P I$-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:

following:

Example:

Related discussion of Dirac mass terms is in:

Some of the above material is taken from

### Anyons in momentum-space

On non-trivial braiding of modal points in the Brillouin torus of 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 $\mathcal{C}_2\mathcal{T}$ 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.

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.

• Bo Peng, Adrien Bouhon, Bartomeu Monserrat, Robert-Jan Slager,Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates, Nature Communications volume 13, Article number: 423 (2022) (doi:10.1038/s41467-022-28046-9)

it is possible to controllably braid Kagome band nodes in monolayer $\mathrm{Si}_2 \mathrm{O}_3$ using strain and/or an external electric field.

• Haedong Park, Wenlong Gao, Xiao Zhang, Sang Soon Oh, Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems, Nanophotonics (2022) (doi:10.1515/nanoph-2021-0692)

• Adrien Bouhon, Robert-Jan Slager, Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs $[$arXiv:2203.16741$]$

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, ….

Last revised on May 25, 2022 at 10:42:47. See the history of this page for a list of all contributions to it.