Topological Semimetals and Nodal Surface States

Topological semimetals are gapless in the bulk but gapped on the lower-dimensional object that supports their node invariant: an enclosing surface for an isolated point node or a linking loop for a nodal line. Surface arcs or drumheads then depend both on the bulk charge and on how nodes project onto the chosen termination; they are not closed-system Fermi surfaces with an autonomous two-dimensional topology.

Required background. Berry geometry supplies node charges on enclosing surfaces; protected interfaces supplies oriented boundary counting.

Helpful background. Chiral and gauge anomalies supplies the qualified response analogy for Weyl nodes.

Weyl charge and node stability

Near an isolated two-band crossing in three dimensions,

$$H(\mathbf q)=\sum_{i,j}v_{ij}q_i\sigma_j,$$

with nonsingular velocity matrix $v$. On a small oriented sphere $S^2$ enclosing the node, the lower-band Chern number is

$$\chi=\frac{1}{2\pi}\oint_{S^2}\boldsymbol\Omega\cdot d\mathbf S =\operatorname{sgn}\det v$$

for the chapter’s Berry convention and outward sphere orientation. The charge cannot change unless the enclosing surface becomes gapless, so an isolated Weyl node can move but only annihilates with opposite total charge. Periodicity of the Brillouin zone forces the total Weyl charge to vanish Wan et al. 2011.

A Dirac point superposes opposite chiralities and therefore needs crystalline, inversion, time-reversal, or other symmetry to prevent hybridization. A nodal line is protected when a loop linking it carries a quantized Berry phase, often $0$ or $\pi$ under combined symmetry. Stating “band crossing” without codimension and protecting symmetry does not establish a stable node.

Surface arcs and slice Chern numbers

Between two Weyl nodes separated, for example, along $k_z$, each two-dimensional $k_z$ slice is gapped and has a Chern number that jumps by the node charge. Its edge state assembles into a surface Fermi arc. The arc can deform and merge with trivial surface states; only its connectivity modulo such reconstruction and the net slice chirality are constrained. If opposite nodes project to the same surface momentum, a visible open arc need not survive that termination.

For a nodal line, a surface drumhead may occupy the projected interior, but its dispersion and even spectral visibility depend strongly on termination. The robust bulk datum is the linking-loop invariant, not a perfectly flat surface band.

Disorder, interactions, and response limits

Weak smooth disorder broadens quasiparticles, while stronger disorder can produce rare-region density of states, diffusive metals, or node annihilation; translation-dependent node separation then loses a sharp meaning. Interactions may renormalize velocities, generate symmetry breaking, split Dirac nodes, or create Green-function zeros. The minimal band invariant is reliable only while its enclosing surface or linking loop remains nonsingular.

Electromagnetic formulas involving Weyl-node separation require equilibrium, lattice regularization, filled-band contributions, and transport order of limits. A formal continuum “chiral magnetic” term is not by itself an equilibrium current prediction Armitage, Mele, and Vishwanath 2018, §§ III–V.

Exercise

Two Weyl nodes of charges $+1$ and $-1$ approach one another. What condition allows them to gap out?

Solution

They must meet at the same crystal momentum modulo a reciprocal vector while any symmetry that forbids their mixing is absent or permits the mass term. A surface enclosing both has total charge zero, so once they coincide it can remain gapped as the pair annihilates.

References

• N. P. Armitage, E. J. Mele, and Ashvin Vishwanath, “Weyl and Dirac Semimetals in Three-Dimensional Solids,” Reviews of Modern Physics 90 (2018) 015001, doi:10.1103/RevModPhys.90.015001.
• Xiangang Wan, Ari M. Turner, Ashvin Vishwanath, and Sergey Y. Savrasov, “Topological Semimetal and Fermi-Arc Surface States in the Electronic Structure of Pyrochlore Iridates,” Physical Review B 83 (2011) 205101, doi:10.1103/PhysRevB.83.205101.