The Fermi gerbe of Weyl semimetals

Abstract

In the gap topology, the unbounded self-adjoint Fredholm operators on a Hilbert space have third homotopy group the integers. We realise the generator explicitly, using a family of Dirac operators on the half-line, which arises naturally in Weyl semimetals in solid-state physics. A “Fermi gerbe” geometrically encodes how discrete spectral data of the family interpolate between essential spectral gaps. Its non-vanishing Dixmier–Douady invariant protects the integrity of the interpolation, thereby providing topological protection of the Weyl semimetal’s Fermi surface.

A Quaternion conventions

The quaternion algebra ${\mathbb {H}}$ is generated by three anticommuting square roots of $-1$, labelled I, J, K. It can be represented with $2\times 2$ complex matrices, e.g.

$$\begin{aligned} {\mathbb {H}}\ni q=\underbrace{q_r}_{\mathrm{Re}(q)}+\underbrace{bI+cJ+dK}_{i\cdot \mathrm{Im}(q)} \longleftrightarrow \begin{pmatrix} q_r+ib &{} -c-id \\ c-id &{} q_r-ib \end{pmatrix}. \end{aligned}$$

(10)

In terms of Pauli matrices, Eq. (5), the imaginary part of q is $\mathrm{Im}(q)=\varvec{q}\cdot \varvec{\sigma }\equiv \sum _{j=1}^3 q_j\sigma _j$, where $\varvec{q}=(-d,-c,b)$.

Quaternion conjugation $q\mapsto {\overline{q}}=q_r-i\varvec{q}\cdot \varvec{\sigma }$ corresponds to the Hermitian adjoint in this representation. The norm is given by $|q|^2=q{\overline{q}}={\overline{q}}q=q_r^2+|\varvec{q}|^2$, and the 3-sphere of unit quaternions is then identified with $\mathrm{SU}(2)$. On ${\mathbb {C}}^2$, there is a standard quaternionic structure given by the operator $\Theta =-i\sigma _2\circ \kappa$ (where $\kappa$ denotes complex conjugation), which is antiunitary, squares to $-1$, and commutes with the left multiplication by q in Eq. (10). In physics, $\Theta$ may be interpreted as a fermionic time-reversal operator. A quaternionic basis vector for ${\mathbb {C}}^2$ is a vector ${\mathsf {e}}$ such that $\{{\mathsf {e}},\Theta {\mathsf {e}}\}$ is an orthonormal basis (over ${\mathbb {C}}$) for ${\mathbb {C}}^2$. It allows us to view ${\mathbb {C}}^2$ as a quaternionic vector space on which $i,\Theta$ generate the (right) quaternionic scalar multiplication, and also to identify $\mathrm{SU}(2)\cong \mathrm{Sp}(1)$. Similarly, a quaternionic structure on ${\mathbb {C}}^{2n}$ is an antiunitary squaring to $-1$, and a quaternionic basis $\{{\mathsf {e}}_j\}_{j=1}^n$ gives an identification ${\mathbb {C}}^{2n}\cong {\mathbb {H}}^n$. One should not confuse the role of ${\mathbb {H}}\subset M_2({\mathbb {C}})$ as an algebra of operators, and as a vector space ${\mathbb {H}}\cong {\mathbb {C}}^2$.

1.1 B Bundle gerbes

Bundle gerbes are, in a sense, a generalisation of line bundles. We recall the local description of bundle gerbes and refer to [15, 19, 20] for detailed treatments. Let ${\mathcal {U}}=\{U_i\}_{i\in I}$ be an open cover of X. The data of a bundle gerbe over X comprise, for each pair $U_i, U_j\in {\mathcal {U}}$, a Hermitian line bundle ${\mathcal {L}}_{ij}$ over $U_i\cap U_j$. It is assumed that ${\mathcal {L}}_{ii}$ is trivial for each $i\in I$ and that on each triple overlap $U_i\cap U_j\cap U_k$, there is a unitary isomorphism $\phi _{ijk}:{\mathcal {L}}_{ij} \otimes {\mathcal {L}}_{jk}\rightarrow {\mathcal {L}}_{ik}$ such that “associativity” holds on quadruple overlaps $U_i\cap U_j\cap U_k\cap U_l$,

In particular, ${\mathcal {L}}_{ij}\cong {\mathcal {L}}_{ji}^*$. The isomorphisms $\phi _{ijk}$ may be specified by a Čech 2-cocycle, and define (after passing to refinements) a cohomology class in $H^3(X,{\mathbb {Z}})$ called the Dixmier–Douady invariant of the bundle gerbe. It is the analogue for bundle gerbes of the Chern class of a line bundle.