Time-Reversal-Invariant Z2 Topological Insulators

A time-reversal-invariant topological insulator has vanishing total Chern number but a nontrivial Z2 obstruction in its occupied Kramers bundle. The invariant is defined only after specifying $T^2=-1$, a bulk gap or mobility gap, and the occupied subspace. Its boundary statement is a parity constraint, not a promise that every termination displays a clean Dirac cone.

Required background. Characteristic classes supplies bundle invariants; invertible responses supplies response language; antiunitary symmetries supplies the $T^2$ convention.

Helpful background. Anomaly inflow supplies the boundary obstruction viewpoint.

Kramers sewing and the Z2 index

For spinful electrons, time reversal $\Theta$ is antiunitary and $\Theta^2=-1$. It maps the occupied space at $\mathbf k$ to that at $-\mathbf k$. The sewing matrix

$$w_{mn}(\mathbf k)=\langle u_{m,-\mathbf k}|\Theta|u_{n,\mathbf k}\rangle$$

is antisymmetric at a time-reversal-invariant momentum $\Lambda=-\Lambda+\mathbf G$. A gauge-invariant Z2 index can be written through Pfaffians, a half-Brillouin-zone curvature formula, or the parity of partner switching in a Wilson-loop spectrum Kane and Mele 2005. These representations agree when their gauges and orientations are implemented consistently.

If inversion $I$ is also present and commutes with $\Theta$, the two members of a Kramers pair share a parity eigenvalue $\xi_m(\Lambda)=\pm1$. In two dimensions the shortcut is

$$(-1)^\nu=\prod_{\Lambda\in\mathrm{TRIM}} \prod_{m=1}^{N_{\rm occ}/2}\xi_m(\Lambda).$$

The product includes one state from each occupied Kramers pair, not both. It ceases to be available when inversion is broken, although the Z2 phase itself can persist Fu and Kane 2007.

Boundary parity and response

For a two-dimensional noninteracting Z2-odd insulator, any time-reversal-preserving edge has an odd parity of Kramers pairs crossing the Fermi energy between boundary time-reversal momenta. Extra even pairs may appear or annihilate. Elastic single-particle backscattering within one helical pair is forbidden by $T^2=-1$, but inelastic processes, interactions, disorder, and additional channels can alter transport.

In three dimensions, four indices $(\nu_0;\nu_1\nu_2\nu_3)$ distinguish a strong index $\nu_0$ from translation-dependent weak indices. A symmetry-preserving surface of a strong free-fermion phase has an odd number of Dirac cones modulo pairs. A magnetic surface mass gaps them and can support a half-quantized surface Hall contribution only after the bulk and regulator contributions are treated consistently Hasan and Kane 2010, §§ IV–V.

Limits of the band invariant

Nonmagnetic disorder is compatible with the phase while a mobility gap and time reversal survive; momentum-space shortcuts must then be replaced by twisted-boundary or scattering formulations. Interactions can reduce free classifications or permit a symmetric gapped surface with intrinsic topological order. Such a surface does not trivialize the bulk: its anyons carry the anomaly. Conversely, applying a free-band Z2 number to a bulk with intrinsic topological order is a category error.

Exercise

Four inversion-invariant momenta have occupied Kramers-pair parity products $(-1,+1,+1,+1)$. Find $\nu$.

Solution

Their total product is $-1$, so $(-1)^\nu=-1$ and $\nu=1$. Multiplying both states in each Kramers pair would incorrectly square every parity and always give $+1$.

References

• Liang Fu and Charles L. Kane, “Topological Insulators with Inversion Symmetry,” Physical Review B 76 (2007) 045302, doi:10.1103/PhysRevB.76.045302.
• M. Zahid Hasan and Charles L. Kane, “Colloquium: Topological Insulators,” Reviews of Modern Physics 82 (2010) 3045–3067, doi:10.1103/RevModPhys.82.3045.
• Charles L. Kane and Eugene J. Mele, “Z2 Topological Order and the Quantum Spin Hall Effect,” Physical Review Letters 95 (2005) 146802, doi:10.1103/PhysRevLett.95.146802.