Time-reversal topological insulators

In a spin-half basis, time reversal can be written as i sy times complex conjugation. It reverses spin and momentum while squaring to minus identity. A state and its time reverse are orthogonal when they belong to the same invariant energy sector. In a translationally invariant crystal, generic k maps to −k, so equal energies at these two momenta do not by themselves imply a twofold degeneracy at each individual k.

At a time-reversal-invariant momentum, k and −k differ only by a reciprocal vector, and the partner lies at the same crystal momentum. Kramers degeneracy is then unavoidable for the spinful single-particle Hamiltonian. Inversion symmetry can impose additional degeneracy away from those momenta, but it is not assumed here. Spinless time reversal with square plus one does not supply the same Kramers theorem or the same band classification.

If one spin component is conserved, the model separates into opposite-spin blocks with equal and opposite Chern numbers. Their charge Hall responses cancel, but the parity of one block's Chern number gives the two-dimensional Z₂ index. An odd value gives an odd helical edge-pair count. Two copies of an odd phase have even parity and can become topologically trivial through a symmetry-preserving coupling that mixes their channels.

When spin is not conserved, assigning independent spin-block Chern numbers is no longer justified. The Z₂ invariant nevertheless survives for the gapped occupied subspace through a more general time-reversal-constrained construction. Spin Hall conductivity is then not generally quantized. Its loss of exact quantization should not be confused with loss of the insulating phase's Z₂ distinction, provided the protecting symmetry and bulk gap remain intact.

A single ideal helical edge has opposite velocities for its two Kramers partners. A constant transverse spin mass would mix them and open a gap, but that term is odd under time reversal. A scalar potential preserves the symmetry and can shift the crossing energy without providing this mass. Weak nonmagnetic elastic single-particle backscattering is consequently constrained in the simple edge theory, although contacts and inelastic processes can still affect measured transport.

These statements are about a symmetry-protected band phase. They do not assert intrinsic topological order, fractional anyons, or topology-dependent ground-state degeneracy on a closed surface. Strong interactions can produce additional boundary behaviour, including spontaneous symmetry breaking, beyond the noninteracting edge argument. The historical phrase Z₂ topological order in early band-insulator papers must therefore be distinguished from the modern intrinsic many-body notion introduced later in this module.