3
$\begingroup$

Is there such thing as a 3D Chern invariant (or some other quantity) that I can use to test an insulating quasiparticle spectrum is a topologically trivial or non-trivial insulator?

Does one exist for a state with broken time reversal symmetry, i.e. a chiral spin liquid?

Specifically I'm looking at a 3D condensed matter system, where a chiral spin liquid is assumed and a gap opens when I add certain terms.

$\endgroup$
  • 2
    $\begingroup$ @induvidyul Unfortunately your question is closed here. So I answered your question at PhysicsOverflow instead physicsoverflow.org/33821/… $\endgroup$ – Everett You Oct 24 '15 at 18:51
2
$\begingroup$

Yes, there is an bulk invariant for 3D topological insulators known as the second Chern parity $P_3$ [1-3], as the integral of the Chern-Simons 3-form of the (presumably non-Abelian) Berry connection $\mathcal{A}$ (in the momentum space) over the Brillouin zone (BZ). Note that now the Brillouin zone is a 3 dimensional manifold (as a 3D torus).

$$P_3=\frac{1}{16\pi^2}\int_\text{BZ}\mathrm{Tr}(\mathcal{F}\wedge\mathcal{A}-\tfrac{1}{3}\mathcal{A}\wedge\mathcal{A}\wedge\mathcal{A}),$$

where $\mathcal{F}=\mathrm{d}\mathcal{A}+\mathcal{A}\wedge\mathcal{A}$ is the Berry curvature. The Berry connection $\mathcal{A}$ can be obtained from the Bloch wave functions in the occupied bands. Let $|n k\rangle$ be the Bloch wave function of electron in the $n$th band at the quasi-momentum $k$ (here $k=(k_x,k_y,k_z)$ is a 3-component vector). $\mathcal{A}$ is restricted to the subspace of occupied bands, i.e. we only take those $n$ 's such that the single-particle energies $\epsilon_{nk}<0$ are negative.

$$\mathcal{A}_{mn}(k)=-\mathrm{i}\langle mk|\mathrm{d}|nk\rangle,$$

where the differential operator $\mathrm{d}=\partial_{k_\mu}\mathrm{d}k_\mu$ is defined in the momentum space. It was proved that $P_3$ can only be an integer or a half-integer. If $P_3=0$(mod 1), then the insulator is trivial. If $P_3=\tfrac{1}{2}$(mod 1), then the insulator is topological. In fact, $(-1)^{2P_3}$ is the $\mathbb{Z}_2$ index of the 3D topological insulator. There are other equivalent expressions for $P_3$ which can be found in [1-3] and the references therein.

In 3D, all the fermionic symmetry protected topological (SPT) states needs time-reversal symmetry protection (either $\mathcal{T}^2=+1$ or $\mathcal{T}^2=-1$). So there is not a topological nontrivial state which can also be chiral. I think the chiral spin liquid you are looking for does not exist, but $Z_2$ or $U(1)$ spin liquids with topological spinon band structures and anomalous gapless spinon surface modes do exist, and have been discussed a lot recently.

[1] https://physics.aps.org/featured-article-pdf/10.1103/PhysRevB.78.195424

[2] http://arxiv.org/pdf/1004.4229.pdf

[3] http://journals.aps.org/prx/pdf/10.1103/PhysRevX.2.031008

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.