Is there a bulk signature of topological nontriviality for a 3D free fermion band insulator?

Question

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.

This post imported from StackExchange Physics at 2015-10-24 14:38 (UTC), posted by SE-user induvidyul

Comments

"A system". What kind of system are you considering? What is your definition of "topologically trivial"? This question is missing context.

This post imported from StackExchange Physics at 2015-10-24 14:38 (UTC), posted by SE-user ACuriousMind

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@ACuriousMind I have edited the title by specifying the scope to fermion band insulators. "Topological triviality" is defined in the condensed matter context as the ability to be smoothly connected to atomic band insulator. Please release the hold.

This post imported from StackExchange Physics at 2015-10-24 14:38 (UTC), posted by SE-user Everett You

Answer 1

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