Berry's curvature and magnetic moment in TMDCs

The Berry curvature is defined as:

$$ Omega_{munu}(mathbf{k})=partial_{mu}A_{nu}(mathbf{k})-partial_{nu}A_{mu}(mathbf{k}), tag{1} $$

where $A_{mu}(mathbf{k})=langle u_{mathbf{k}}|ipartial_{mu}u_{mathbf{k}}rangle$ is the Berry connection, $|u_{mathbf{k}}rangle$ is a Bloch state, and $partial_muequiv frac{partial}{partial k_mu}$, and $mu,nu=x,y,z$.

Invesion symmetry. Under inversion, $mathbf{k}to-mathbf{k}$, so that applying the inversion operation $mathcal{I}$ on a Bloch state gives $ mathcal{I}|u_{mathbf{k}}rangle=|u_{-mathbf{k}}rangle$. If the system is invariant under inversion, then $|u_{mathbf{k}}rangle$ and $|u_{-mathbf{k}}rangle$ must be the same state up to a global phase, so that:

$$ mathcal{I}|u_{mathbf{k}}rangle=e^{ivarphi_{mathbf{k}}}|u_{mathbf{k}}rangleLongrightarrow |u_{-mathbf{k}}rangle=e^{ivarphi_{mathbf{k}}}|u_{mathbf{k}}rangle.tag{2} $$

For the Berry connection, $mathcal{I}A_{mu}(mathbf{k})=A_{mu}(-mathbf{k})$. If the system has inversion symmetry, then

$$ begin{eqnarray} A_{mu}(-mathbf{k})&=&langle u_{-mathbf{k}}|ipartial_{mu}u_{-mathbf{k}}rangle tag{3}\ &=& langle u_{mathbf{k}}|e^{-ivarphi_{mathbf{k}}}ipartial_{mu}left(e^{ivarphi_{mathbf{k}}}u_{mathbf{k}}right)rangle tag{4}\ &=& langle u_{mathbf{k}}|e^{-ivarphi_{mathbf{k}}}ie^{ivarphi_{mathbf{k}}}partial_{mu}u_{mathbf{k}}rangle + langle u_{mathbf{k}}|e^{-ivarphi_{mathbf{k}}}i^2e^{ivarphi_{mathbf{k}}}u_{mathbf{k}}ranglepartial_{mu}varphi_{mathbf{k}}tag{4} \ &=& langle u_{mathbf{k}}|ipartial_{mu}u_{mathbf{k}}rangle -partial_{mu}varphi_{mathbf{k}} tag{5}\ &=&A_{mu}(mathbf{k})-partial_{mu}varphi_{mathbf{k}},tag{6} end{eqnarray} $$ where in the second line I used the result for the Bloch state in a system with inversion symmetry, and in the third line the chain rule for differentiation. This result means that for a system that is invariant under inversion, then $A_{mu}(mathbf{k})$ and $A_{mu}(-mathbf{k})$ differ at most by a gauge transformation.

We are now ready to look at the Berry curvature. Under inversion, $mathcal{I}Omega_{munu}(mathbf{k})=Omega_{munu}(-mathbf{k}$). If the system has inversion symmetry, then

$$ begin{eqnarray} Omega_{munu}(-mathbf{k})&=&partial_{mu}A_{nu}(-mathbf{k})-partial_{nu}A_{mu}(-mathbf{k}) tag{7}\ &=&partial_{mu}left(A_{nu}(mathbf{k})-partial_{nu}varphi_{mathbf{k}}right)-partial_{nu}left(A_{mu}(mathbf{k})-partial_{mu}varphi_{mathbf{k}}right) tag{8}\ &=&partial_{mu}A_{nu}(mathbf{k})-partial_{nu}A_{mu}(mathbf{k})-partial_{mu}partial_{nu}varphi_{mathbf{k}}+partial_{nu}partial_{mu}varphi_{mathbf{k}} tag{9}\ &=&partial_{mu}A_{nu}(mathbf{k})-partial_{nu}A_{mu}(mathbf{k}) tag{10}\ &=&Omega_{munu}(mathbf{k})tag{11}, end{eqnarray} $$

where in the second line I used the result for the Berry connection in a system with inversion symmetry. This proves that for a system with inversion symmetry, $Omega_{munu}(mathbf{k})=Omega_{munu}(-mathbf{k})$.

Time reversal symmetry. You can use an analogous procedure (I encourage you to try) to prove that for a time reversal invariant system, $Omega_{munu}(mathbf{k})=-Omega_{munu}(-mathbf{k})$. All you need to know is how the time reversal operator acts on a Bloch state, $mathcal{T}|u_{mathbf{k}}rangle=|u_{mathbf{-k}}^{ast}rangle$, and the rest of the proof proceeds in the same way.

Physical interpretation. Berry phase-like quantities look at the evolution of Bloch states at neighbouring $mathbf{k}$-points in the Brillouin zone. As an example, the Berry connection is looking at the overlap between a state $|u_{mathbf{k}}rangle$ and a state infinitessimally away from it, $partial_{mu}|u_{mathbf{k}}rangle$. As such, they are useful for calculation properties that depend on the structure of the Block states across the Brillouin zone. A well-known example is the calculation of topological invariants of materials, which measure the "twists" that the electronic wave function has when crossing the Brillouin zone. I am not familiar with applications in valleytronics, so will leave that for someone more knowledgeable.

Futher reading. An excellent book to learn about Berry phase-like quantities and applications (modern theory of polarization, topological materials, etc.) is David Vanderbilt's book.