Berry Curvature

Formula

For a locally chosen normalized cell-periodic Bloch eigenvector $\lvert u_{n\mathbf k}\rangle$, the Berry connection is

$$\mathcal A_n(\mathbf k) = i \langle u_{n\mathbf k}\rvert \nabla_{\mathbf k} u_{n\mathbf k} \rangle.$$

In vector notation, the Berry curvature is

$$\boldsymbol\Omega_n(\mathbf k) = \nabla_{\mathbf k}\times \mathcal A_n(\mathbf k).$$

In local coordinates,

$$\Omega_{ij}^{(n)} = \partial_{k_i}\mathcal A_j^{(n)} - \partial_{k_j}\mathcal A_i^{(n)}.$$

Using the band projector

$$P_n(\mathbf k) = \lvert u_{n\mathbf k}\rangle \langle u_{n\mathbf k}\rvert,$$

the same curvature can be written gauge independently as

$$\Omega_{ij}^{(n)} = i\,\operatorname{Tr} \left( P_n \left[ \partial_{k_i}P_n, \partial_{k_j}P_n \right] \right).$$

Assumptions

• The band is isolated and nondegenerate on the region where the abelian formula is used.
• The Bloch eigenvectors are normalized with the cell-periodic inner product convention.
• $\mathcal A_n$ depends on a local gauge choice, while $\boldsymbol\Omega_n$ is gauge invariant in the abelian setting.
• Degenerate bands require a nonabelian Berry connection or a projector onto the full isolated subspace.

Validity

Berry curvature is local geometric data in parameter space or Brillouin-zone space. It can affect semiclassical dynamics, anomalous velocities, orbital magnetization, and topological response formulas, but the physical response formula always has additional assumptions.

The curvature can become large near avoided crossings. At an actual degeneracy, the isolated-band formula fails at the degeneracy point.

Common Mistakes

• Treating the Berry connection as gauge invariant.
• Computing curvature from eigenvectors whose phases jump discontinuously across the mesh without using a gauge-stable method.
• Applying a single-band curvature formula through a band crossing.
• Calling a local curvature peak a topological invariant. The appropriately normalized integral may be topological; the local profile is not.

Quick Check

Why is the projector formula often safer than the eigenvector connection formula?

Solution

The eigenvector $\lvert u_{n\mathbf k}\rangle$ changes under the gauge transformation $\lvert u\rangle\mapsto e^{i\chi(\mathbf k)}\lvert u\rangle$. The projector $P_n=\lvert u\rangle\langle u\rvert$ is unchanged because the phases cancel. A formula written in terms of $P_n$ therefore avoids arbitrary local phase choices.

Canonical Links

• Berry Connection as a Mathematical Object
• Berry Phase
• Berry Connection
• Berry Curvature
• Chern Numbers
• Chern Number

References

• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45-57, 1984.
• D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects on electronic properties”, Reviews of Modern Physics 82, 1959-2007, 2010.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.