How to Derive the Expression for Berry Curvature

Berry curvature describes how a quantum eigenstate changes as external parameters vary. It is the curl of the Berry connection, but it also has a useful expression directly in terms of derivatives of the state—or, for an energy eigenstate, in terms of matrix elements of derivatives of the Hamiltonian.

The overall sign depends on convention. This derivation uses the Berry connection $A_a=i\langle n|\partial_a n\rangle$, where $a$ labels a parameter such as a momentum component. With this choice, the curvature is $F_{ab}=\partial_a A_b-\partial_b A_a$.

From the connection to curvature

Let $|n(\boldsymbol{\lambda})\rangle$ be a normalized state that varies smoothly with parameters $\boldsymbol{\lambda}$. Its Berry connection is

$$A_a=i\langle n|\partial_a n\rangle.$$

Take the antisymmetric derivative:

$$\begin{aligned} F_{ab} &=\partial_a A_b-\partial_b A_a \\ &=i\left(\langle\partial_a n|\partial_b n\rangle-\langle\partial_b n|\partial_a n\rangle\right). \end{aligned}$$

The terms involving $\langle n|\partial_a\partial_b n\rangle$ cancel because mixed derivatives commute for a sufficiently smooth state. This is the basic expression for Berry curvature. In two dimensions, the scalar curvature is often written $F_{xy}$; in three dimensions, it is common to package the components as a vector, $\Omega_c=\tfrac12\epsilon_{cab}F_{ab}$.

Why the formula is gauge invariant

The state can be rephased without changing its physical meaning: $|n\rangle\mapsto e^{i\chi(\boldsymbol{\lambda})}|n\rangle$. Under this change, the connection becomes $A_a\mapsto A_a-\partial_a\chi$, while its curl is unchanged. Thus the connection depends on the phase convention, but the curvature does not.

An equivalent form uses the projector orthogonal to the state, $Q=1-|n\rangle\langle n|$:

$$F_{ab}=i\left(\langle\partial_a n|Q|\partial_b n\rangle-\langle\partial_b n|Q|\partial_a n\rangle\right).$$

The projection removes changes that are only along $|n\rangle$, including changes from a local phase choice.

Deriving the energy-eigenstate formula

Suppose $|n\rangle$ is a normalized, nondegenerate eigenstate of a parameter-dependent Hamiltonian:

$$H|n\rangle=E_n|n\rangle.$$

Differentiate with respect to parameter $\lambda_a$ and project onto a different eigenstate $|m\rangle$. For $m\ne n$, this gives

$$\langle m|\partial_a n\rangle=\frac{\langle m|\partial_a H|n\rangle}{E_n-E_m}.$$

The component of $|\partial_a n\rangle$ parallel to $|n\rangle$ does not contribute to the projected curvature. Substituting the components along all other eigenstates into the curvature expression yields

$$F_{ab}^{(n)}=i\sum_{m\ne n}\frac{\langle n|\partial_a H|m\rangle\langle m|\partial_b H|n\rangle-\langle n|\partial_b H|m\rangle\langle m|\partial_a H|n\rangle}{(E_n-E_m)^2}.$$

This spectral formula is especially useful in band theory, where the parameters are often crystal momentum components. It assumes that the eigenvalue is isolated from the other bands in the region of interest; at a degeneracy, the single-state formula is not defined in this form.

Worked example: a two-level Hamiltonian

Consider $H(\mathbf{k})=\mathbf{k}\cdot\boldsymbol{\sigma}$, where $\boldsymbol{\sigma}$ are the Pauli matrices and $\mathbf{k}=(k_x,k_y,k_z)$. The eigenvalues are $E_\pm=\pm k$, with $k=|\mathbf{k}|$. Using the convention above, the upper state has curvature

$$\boldsymbol{\Omega}_+(\mathbf{k})=-\frac{\mathbf{k}}{2k^3}.$$

For example, its $xy$ component is

$$F_{xy}^{(+)}=-\frac{k_z}{2(k_x^2+k_y^2+k_z^2)^{3/2}}.$$

At $\mathbf{k}=(0,0,2)$, this gives $F_{xy}^{(+)}=-1/8$. The lower state has the opposite curvature, $\boldsymbol{\Omega}_-=+\mathbf{k}/(2k^3)$. Both expressions become singular at $\mathbf{k}=0$, where the two energy levels meet and the isolated-band assumption fails.

Common mistakes

• Mixing sign conventions: Some authors define $A_a=-i\langle n|\partial_a n\rangle$. That convention reverses the signs of $F_{ab}$ and the spectral formula.
• Dropping the antisymmetrization: Curvature uses the difference between the $ab$ and $ba$ terms; keeping only one term generally gives a complex quantity rather than the real curvature.
• Using the formula at a degeneracy: The energy denominators vanish when bands meet. A degenerate subspace requires a matrix-valued, or non-Abelian, Berry connection.
• Confusing connection and curvature: $A_a$ changes under a phase redefinition; $F_{ab}$ does not.