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Many texts quote the expression for the Berry curvature for a two-level system, with Hamiltonian $\mathbf{h}(\mathbf{k})=(h_x,h_y,h_z)$ in terms of $\mathbf{k}=(k_x,k_y)$, as something like \begin{equation} F_{ij}=\frac{1}{2h^3}\epsilon_{abc}h_a\frac{\partial h_b}{\partial k_i}\frac{\partial h_c}{\partial k_j} \end{equation} However, I have been unable to find any explanation as to how exactly this is derived from the expression for the Berry connection \begin{equation} A_x=\frac{1}{2h(h-h_z)}\left(h_y\frac{\partial h_x}{\partial k_x}-h_x\frac{\partial h_y}{\partial k_x}\right)\quad (\text{and similar for }A_y) \end{equation} using \begin{equation} F_{ij}=\frac{\partial A_j}{\partial k_i}-\frac{\partial A_i}{\partial k_j} \end{equation} I have tried inserting the expressions for $A_x$ and $A_y$ in the above, but just end up with a huge number of terms that won't reduce to anything like what I am aiming for. I have also tried doing it by manipulating components and using Levi-Civita symbols, but I can't get rid of the prefactors $1/2h(h-h_z)$. I'm sure it is possible to parameterise in terms of spherical polars in order to make things easier, but I would really like to do it in terms of cartesian coordinates to improve my knowledge of index notation and manipulation. (I apologise in advance if this question is more mathematics than physics, but I think it would be helpful to anyone entering the field of topological insulators etc. to see this worked out.)

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  • $\begingroup$ This question is missing some context, in particular the Hamiltonian for the two-level system. Do you mean $H = \mathbf{h}(\mathbf{k}) \cdot \sigma$, where $\sigma = (\sigma^x,\sigma^y,\sigma^z)$ are Pauli operators? $\endgroup$ Jul 28, 2016 at 9:07
  • $\begingroup$ $\uparrow$ Which texts? Which pages? $\endgroup$
    – Qmechanic
    Jul 28, 2016 at 9:25
  • $\begingroup$ Sorry Mark - yes, that is exactly what I meant. $\endgroup$
    – dgwp
    Jul 28, 2016 at 10:41
  • $\begingroup$ Qmechanic - standard books on topological insulators such as Bernevig & Hughes, Franz & Molencamp, plus numerous review and other articles. $\endgroup$
    – dgwp
    Jul 28, 2016 at 10:48
  • $\begingroup$ Just noticed that Nakahara's geometry/topology book makes a similar leap from the Berry connection to curvature without showing any intermediate steps, albeit expressed in terms of differential forms. $\endgroup$
    – dgwp
    Aug 2, 2016 at 18:07

1 Answer 1

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I will write my response as detailed as possible.

We can write the Hamiltonian explicitly as, \begin{align*} H(k)=\vec{h}(k)\cdot\vec{\sigma}=\left(\begin{array}{cc} h_z(k) & h_x(k)-ih_y(k)\\ h_x(k)+ih_y(k) &-h_z(k) \end{array}\right) \end{align*} The eigenvalues for $H(k)$ are $\pm|\vec{h}(k)|$. We assume a gap condition, namely, $|\vec{h}(k)|\neq 0$ for all $k$. In this case, for each $k$, there are two different eigenvalues, associated to one-dimensional orthogonal eigenspaces. Since they are orthogonal to each other, we will focus on, \begin{align*} H(k)\psi=-|\vec{h}(k)|\psi, \end{align*} explicitly, if we write $\psi=(u\ v)^t$, for $u,v\in\mathbb{C}$, we have \begin{align*} \left(\begin{array}{cc} h_z(k)+|\vec{h}(k)| & h_x(k)-ih_y(k)\\ h_x(k)+ih_y(k) &-(h_z(k)-|\vec{h}(k)|) \end{array}\right)\left(\begin{array}{cc} u\\ v \end{array} \right)=0 \end{align*} Giving, if $|\vec{h}(k)|\neq h_z(k)$, \begin{align*} v=\frac{h_x(k)+ih_y(k)}{|\vec{h}(k)|-h_z(k)}u. \end{align*} If we introduce a normalized vector $(x(k),y(k),z(k))=\vec{h}(k)/|\vec{h}(k)|$, then, \begin{align*} v=\frac{x(k)+iy(k)}{1-z(k)}u. \end{align*} This gives a non-trivial solution whenever $u\neq 0$. Notice that this solution is valid if $z(k)\neq 1$. If $z(k)=1$, then $v=1$ and $u=0$ provides the right solution. Since $\psi(k)$ generates a one-dimensional subspace, we have the freedom of rescaling the solution. In particular, we can set $u\equiv 1$, giving, \begin{align*} v(k)=\frac{x(k)+iy(k)}{1-z(k)}. \end{align*} Hence we get a local solution, valid if $z(k)\neq 1$, \begin{align*} \psi(k)=(1, v(k)). \end{align*} Actually, $v=x-iy/(1-z)$ is associated to a local complex coordinate on the sphere $S^2$ coming from stereographic projection from the North pole and $v(k)$ is the local form of the map $k\mapsto \vec{h}(k)/|\vec{h}(k)|\in S^2$ in terms of this local coordinate system. Now the Berry connection acts on the local ``frame'' provided by $\psi$ (any vector on the associated eigenspace at $k$ can be written as a multiple of $\psi(k)$, whenever $\psi$ is defined), as\begin{align*} \nabla \psi= \frac{\langle\psi |d\psi\rangle}{\langle \psi|\psi\rangle} \psi\equiv \mathcal{A}\cdot \psi, \end{align*} Hence, \begin{align*} \mathcal{A}=\frac{\bar{v}(k)dv(k)}{1+|v(k)|^2}. \end{align*} The Berry curvature is obtained by taking the exterior derivative of $\mathcal{A}$, \begin{align*} \mathcal{F}&=d\mathcal{A}=\frac{d\bar{v}(k)\wedge dv(k)}{(1+|v(k)|^2)}-\frac{|v(k)|^2d\bar{v}(k)\wedge dv(k)}{(1+|v(k)|^2)^2}\\ &=\frac{d\bar{v}(k)\wedge dv(k)}{(1+|v(k)|^2)^2}. \end{align*} In order to relate this to the expression wanted, we will use spherical coordinates on the sphere $S^2$, namely $x=\sin\theta\cos\phi$, $y=\sin\theta\sin\phi$, $z=\cos\theta$, then, \begin{align*} v=\frac{x+iy}{1-z}=\frac{\sin\theta}{1-\cos\theta}e^{i\phi}=\frac{\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)}{\sin^2\left(\frac{\theta}{2}\right)}e^{i\phi}=\cot\left(\frac{\theta}{2}\right)e^{i\phi} \end{align*} and, \begin{align*} dv&=-\frac{1}{2}\left(1+\cot\left(\frac{\theta}{2}\right)^2\right)d\theta e^{i\phi}+ivd\phi=-\frac{1}{2}(1+|v|^2)d\theta \frac{v}{|v|} +ivd\phi,\\ d\bar{v}&=-\frac{1}{2}(1+|v|^2)d\theta \frac{\bar{v}}{|v|} -i\bar{v}d\phi \end{align*} so that, \begin{align*} \frac{d\bar{v}\wedge dv}{(1+|v|^2)^2}=-i\frac{|v|d\theta\wedge d\phi}{(1+|v|^2)}, \end{align*} now \begin{align*} \frac{|v|}{1+|v|^2}=\frac{|\cot\left(\frac{\theta}{2}\right)|}{1+\cot^2\left(\frac{\theta}{2}\right)}=|\cot\left(\frac{\theta}{2}\right)|\sin^2\left(\frac{\theta}{2}\right)=|\cos\left(\frac{\theta}{2}\right)|\sin\left(\frac{\theta}{2}\right)=\frac{1}{2}\sin\theta, \end{align*} since $\sin(\theta)>0$ for $\theta\in(0,\pi)$. Therefore, \begin{align*} \mathcal{F}=-\frac{i}{2}\sin(\theta(k)) d\theta(k)\wedge d\phi(k). \end{align*} Notice that $\sin\theta d\theta\wedge d\phi$ is nothing but the Riemannian volume form on the sphere $S^2$, which can be written as \begin{align*} \sin\theta d\theta\wedge d\phi&=\left(\vec{x}\cdot \frac{\partial \vec{x}}{\partial \theta}\times \frac{\partial\vec{x}}{\partial \theta}\right)d\theta\wedge d\phi=\varepsilon_{ijk}x^i\frac{\partial x^j}{\partial\theta}\frac{\partial x^j}{\partial\phi}d\theta\wedge d\phi\\ &=\frac{1}{2}\varepsilon_{ijk}x^i dx^j\wedge dx^k. \end{align*} If we take in the previous expression $x^i\rightarrow x^i(k)$ (mathematically, it means that we are taking the pull-back by the map $k\mapsto \vec{x}(k)= \vec{h}(k)/|\vec{h}(k)|\in S^2$), we get \begin{align*} \varepsilon_{ijk}x^i\frac{\partial x^j}{\partial k_1}\frac{\partial x^j}{\partial k_2}dk_1\wedge dk_2=\frac{1}{||\vec{h}||^3}\varepsilon_{ijk}h^i\frac{\partial h^j}{\partial k_1}\frac{\partial h^k}{\partial k_2}dk_1\wedge dk_2, \end{align*} where we used skew-symmmetry of $\varepsilon_{ijk}$. Therefore, \begin{align*} \mathcal{F}=-\frac{i}{2}\varepsilon_{ijk}x^i(k)\frac{\partial x^j}{\partial k_1}\frac{\partial x^k}{\partial k_2}dk_1\wedge dk_2=-\frac{i}{2||\vec{h}(k)||^3}\varepsilon_{ijk}h^i(k)\frac{\partial h^j}{\partial k_1}\frac{\partial h^k}{\partial k_2}dk_1\wedge dk_2. \end{align*} It is now clear that \begin{align*} \mathcal{F}_{ab}(k)=-\frac{i}{2||\vec{h}(k)||^3}\varepsilon_{ijk}h^i(k)\frac{\partial h^j}{\partial k_a}\frac{\partial h^k}{\partial k_b}. \end{align*} The factor of $-i$ is just a matter of convention: usually, in the physics literature, the Berry connection is taken to be $i\mathcal{A}$. The expression found for the curvature also shows, explicitly, that the first Chern number is the degree/winding number of the map $\Phi:k\mapsto \vec{x}(k)= \vec{h}(k)/|\vec{h}(k)|\in S^2$, \begin{align*} \int_{\mathbb{T}^2}\frac{i\mathcal{F}}{2\pi}=\frac{1}{4\pi}\int_{\mathbb{T}^2}\Phi^{*}(\sin\theta d\theta\wedge d\phi)=\text{deg}(\Phi), \end{align*} where $\mathbb{T}^2$ is the first Brillouin zone (a two-dimensional torus).

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  • $\begingroup$ I have a quick question on this derivation. In your definition of the Berry connection, it seems as if you perform the normalization of the wave function after you take the derivative. Am I correct to say that you are exploiting the fact that the norm is time-independent, and therefore normalization can be performed after the calculate of the Berry connection with the un-normalized wave function to simplify the calculation? $\endgroup$ Aug 10, 2022 at 16:38

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