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Why is the Hall conductivity in a 2D material $$\tag{1} \sigma_{xy}=\frac{e^2}{2\pi h} \int dk_x dk_y F_{xy}(k)$$ where the integral is taken over the Brillouin Zone and $F_{xy}(k)$ is the Berry curvature of the filled bands? What is the physical interpretation of this equation?

Also, can we re-parametrize all of the filled states by another pair of variables $A$ and $B$ and conclude that $$\tag{2} \sigma_{xy}=\frac{e^2}{2\pi h} \int F(A,B)dAdB$$ where $F(A,B)$ is the Berry curvature with respect to the $A$ and $B$ parameter space?

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The formula follows from the Kubo formula of conductivity (based on the linear response theory), which is discussed in this question: Kubo Formula for Quantum Hall Effect and in the references therein. Starting from the Kubo formula (set $e=\hbar=1$) $$\tag{1}\sigma_{xy}=i\sum_{E_m<0<E_n}\frac{\langle m|v_x|n\rangle\langle n|v_y|m\rangle-\langle m|v_y|n\rangle\langle n|v_x|m\rangle}{(E_m-E_n)^2},$$ where $|m\rangle$ is the single particle eigen state of the eigen energy $E_m$, i.e. $$\tag{2} H|m\rangle = E_m|m\rangle.$$ Let us take the momentum derivative $\partial_\boldsymbol{k}$ on both sides of Eq. (2), we have $$\tag{3}(\partial_{\boldsymbol{k}}H)|m\rangle + H\partial_{\boldsymbol{k}}|m\rangle = (\partial_{\boldsymbol{k}}E_m)|m\rangle + E_m \partial_{\boldsymbol{k}}|m\rangle.$$ Then overlap with $\langle n|$ from left, Eq. (3) becomes $$\tag{4}\langle n|(\partial_{\boldsymbol{k}}H)|m\rangle + E_n\langle n|\partial_{\boldsymbol{k}}|m\rangle = (\partial_{\boldsymbol{k}}E_m)\langle n|m\rangle + E_m \langle n|\partial_{\boldsymbol{k}}|m\rangle.$$ Here we have used $\langle n|H = E_n\langle n|$. If $|m\rangle$ and $|n\rangle$ are different eigen states (for $E_m\neq E_n$ in Eq. (1)), their overlap should vanish, i.e. $\langle n|m\rangle=0$. Also note that $\partial_{\boldsymbol{k}} H$ is nothing but the velocity operator $\boldsymbol{v}=\partial_{\boldsymbol{k}} H$ by definition. So Eq. (4) can be reduced to $$\tag{5} \langle n|\boldsymbol{v}|m\rangle = (E_m - E_n) \langle n|\partial_{\boldsymbol{k}}|m\rangle.$$ Substitute Eq. (5) to Eq. (1) (restoring the $x$, $y$ subscript), we have $$\tag{6} \sigma_{xy}=-i\sum_{E_m<0<E_n}\big(\langle m|\partial_{k_x}|n\rangle\langle n|\partial_{k_y}|m\rangle - \langle m|\partial_{k_y}|n\rangle\langle n|\partial_{k_x}|m\rangle\big).$$

On the other hand, the Berry connection is defined as $\boldsymbol{A}=i\langle m|\partial_{\boldsymbol{k}}|m\rangle$, and the Berry curvature is $F_{xy}=(\partial_{\boldsymbol{k}}\times\boldsymbol{A})_z=\partial_{k_x}A_{y}-\partial_{k_y}A_{x}$. Given that $(\partial_\boldsymbol{k}\langle m|)|n\rangle = - \langle m|\partial_\boldsymbol{k}|n\rangle$ (integration by part), we can see $$\tag{7} F_{xy}= -i \sum_n\big( \langle m|\partial_{k_x}|n\rangle\langle n|\partial_{k_y}|m\rangle - \langle m|\partial_{k_y}|n\rangle\langle n|\partial_{k_x}|m\rangle\big) +i \langle m|\partial_{k_x}\partial_{k_y}-\partial_{k_y}\partial_{k_x}|m\rangle.$$ The last term will vanish as the partial derivatives commute with each other. So, by comparing with Eq. (6), we end up with $$\tag{8} \sigma_{xy}=\sum_{E_m<0}F_{xy}\sim\int_\text{BZ} d^2k F_{xy}.$$ This means that the Hall conductance is simply the sum of the Chern numbers, i.e. the total Berry flux through the Brillouin zone (BZ), for all the occupied bands. Of course, we are free to re-parameterize the momentum space by another pair of variables and the total Berry flux through the BZ will not change (as it is coordinate independent).

So what is the physical meaning of $F_{xy}$? $F_{xy}$ is an effective magnetic field in the momentum space (perpendicular to the $xy$-plane along the $z$-direction). We know that for the magnetic field $\boldsymbol{B}$ in the real space, a charged particle moving in it will experience the Lorentz force, such that the equation of motion reads $\dot{\boldsymbol{k}}=\dot{\boldsymbol{r}}\times \boldsymbol{B}$. Now switching to the momentum space, we just need to interchange the momentum $\boldsymbol{k}$ and the coordinate $\boldsymbol{r}$, and replace $\boldsymbol{B}$ by $\boldsymbol{F}$ (note that the symbol $\boldsymbol{F}$ here denotes the Berry curvature, not the force), which leads to $$\tag{9} \dot{\boldsymbol{r}}=\dot{\boldsymbol{k}}\times \boldsymbol{F}$$ So what is $\dot{\boldsymbol{r}}$? It is the velocity of the electron, which is proportional to the electric current $\boldsymbol{j}$. And what is $\dot{\boldsymbol{k}}$? It is the force acting on the electron (because the force is the rate that the momentum changes with time), which is proportional to the electric field strength $\boldsymbol{E}$, so Eq. (9) implies $$\tag{10} \boldsymbol{j} \sim \boldsymbol{E}\times \boldsymbol{F}.$$ Therefore the Berry curvature $F_{xy}$ at each momentum point simply gives the Hall response of the single-particle state at that momentum. So the Hall conductivity of the whole electron system should be the sum of the Berry curvature over all occupied states, which is stated in Eq. (8).

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  • $\begingroup$ A fantastic answer! Although I was looking for pictorial intuition, your equations (9) and (10) were a beautiful way at looking at the relationship between conductivity and Berry curvature. However, where do the edge states enter in this formalism? And also, I've never seen the Kubo formula expressed as your equation (1). Finally, even though the Kubo formula is an approximation, your equation (6) should be exact by Laughlin's Argument, right? $\endgroup$ – ChickenGod Nov 16 '13 at 13:44
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    $\begingroup$ @ChickenGod The edge state does not enter in this formalism. The Kubo formula describes the bulk response. All the information needed to calculate the conductivity is the single particle wave functions in the bulk. Presumably the wave function is subject to periodic boundary conditions, so there is no edge in this formalism at all. But the amazing thing is that there is this bulk-boundary duality, that the bulk response can be encoded in the boundary. But that is another long story then. $\endgroup$ – Everett You Nov 16 '13 at 13:54
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    $\begingroup$ Kubo formula is indeed an approximation, so as Eq. (6), which is formulated in terms of free fermions. The interaction and disorder effects are not included. So the amazing thing is that even with electron interaction, the Hall conductivity is still quantized exactly to the same integer value, which was shown by Laughlin's argument. The point is that for gapped electron system, the Hall conductance is a topological character, which is invariant under any deformation as long as the gap is not closed. So the quantization is always exact, which is beyond the description of Kubo formula. $\endgroup$ – Everett You Nov 16 '13 at 14:08
  • $\begingroup$ @Everett You : The fractional quantum hall effect is a counter example. $\endgroup$ – jjcale Nov 16 '13 at 15:53
  • $\begingroup$ @jjcale Do you mean FQH is a counter example of the bulk-boundary correspondence or a counter example of the topological robustness? $\endgroup$ – Everett You Nov 16 '13 at 19:10

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