Lecture 8

1. Wavepacket

$t=0$

\begin{matrix} (1) & \begin{aligned} w (x, 0) = \exp [i k_{c} (x - x_{c})] \exp [- \frac{1}{2 σ^{2}} {(x - x_{c})}^{2}] \\ = \frac{σ}{2 π} \int d k \exp [i k (x - x_{c})] \exp [- \frac{σ^{2}}{2} {(k - k_{c})}^{2}] \end{aligned} \end{matrix}

The packet evolves as

\begin{matrix} (2) & w (x, t) = \frac{σ}{2 π} \int d k \exp [i (k (x - x_{c}) - ω (k) t] \exp [- \frac{σ^{2}}{2} {(k - k_{c})}^{2}] \end{matrix}

$\hbar \omega(k)$ is the dispersion relation. If we expand around the center of wavepacket

\begin{matrix} (3) & ω = ω_{c} + ω_{c}^{'} (k - k_{c}) + \frac{1}{2} ω_{c}^{''} {(k - k_{c})}^{2} + \dots \end{matrix}

we find

\begin{matrix} (4) & w (k, t) = e^{i ϕ (t)} \exp [i k_{c} (x - x_{(} (t))] \exp [- \frac{1}{2 σ (t)^{2}} {(x - x_{c} (t))}^{2}] \end{matrix}

where

\begin{matrix} (5) & ϕ (t) = k_{c} (v_{p} - v_{g}) t, σ (t) = \sqrt{σ^{2} + {(\frac{ω_{c}^{''} t}{σ})}^{2}} \end{matrix}

$v_p=\omega_c/k_c$ $v_g=\omega_c'=(\text d\omega/\text dk)_{k_c}$ is the group velocity.

\begin{matrix} (6) & \begin{aligned} {\dot{x}}_{c} (t) & = \frac{d}{d t} ⟨ w (t) | x | w (t) ⟩ \\ = ⟨ w (0) | \frac{i}{ℏ} [H, x (t)) | w (0) ⟩ \\ = ⟨ w (0) | \frac{p (t)}{m} | w (0) ⟩ = ⟨ w (t) | \frac{p}{m} | w (t) ⟩ \end{aligned} \end{matrix}

\begin{matrix} (7) & \frac{d}{d t} ⟨ p ⟩ = ⟨ \frac{\partial p}{\partial t} + \frac{i}{ℏ} [H, p] ⟩ = - ⟨ \frac{\partial H}{\partial x} ⟩ \end{matrix}

\begin{matrix} (8) & \begin{matrix} x_{c} (t) = ⟨ \frac{\partial H}{\partial p} ⟩ \\ ℏ {\dot{k}}_{c} (t) = - ⟨ \frac{\partial H}{\partial x} ⟩ \end{matrix} \end{matrix}

This is the so-called Ehrenfest theorem.

That is to say, if we talk about a wavepacket in terms of its center of mass and center of momentum, we may recover the classical dynamics. This is the key idea of semiclassical dynamics.

We want to apply this trick to Bloch electrons, so we need wavepacket that is localized in both real and k-space

\begin{matrix} (9) & | W_{0} ⟩ = \frac{1}{\sqrt{N}} \sum_{k} w (k) | ψ_{k} ⟩ \end{matrix}

$\int_{\text{uc}}\text d^3r |u_k(r)|^2=1$

\begin{matrix} (10) & ⟨ W_{0} | W_{0} ⟩ = 1 \Leftrightarrow \sum_{k} | w (k) |^{2} = 1 \end{matrix}

$\boldsymbol r_c=\langle W_0|\boldsymbol r|W_0\rangle$
$\boldsymbol k_c = \sum_\boldsymbol k\boldsymbol k|w(\boldsymbol k)|^2$

$\boldsymbol r_c$ ? need position operator in the Bloch representation (布洛赫表象)

\begin{matrix} (11) & \begin{aligned} r | W_{0} ⟩ & = \frac{1}{\sqrt{N}} \sum_{k} w (k) r | ψ_{k} ⟩ \\ = \frac{1}{\sqrt{N}} \sum_{k} w (k) (- i \partial_{k} e^{i k \cdot r}) | u_{k} ⟩ \\ = \frac{1}{\sqrt{N}} \sum_{k} w (k) e^{i k \cdot r} i \partial_{k} | u_{k} ⟩ + (i \partial_{k} w (k)) | ψ_{k} ⟩ \end{aligned} \end{matrix}

In the last step, we have performed an integration by parts (分部积分). We now introduce a quantity call the Berry connection

\begin{matrix} (12) & A (k) = ⟨ u_{k} | i \nabla_{k} | u_{k} ⟩ \end{matrix}

So the center of the wavepacket is

\begin{matrix} (13) & r_{c} = \sum_{k} w (k)^{*} (i \frac{\partial}{\partial k} + A (k)) w (k) \end{matrix}

that is simply stating that in the Bloch representation

\begin{matrix} (14) & r \to i \frac{\partial}{\partial k} + A (k) \end{matrix}

$U(1)$ gauge freedom. That is,

\begin{matrix} (15) & ψ_{k} \mapsto e^{i χ (k)} ψ_{k} \end{matrix}

is still the same Bloch state. Under this change of basis, we find

\begin{matrix} (16) & A (k) \mapsto A (k) - \nabla_{k} χ \end{matrix}

which ensures that the center of a wavepacket is invariant.

2. Semiclassical equations of motion

$\boldsymbol a(\boldsymbol r,t)$ for uniform magnetic and electric fields

\begin{matrix} (17) & B = \nabla \times a, E = - \partial_{t} a . \end{matrix}

After a careful calculation¹

\begin{matrix} (18) & \begin{matrix} L (r_{c}, k_{c}, {\dot{r}}_{c}, {\dot{k}}_{c}) = ⟨ w | i ℏ \frac{d}{d t} - H | w ⟩ \\ = - e a (r_{c}, t) \cdot r_{c} + ℏ k_{c} \cdot {\dot{r}}_{c} + A \cdot ℏ {\dot{k}}_{c} - E \end{matrix} \end{matrix}

where

\begin{matrix} (19) & \begin{matrix} E = ε (k_{c}) + B \cdot M (k_{c}) \\ M = \frac{e}{2 m} ⟨ w | (r - r_{c}) \times p | w_{0} ⟩ \end{matrix} \end{matrix}

$\boldsymbol M$ is the orbital moment from the self-rotation of a wavepacket.

$c$ . Consider

\begin{matrix} (20) & L = - ε (k) - e a (r, t) \cdot \dot{r} + ℏ k \cdot \dot{r} \end{matrix}

For a uniform magnetic field,

\begin{matrix} (21) & a (r) = \frac{1}{2} B \times r \end{matrix}

So this leads to the EOM

\begin{matrix} (22) & \dot{r} = \frac{\partial ε}{ℏ \partial k}, ℏ \dot{k} = - e E - e \dot{r} \times B \end{matrix}

$\boldsymbol A\cdot \hbar \dot{\boldsymbol k}$ , with which we have

\begin{matrix} (23) & \dot{r} = \frac{\partial ε}{ℏ \partial k} - \dot{k} \times Ω (k), ℏ \dot{k} = - e E - e \dot{r} \times B \end{matrix}

$\boldsymbol \Omega$ is defined via

\begin{matrix} (24) & Ω_{α β} = \frac{\partial A_{β}}{\partial k_{α}} - \frac{\partial A_{α}}{\partial k_{β}} = ϵ_{α β γ} Ω_{γ} . \end{matrix}

$\boldsymbol \Omega$ $\dot{\boldsymbol r}(\boldsymbol k)$ is called the anomalous velocity (反常速度).

3. Anomalous Hall effect

Under an electric field, the anomalous velocity can cause a transverse current, even without an external magnetic field. This is called the anomalous Hall effect (反常霍尔效应).

Historically, there had been an debate over the origin of the anomalous Hall effect (AHE). If the anomalous Hall current comes from the Berry curvature of the Bloch bands, as described above, it is called the intrinsic AHE since it depends on the band structure and not on scattering. Correspondingly, the AHE conducitivtiy is

\begin{matrix} (25) & \begin{matrix} σ^{α β} = - \frac{e^{2}}{ℏ} \sum_{n} \int_{BZ} [d k] f^{0} (ε_{n k}) Ω_{n k}^{α β} \\ = - ϵ^{α β γ} \frac{e^{2}}{ℏ} \sum_{n} \int_{BZ} [d k] f^{0} (ε_{n k}) Ω_{n k}^{γ} . \end{matrix} \end{matrix}

Take a two-dimensional system as an example,

\begin{matrix} (26) & ρ_{x y} \sim \frac{σ_{x y}}{σ_{x x}^{2}} \sim ρ^{2} \end{matrix}

² $\beta\approx 1.9$ . However, there had been a long debate, as at finite temperatures inelastic scattering by bosons (phonon or magnon) makes a complete microscopic theory difficult. Other origins of the AHE have been suggested. These theories are based certain asymmetry in impurity scattering, such as side jump and skew scattering, which are called extrinic mechanisms. We will not go into details of these extrinsic mechanism, interested readers can consult Nagaosa's review.³

$\rho_{xy}$ $H_z$ is small, but then saturates (饱和) at high fields. Empirically, peopel have found

\begin{matrix} (27) & ρ_{x y} = R_{0} H_{z} + R_{s} M_{z} \end{matrix}

References and notes

1 Ming-Che Chang and Qian Niu. Phys. Rev. B 53, 7010 (1996). ↩

2 Kooi, C., 1954, Phys. Rev. 95, 843 ↩

3 Nagaosa et al.: Anomalous Hall effect. Rev. Mod. Phys. 82, 1539 (2010). ↩