Problem set 4

Solid State theory, spring '23

Due May 10, M. Before class.

1. Berry phase of a 2-level system

A Hamiltonian of a 2-level system reads

\begin{matrix} (1) & H = f_{x} σ^{x} + f_{y} σ^{y} + f_{z} σ^{z} \end{matrix}

$\boldsymbol{f}=f(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta), \text { with } f>0$ .

(1) Show that

\begin{matrix} (2) & \begin{matrix} | ψ ⟩ = e^{i χ (θ, ϕ)} [\begin{matrix} e^{- i ϕ / 2} \cos θ / 2 \\ e^{i ϕ / 2} \sin θ / 2 \end{matrix}] \end{matrix} \end{matrix}

is an eigenstate of the upper band.

$A_\theta$ $A_\phi$ $\Omega_{\theta \phi}$ .

$\chi = 0$ . Find the discontinuity in the above wavefunction given in (1).

$\chi = ϕ/2$ . Find the problem in the above wavefunction given in (1).

$\chi = -\phi/2$ . Find the problem in the above wavefunction given in (1).

(6) Take the linear combination of (4) and (5),

\begin{matrix} (3) & | ψ_{1} ⟩ = (e^{- i μ} e^{- i ϕ / 2} \sin \frac{θ}{2} + e^{i ϕ / 2} \cos \frac{θ}{2}) | ψ ⟩ . \end{matrix}

What is the problem with the new wavefunction?

2. Sum of Berry curvature

Show that wen summed over all bands, the total Berry curvature vanishes

\begin{matrix} (4) & \sum_{n}^{all bands} Ω_{n}^{a b} = 0 . \end{matrix}

3. Winding number 2

Design a 1-dimensional 2-band model like the Su-Schrieffer-Heeger model, but with a winding number of 2. Your model should not be separable into uncoupled sublattices.

4. Exercises in Girvin & Yang

13.1, 13.2, 13.3, 13.4, 13.5, 13.9, 13.10.