Topological Insulators and Semimetals

13.1 Berry Phase

When an electron adiabatically traverses a closed loop in $\mathbf{k}$-space, its Bloch state acquires a geometric phase:

$\gamma_n(\mathcal{C}) = i\oint_{\mathcal{C}} \langle u_{n\mathbf{k}}|\nabla_{\mathbf{k}} u_{n\mathbf{k}}\rangle \cdot d\mathbf{k}$

The Berry curvature is the $\mathbf{k}$-space analog of a magnetic field:

$\boldsymbol{\Omega}_n(\mathbf{k}) = \nabla_{\mathbf{k}} \times \langle u_{n\mathbf{k}}|i\nabla_{\mathbf{k}} u_{n\mathbf{k}}\rangle$

The Berry phase for a loop $\mathcal{C}$ enclosing area $\mathcal{A}$ is:

$\gamma = \int_{\mathcal{A}} \boldsymbol{\Omega} \cdot d\mathcal{A}$

For graphene near a Dirac point, the Berry phase is $\gamma = \pi$ (a half-flux quantum), which leads to the absence of backscattering and contributes to the high mobility of graphene.

13.2 Topological Insulators

A topological insulator (TI) is an insulator in the bulk but has conducting states on its surface. These surface states are topologically protected: they cannot be removed by surface impurities or disorder (as long as time-reversal symmetry is preserved).

Key properties:

• Bulk has a band gap, but the surface has gapless Dirac-like states
• Surface states have a single Dirac cone (spin-momentum locking)
• The $Z_2$ topological invariant $\nu = 1$ distinguishes TIs ($\nu = 1$) from trivial insulators ($\nu = 0$)

2D topological insulator (quantum spin Hall insulator): Time-reversal-symmetric 2D system with helical edge states. The conductance is quantised: $G = 2e^2/h$ (one channel per edge, with opposite spins moving in opposite directions).

Examples: Bi$_2$Se$_3$Bi$_2$Te$_3$Sb$_2$Te$_3$ (3D TIs); HgTe/CdTe quantum wells (2D TIs).

13.3 Weyl and Dirac Semimetals

Weyl semimetals have band touchings at discrete points (Weyl nodes) in the Brillouin zone where the dispersion is linear in all three directions:

$\varepsilon(\mathbf{k}) = \pm\hbar v_F |\mathbf{k} - \mathbf{k}_W|$

Weyl nodes come in pairs of opposite chirality and are topologically protected. Key signatures:

• Fermi arcs: Surface states connecting projections of Weyl nodes of opposite chirality
• Chiral anomaly: In parallel $\mathbf{E}$ and $\mathbf{B}$ fields, charge is pumped between Weyl nodes, giving negative magnetoresistance
• Anomalous Hall effect: Even without magnetic order

Dirac semimetals have fourfold-degenerate Dirac points (two overlapping Weyl points of opposite chirality). Examples: Na$_3$Bi, Cd$_3$As$_2$.