Topological Insulators and Semimetals

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.

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).

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$_2$.

• Confusing topological protection with robustness to all perturbations. Surface states are protected only as long as the symmetry (e.g., time-reversal) that defines the topological phase is preserved. Magnetic impurities break time-reversal symmetry and can gap the surface states.
• Assuming all surface states are topological. Surface states can also arise from trivial band-bending effects. The hallmark of topological surface states is their helical spin texture and the fact that they span the bulk band gap.
• Thinking the Berry phase is always quantised. The Berry phase is quantised only when the loop encloses a degeneracy point or when protected by symmetry. In general it can take any value.
• Confusing Weyl and Dirac semimetals. Weyl nodes require breaking either inversion or time-reversal symmetry. Dirac nodes require both symmetries to be present and are less robust.

• Spintronics: The spin-momentum locking in TI surface states enables efficient spin-to-charge conversion without magnetic materials, promising for low-power spintronic devices.
• Quantum computing: Majorana zero modes can arise at the interface between a TI and a superconductor, forming the basis for topological quantum computation.
• Photodetectors: TIs exhibit broadband photoresponse from terahertz to visible due to their gapless surface states, enabling high-sensitivity photodetection.
• Thermoelectrics: The large Seebeck coefficient and low thermal conductivity of topological materials like Bi$_2$Te$_3$ make them excellent thermoelectric candidates.