Chapter 16 — Topological materials and topology in band theory

The band structure of a crystal can carry an integer that no continuous deformation can change — and that integer controls whether a current flows around the edge of an otherwise insulating sample.

In chapter 07 we saw that a perfect crystal’s Hamiltonian commutes with every lattice translation, and the eigenfunctions therefore factorise into a plane wave and a cell-periodic piece, the Bloch function. In chapter 11 we saw that those Bloch functions produce a band structure $\varepsilon_n(\mathbf k)$ that we plot, density-of-state, and project onto atomic orbitals. The missing chapter in that story is the geometric one: a Bloch function $|u_{n\mathbf k}\rangle$ is a vector in Hilbert space, and as we move $\mathbf k$ around in the Brillouin zone the vector traces out a path in Hilbert space. The geometry of that path — its twist, its curvature, its winding number — is encoded in three objects we will spend most of this chapter on: the Berry phase, the Berry curvature, and the Chern number they integrate to. The Chern number is an integer that classifies an isolated band; it cannot change without the band gap closing. And it is not a mathematical curiosity: it is the bulk topological invariant that determines the quantised Hall conductivity of chapter 15’s two-dimensional electron gas, the protected edge currents of the quantum Hall effect, the helical surface states of the 3D topological insulators, and the open Fermi arcs of the Weyl semimetals. The bulk–boundary correspondence — “non-trivial bulk topology forces gapless boundary states” — is the through-line of the entire chapter.

Reading note. This chapter assumes chapter 07 (Bloch’s theorem, the Brillouin zone, reciprocal lattice), chapter 11 (Kohn–Sham band structure), and the relativistic background of chapter 15 (spin–orbit coupling and the spin Hall effect) — every non-trivial 2D and 3D TI in a real material relies on SOC to invert the band ordering. No time-dependent perturbation theory is needed; the linear-response machinery of chapter 12 is touched on only in the worked example.

16.1 The claim

The claim of the chapter, boxed, is the bulk–boundary correspondence in its sharpest form.

Claim. A gapped band structure is characterised by integer topological invariants $\{ C_n \}$ (Chern numbers) and, when TR-symmetric, a set of $\mathbb Z_2$ indices ${\nu}$. These integers are robust — they cannot change under any continuous deformation of the Hamiltonian that keeps the gap open — and observable: they fix the quantised Hall conductivity, the existence of protected edge / surface states, the parity of the surface Dirac cone, and the helicity of the chiral edge modes. The bulk–boundary correspondence states

\[\label{eq:ch-16-bulk-boundary} \boxed{ \sigma_{xy} = \frac{e^2}{h} \sum_{n\,:\,\text{occupied}} C_n , \qquad \text{edge states} = \text{``hologram`'' of the bulk Chern number}. }\]

The first equation is the TKNN formula (Thouless, Kohmoto, Nightingale, den Nijs, 1982): the off-diagonal Hall conductivity of an insulator is the sum of the Chern numbers* of its occupied bands, in units of $e^2/h$. The second is the bulk–boundary correspondence (Halperin, 1982): a non-zero bulk Chern number is not consistent with a gapped, periodic boundary condition in real space, and the inconsistency is resolved by the appearance of chiral edge states at the boundary between a non-trivial insulator and the vacuum (or between a non-trivial and a trivial insulator). The generalisation to time-reversal-symmetric systems replaces the Chern number by the $\mathbb Z_2$ index $\nu \in {0, 1}$ and the chiral edge state by a Kramers pair of counter- propagating helical edge states.

Equation \eqref{eq:ch-16-bulk-boundary} is the headline. Every section of the chapter builds toward it. The plan: in §16.2 we introduce the mathematical machinery (Berry phase, Berry connection and curvature, Chern number, $\mathbb Z_2$ index); in §16.3 we apply it to the integer quantum Hall effect (IQHE); in §16.4 to the quantum spin Hall (QSH) effect and 2D topological insulators; in §16.5 to 3D topological insulators and topological crystalline insulators; in §16.6 to the gapless cousins — Dirac, Weyl, and nodal-line semimetals; in §16.7 to the practical computation of topological invariants from a DFT band structure; in §16.8 to the modern extension to higher-order topological insulators (HOTIs). We close with two worked examples (§16.9, the Kane–Mele model on the honeycomb lattice; §16.10, the Bi$_2$Se$_3$ surface Dirac cone from DFT+SOC), three graded problems (§16.11), a list of cross-references (§16.12), and an honest catalogue of what we left out (§16.13).

The roadmap in one diagram:

%%{init: {'flowchart': {'htmlLabels': true, 'curve': 'basis'}}}%%
graph TD
  C16["Chapter 16 — Topology<br/>in band theory"]
  C162["§16.2 Mathematical<br/>machinery<br/>(Berry, Chern, Z2)"]
  C163["§16.3 Integer QHE<br/>(Chern = 1)"]
  C164["§16.4 Quantum spin Hall<br/>(2D Z2 TIs)"]
  C165["§16.5 3D topological<br/>insulators (Bi2Se3, TCIs)"]
  C166["§16.6 Dirac / Weyl<br/>nodal-line semimetals"]
  C167["§16.7 Computing topology<br/>from DFT (Wannier, WCC)"]
  C168["§16.8 Higher-order TIs<br/>(HOTIs)"]
  C169["§16.9 Worked:<br/>Kane-Mele"]
  C1610["§16.10 Worked:<br/>Bi2Se3 + SOC"]
  C1611["§16.11 Problems"]
  BULK["Bulk topological<br/>invariant: Chern or Z2"]
  EDGE["Boundary state:<br/>chiral or helical"]
  BBC["Bulk-boundary<br/>correspondence"]

  C16 --> C162
  C162 --> C163
  C162 --> C164
  C163 --> C165
  C164 --> C165
  C162 --> C166
  C165 --> C167
  C166 --> C167
  C167 --> C168
  C162 --> BULK
  BULK --> BBC
  BBC --> EDGE
  C167 --> C169
  C167 --> C1610
  C162 --> C1611

The two columns on the right are the “through-line” of the chapter: the bulk topological invariant (Chern number or $\mathbb Z_2$ index) determines, via the bulk–boundary correspondence \eqref{eq:ch-16-bulk-boundary}, the existence and structure of the boundary state (chiral for Chern insulators, helical for $\mathbb Z_2$ TIs). Every section of the chapter builds toward this connection.

16.2 The mathematical machinery

The whole subject rests on four objects, defined and connected below. We use atomic units ($\hbar = e = 1$) throughout, and work at zero temperature.

16.2.1 The Berry phase

Consider a quantum system whose Hamiltonian $\hat H(\mathbf R)$ depends on external parameters $\mathbf R \in \mathcal M$, with a non-degenerate ground state $|n(\mathbf R)\rangle$ that varies smoothly with $\mathbf R$. Under an adiabatic change of $\mathbf R$, the system stays in the instantaneous ground state, acquiring only a dynamical phase

\[\label{eq:ch-16-dynamical-phase} \theta_\text{dyn}(T) = -\int_0^T E_n(\mathbf R(t))\, dt .\]

Berry (1984) pointed out that on top of the dynamical phase there is also a geometric phase that depends only on the path $\mathcal C : [0, T] \to \mathcal M$ in parameter space:

\[\label{eq:ch-16-berry-phase} \boxed{ \gamma_n[\mathcal C] \;=\; i \oint_{\mathcal C} \langle n(\mathbf R) \mid \nabla_{\mathbf R} n(\mathbf R) \rangle \cdot d\mathbf R . }\]

Equation \eqref{eq:ch-16-berry-phase} is the Berry phase. It is gauge-invariant and geometric: it depends only on the path $\mathcal C$, not on how fast the path is traversed.

For our application, the parameter space is the Brillouin zone, $\mathbf R \to \mathbf k$, and the ground state at each $\mathbf k$ is the cell-periodic Bloch function $|u_{n\mathbf k}\rangle$ of chapter 07. The Berry phase of band $n$ along a 1-D closed loop $\mathcal C$ in the BZ is

\[\label{eq:ch-16-berry-1d} \gamma_n[\mathcal C] \;=\; i \oint_{\mathcal C} \langle u_{n\mathbf k} \mid \nabla_{\mathbf k} u_{n\mathbf k} \rangle \cdot d\mathbf k .\]

Note. The Berry phase is defined modulo $2\pi$. The convention we use throughout the chapter reports it in the principal range $(-\pi, \pi]$. Because it is gauge-invariant and only defined modulo $2\pi$, the parity of $\gamma_n/2\pi$ is the gauge-invariant content of a 1-D Berry phase: this is the $\mathbb Z_2$ invariant of §16.2.4. ### 16.2.2 The Berry connection and curvature

Define the Berry connection (also called the “Berry vector potential”) of band $n$ as the 1-form

\[\label{eq:ch-16-berry-connection} \boxed{ \mathcal A_n(\mathbf k) \;=\; i \langle u_{n\mathbf k} \mid \nabla_{\mathbf k} u_{n\mathbf k} \rangle . }\]

The Berry connection is gauge-dependent: under a $\mathbf k$-dependent phase rotation $|u_{n\mathbf k}\rangle \to e^{i \chi(\mathbf k)} |u_{n\mathbf k}\rangle$ it shifts as $\mathcal A_n \to \mathcal A_n - \nabla_{\mathbf k} \chi$, exactly like the electromagnetic vector potential under a gauge transformation. The Berry phase \eqref{eq:ch-16-berry-1d} is the integral of the Berry connection along a closed loop and is gauge-invariant because the gradient integrates to zero on a closed loop.

The Berry curvature is the exterior derivative of the Berry connection,

\[\label{eq:ch-16-berry-curvature} \boxed{ \boldsymbol \Omega_n(\mathbf k) \;=\; \nabla_{\mathbf k} \times \mathcal A_n(\mathbf k) . }\]

The Berry curvature is a gauge-invariant pseudo-vector in $\mathbf k$-space. In components,

\[\label{eq:ch-16-berry-curvature-comp} \Omega_{n,a}(\mathbf k) \;=\; \epsilon_{abc}\, \partial_{k_b} \mathcal A_{n,c}(\mathbf k) , \qquad a, b, c \in \{x, y, z\}.\]

Unlike the connection, the curvature is a physical (gauge-invariant) object: it enters the semi-classical equations of motion of a wave packet in band $n$ as an anomalous velocity term

\[\label{eq:ch-16-anomalous-velocity} \dot{\mathbf r}_n = \frac{1}{\hbar}\frac{\partial \varepsilon_n}{\partial \mathbf k} - \dot{\mathbf k} \times \boldsymbol \Omega_n(\mathbf k) , \qquad \hbar \dot{\mathbf k}_n = -\frac{\partial \varepsilon_n}{\partial \mathbf r} ,\]

The second term of the velocity equation is the anomalous velocity, perpendicular to $\dot{\mathbf k}$ and proportional to $\boldsymbol \Omega_n$. Note that the Berry curvature enters only the velocity equation: the crystal-momentum equation is the plain Lorentz-force statement (in an electromagnetic field, $\hbar \dot{\mathbf k} = -e(\mathbf E + \dot{\mathbf r} \times \mathbf B)$), with no curvature term (Xiao, Chang & Niu, Rev. Mod. Phys. 82, 1959 (2010), eq. (1)). The anomalous term is the microscopic origin of the intrinsic anomalous Hall effect of a magnetic metal and the chiral edge current of a Chern insulator.

For an isolated non-degenerate band, the curvature can also be written explicitly in terms of the other bands:

\[\label{eq:ch-16-curvature-other-bands} \Omega_{n,ab}(\mathbf k) \;=\; i \sum_{m \neq n} \frac{\langle u_{n\mathbf k} \mid \partial_{k_a} \hat H \mid u_{m\mathbf k} \rangle \langle u_{m\mathbf k} \mid \partial_{k_b} \hat H \mid u_{n\mathbf k} \rangle - (a \leftrightarrow b)}{(\varepsilon_{m\mathbf k} - \varepsilon_{n\mathbf k})^2} .\]

Equation \eqref{eq:ch-16-curvature-other-bands} is sometimes called the “sum-over-states” form of the Berry curvature. The denominator is the square of the band gap; the curvature is large where the gap is small, which is the reason topological transitions are tied to gap closings.

16.2.3 The Chern number

The integral of the Berry curvature over a closed 2-D surface in $\mathbf k$-space is, by Stokes’ theorem, the Berry phase around the boundary of the surface. For an isolated band $n$ and a closed 2-D Brillouin zone (e.g. a 2-D crystal), the relevant closed surface is the 2D BZ itself (a torus, $T^2$). Stokes’ theorem on a torus gives

\[\label{eq:ch-16-stokes-torus} \oint_{\partial(\text{BZ})} \mathcal A_n \cdot d\mathbf k \;=\; \int_{\text{BZ}} \Omega_{n,z}(\mathbf k)\, d^2k \;=\; 0 \pmod{2\pi} .\]

The “modulo $2\pi$” is the only place the topology of the torus enters. The integral of the curvature is an integer multiple of $2\pi$, and the integer is the Chern number of the band:

\[\label{eq:ch-16-chern-number} \boxed{ C_n \;=\; \frac{1}{2\pi} \int_{\text{BZ}} \Omega_{n,z}(\mathbf k)\, d^2 k \;=\; \frac{1}{2\pi} \oint_{\partial(\text{BZ})} \mathcal A_n \cdot d\mathbf k \in \mathbb Z . }\]

The two expressions in \eqref{eq:ch-16-chern-number} are equal as integers (Stokes on a torus); the second is sometimes called the “non-Abelian” form when there are degeneracies. The Chern number is the parent of all the topological invariants we will meet. It is the topological invariant of the integer quantum Hall state (§16.3) and of the quantum anomalous Hall state (a Chern insulator without Landau levels).

Tip. The Chern number can also be written as a winding number. Define the “projector” $P_n(\mathbf k) = |u_{n\mathbf k}\rangle \langle u_{n\mathbf k}|$ onto band $n$, and the BZ Berry connection as $\mathcal A = i\,\text{Tr}[P \nabla_{\mathbf k} P]$. Then $C = (1/2\pi) \int \text{Tr}[P\, dP \wedge dP]$, a 2-form that depends only on the subspace spanned by the band, not on the basis within it. This is the form used in practical codes (Z2Pack, see §16.7.4).

For an insulator with several occupied bands $\{ n \}$ (the cumulative Chern number is what enters the Hall conductivity), we generalise to a multi-band projector $P(\mathbf k) = \sum_{n \in \text{occ}} |u_{n\mathbf k}\rangle \langle u_{n\mathbf k}|$ and define

\[\label{eq:ch-16-multiband-chern} C \;=\; \frac{1}{2\pi} \int_{\text{BZ}} \text{Tr}\Bigl[ P\, dP \wedge dP \Bigr] \;\in\; \mathbb Z .\]

Equation \eqref{eq:ch-16-multiband-chern} is invariant under continuous unitary rotations within the occupied subspace — the same $U(N)$ freedom that appears in the Wannier construction of §16.7.1. ### 16.2.4 Time-reversal symmetry and the $\mathbb Z_2$ invariant

In a system with time-reversal symmetry (TRS), the Chern number is forced to zero (the Berry curvature is odd under TRS, $\boldsymbol \Omega_n(\mathbf k) = -\boldsymbol \Omega_n(-\mathbf k)$, so its BZ integral vanishes), but a weaker invariant can still be non-trivial:

\[\label{eq:ch-16-z2-definition} \nu \;\in\; \{0, 1\}.\]

The physical meaning of $\nu = 1$: there is an odd number of Kramers pairs of helical edge (2D) or surface (3D) states at the boundary.

Fu–Kane formula for 2D. For a 2D TR-symmetric insulator with $N$ occupied Kramers pairs, the $\mathbb Z_2$ index can be computed from the sewing matrix $w(\mathbf k)$ that relates the Kramers partners at $\mathbf k$ and at $-\mathbf k$:

\[\label{eq:ch-16-sewing} w_{mn}(\mathbf k) \;=\; \langle u_{m,\mathbf k} \mid \Theta \mid u_{n, -\mathbf k} \rangle , \qquad m, n \in \{1, \ldots, 2N\}.\]

The sewing matrix is antisymmetric at the four **time- reversal invariant momenta (TRIM) $\Lambda_a$ in the 2D BZ, $\mathbf \Lambda_a = (1/2)(n_1 \mathbf b_1 + n_2 \mathbf b_2)$ with $n_1, n_2 \in {0, 1}$, so its Pfaffian is well-defined at those four points. The Fu–Kane formula is

\[\label{eq:ch-16-fu-kane-2d} \boxed{ (-1)^\nu \;=\; \prod_{a=1}^{4} \frac{\sqrt{\det\Bigl[w(\Lambda_a)\Bigr]}}{\text{Pf}\Bigl[w(\Lambda_a)\Bigr]} . }\]

Each factor in the product is $\pm 1$ (the ratio of the square root of a determinant to a Pfaffian of an antisymmetric matrix is always $\pm 1$); the product of the four factors is the $\mathbb Z_2$ invariant. This is the working tool for computing $\nu$ from a DFT Wannier Hamiltonian.

Wilson-loop interpretation. Equation \eqref{eq:ch-16-fu-kane-2d} has an equivalent and more geometric form. Define the Wilson loop of the occupied subspace along $b_2$ as a function of $k_1$:

\[\label{eq:ch-16-wilson-loop} W(k_1) \;=\; \mathcal P \exp\!\left( i \int_{0}^{2\pi} \mathcal A_{k_2}(k_1, k_2)\, dk_2 \right) ,\]

where $\mathcal P$ is path-ordering and $\mathcal A_{k_2}$ is the non-Abelian Berry connection. The eigenvalues of $W(k_1)$ are $e^{i \theta_n(k_1)}$; the $\theta_n(k_1)$ are the Wannier charge centres (WCC). Plotting $\theta_n(k_1)$ as a function of $k_1$, a trivial insulator gives closed loops (each $\theta_n$ winds from $0$ to $2\pi$); a non-trivial insulator gives a set of loops with an odd number of them not closing:

\[\label{eq:ch-16-z2-wilson} \nu \;=\; \left( \frac{1}{2\pi} \sum_n \oint dk_1\, \partial_{k_1} \theta_n(k_1) \right) \!\!\mod 2 .\]

Equation \eqref{eq:ch-16-z2-wilson} is the practical $\mathbb Z_2$ test: count the WCC windings and take the parity. We demonstrate this in §16.7.2 and §16.9. Fu–Kane formula for 3D. A 3D TR-symmetric insulator has eight TRIM, $\Lambda_i = (1/2)(n_1 \mathbf b_1 + n_2 \mathbf b_2 + n_3 \mathbf b_3)$ with $n_1, n_2, n_3 \in {0, 1}$. The sewing-matrix product over all eight gives the strong index:

\[\label{eq:ch-16-fu-kane-3d-strong} (-1)^{\nu_0} \;=\; \prod_{i=1}^{8} \frac{\sqrt{\det\Bigl[w(\Lambda_i)\Bigr]}}{\text{Pf}\Bigl[w(\Lambda_i)\Bigr]} ,\]

and three weak indices $\nu_1, \nu_2, \nu_3 \in {0, 1}$ are defined by restricting the product to TRIM with a common value of $n_k$:

\[\label{eq:ch-16-fu-kane-3d-weak} (-1)^{\nu_k} \;=\; \prod_{i\,:\,n_k = 1} \frac{\sqrt{\det\Bigl[w(\Lambda_i)\Bigr]}}{\text{Pf}\Bigl[w(\Lambda_i)\Bigr]} , \qquad k = 1, 2, 3 .\]

A 3D TI with $\nu_0 = 1$ is a strong TI and has a single robust Dirac cone on every surface; a 3D TI with $\nu_0 = 0$ but some $\nu_k = 1$ is a weak TI, equivalent to a stack of 2D QSH layers, and its surface states are not robust to disorder that breaks the layer structure. Bi$_2$Se$_3$ (§16.5.2) is a strong TI with $\nu_0 = 1$, $\nu_1 = \nu_2 = \nu_3 = 0$.

Tip. The $\mathbb Z_2$ invariants are the unique robust topological classification of a TR-symmetric gapped band structure. Other classifications (topological crystalline insulators, §16.5.3) require additional crystal symmetries.

16.3 The integer quantum Hall effect

The IQHE is the original topological phenomenon in condensed-matter physics, and the simplest setting in which the Chern number appears as a measurable integer.

16.3.1 The setup

Consider a 2D electron gas (2DEG) — the $xy$-plane of a semiconductor heterostructure such as a GaAs/AlGaAs interface — in a perpendicular magnetic field $\mathbf B = B \hat{\mathbf z}$. The single-particle Hamiltonian is

\[\label{eq:ch-16-2deg-hamiltonian} \hat H \;=\; \frac{1}{2m} \left( \mathbf p + e \mathbf A \right)^2 , \qquad \mathbf A = (0, B x, 0)\]

in the Landau gauge. The eigenstates are the Landau levels

\[\label{eq:ch-16-landau-levels} E_n \;=\; \hbar \omega_c \left( n + \tfrac{1}{2} \right) , \qquad \omega_c = \frac{eB}{m} ,\]

each with a macroscopic degeneracy per unit area

\[\label{eq:ch-16-ll-degeneracy} n_B \;=\; \frac{eB}{h}\]

(the number of flux quanta per unit area). The filling factor is

\[\label{eq:ch-16-filling} \nu \;=\; \frac{n_s}{n_B} \;=\; \frac{n_s h}{eB} ,\]

where $n_s$ is the 2D electron density. The Hall conductivity at integer $\nu$ is the quantised value

\[\label{eq:ch-16-iqhe-sigma} \sigma_{xy} \;=\; \nu \frac{e^2}{h} , \qquad \rho_{xy} = \frac{h}{\nu e^2} .\]

This is the integer quantum Hall effect (von Klitzing, 1980; Nobel Prize 1985). The longitudinal resistance $\rho_{xx}$ vanishes at the same time; the system is a perfect Hall conductor.

16.3.2 The TKNN invariant

Thouless, Kohmoto, Nightingale, and den Nijs (1982) showed that the integer $\nu$ in \eqref{eq:ch-16-iqhe-sigma} is the sum of the Chern numbers of the occupied Landau levels:

\[\label{eq:ch-16-tknn} \boxed{ \sigma_{xy} \;=\; \frac{e^2}{h} \sum_{n\,:\,\text{occupied}} C_n . }\]

This is the TKNN invariant, the first explicit appearance of the Chern number in an observable. The RHS of \eqref{eq:ch-16-tknn} is the bulk band structure computed via \eqref{eq:ch-16-chern-number}; the LHS is the transport measurement. The bulk– boundary correspondence \eqref{eq:ch-16-bulk-boundary} is the reason the bulk Chern number fixes the transport coefficient: at the edge of the sample the Landau level bends up to higher energy (to match the vacuum), and the Fermi energy crosses the bent level an integer number of times, giving the chiral edge states that carry the quantised current.

Why is the IQHE topological? Two ways to see it. (i) Robustness to disorder: a smooth, local perturbation to $\hat H$ cannot change $C_n$ without closing the band gap; the disorder potential in a real 2DEG is far too weak to close the Landau gap of $\sim 10$ K, so the quantisation survives to ppm precision. (ii) The topological origin of the edge states: the chiral edge mode is a topologically- protected, uni-directional channel at the boundary; back- scattering is forbidden because there is no counter- propagating edge state to back-scatter into. The result is dissipationless transport.

16.3.3 The Hofstadter butterfly

A different manifestation of the topology of a 2D crystal in a magnetic field is the Hofstadter butterfly (Hofstadter, 1976): the energy spectrum of a tight-binding electron on a 2D lattice in a uniform magnetic field, as a function of the flux $\Phi$ per unit cell (in units of the flux quantum $\Phi_0 = h/e$). The spectrum is a fractal: a self-similar pattern of sub-bands and gaps that recursively splits as the flux per unit cell is increased.

The spectrum is the solution of the Harper equation

\[\label{eq:ch-16-harper} \psi_{n+1} + \psi_{n-1} + 2 \cos\!\Bigl( 2\pi n \phi + k_y \Bigr) \psi_n \;=\; \varepsilon_n(k_y)\, \psi_n ,\]

where $\phi = \Phi/\Phi_0$. The bands and gaps are labelled by a pair of integers $(s, t)$ via the Diophantine equation

\[\label{eq:ch-16-diophantine} r \;=\; s \phi + t ,\]

where $r$ is the number of sub-bands below the gap and $(s, t)$ are topological integers. $s$ is the Chern number of the sub-band just below the gap, and $t$ is related to the Hall conductivity in the gap. The Diophantine equation is the gap-labelling theorem: every gap in the Hofstadter spectrum is labelled by an integer pair $(s, t)$ that satisfies \eqref{eq:ch-16-diophantine} for some integer $r$.

Tip. The Hofstadter butterfly is realised in moiré superlattices of bilayer graphene (h-BN-encapsulated, twisted at the magic angle) and in optical lattices of cold atoms. The Chern numbers of the sub-bands in twisted-bilayer graphene at certain flux values are the origin of the fractional Chern insulator states observed in 2023. ## 16.4 The quantum spin Hall effect and 2D topological insulators

In 2005, Kane and Mele showed that a 2D time-reversal- symmetric insulator can be non-trivial in a way the IQHE is not. The quantum spin Hall (QSH) effect is the TR-symmetric analogue of the QAH: a bulk insulator with helical edge states protected by TRS. The 2007 work of Bernevig, Hughes, and Zhang (BHZ) predicted the first realistic material platform — the HgTe/CdTe quantum well — and the experimental confirmation by König et al. in 2007 launched the field of 2D and 3D TIs.

16.4.1 The Kane–Mele model

The Kane–Mele model is the spin–orbit extension of the honeycomb-lattice tight-binding model of graphene (see chapter 11’s worked example). The model has four terms:

\[\label{eq:ch-16-kane-mele-hamiltonian} \hat H \;=\; \hat H_0 + \hat H_\text{SO} + \hat H_R + \hat H_\text{staggered} .\]

The four terms in detail:

\[\begin{align} \label{eq:ch-16-kane-mele-terms} \hat H_0 &= -t \sum_{\langle i j \rangle, \sigma} c^\dagger_{i\sigma} c_{j\sigma} , \\\\[4pt] \hat H_\text{SO} &= i \lambda_\text{SO} \sum_{\langle\langle i j \rangle\rangle, \sigma} \nu_{ij}\, c^\dagger_{i\sigma} \sigma^z_{\sigma\sigma'}\, c_{j\sigma'} , \\\\[4pt] \hat H_R &= i \lambda_R \sum_{\langle i j \rangle, \sigma\sigma'} c^\dagger_{i\sigma} \Bigl( \mathbf s_{\sigma\sigma'} \times \mathbf d_{ij} \Bigr)_z\, c_{j\sigma'} , \\\\[4pt] \hat H_\text{staggered} &= \lambda_v \sum_{i, \sigma} \nu_i\, c^\dagger_{i\sigma} c_{i\sigma} . \end{align}\]

In \eqref{eq:ch-16-kane-mele-terms}:

The Kane–Mele model is the paradigm of a 2D Z$2$ topological insulator. In the limit $\lambda_R = 0$ and $\lambda_v = 0$, the Hamiltonian is block-diagonal in spin (the spin-up and spin-down sectors are decoupled copies of the Haldane model with fluxes $\pm \Phi$ on the two sublattices), and each block has a Chern number $C\uparrow = +1$, $C_\downarrow = -1$. The total Chern number vanishes (TRS is preserved), but the spin Chern number $C_s = (C_\uparrow - C_\downarrow)/2 = 1$ is non-zero — one of several equivalent characterisations of the $\mathbb Z_2$ index in TR-symmetric systems.

Topological phase diagram. The $\lambda_\text{SO}$– $\lambda_v$ plane has three phases:

Phase $\lambda_\text{SO}$ vs. $\lambda_v$ $\nu$ Surface state
Quantum spin Hall $\lambda_\text{SO} > 3\sqrt{3} \lambda_v$ $1$ Helical edge modes
Band insulator $\lambda_\text{SO} < 3\sqrt{3} \lambda_v$ $0$ None
Gapless $\lambda_\text{SO} = 3\sqrt{3} \lambda_v$ — Dirac point closes

The bulk gap at $\lambda_v = 0$ is $\Delta = 6\sqrt{3}\, \lambda_\text{SO}$; the gap closes at the critical ratio $\lambda_\text{SO} = 3\sqrt{3}\, \lambda_v$.

16.4.2 The Bernevig–Hughes–Zhang (BHZ) model

The BHZ model (Bernevig, Hughes, Zhang, 2006) is a four- band effective Hamiltonian for the HgTe/CdTe quantum well. The two relevant orbitals are the $s$-like $\Gamma_6$ and the $p$-like $\Gamma_8$ (heavy-hole), with spin degeneracy giving a four-band model:

\[\label{eq:ch-16-bhz} H(\mathbf k) \;=\; \begin{pmatrix} h(\mathbf k) & 0 \\\\ 0 & h^*(-\mathbf k) \end{pmatrix} , \qquad h(\mathbf k) \;=\; \mathbf d(\mathbf k) \cdot \boldsymbol \sigma ,\]

with $h(\mathbf k)$ acting in the orbital subspace ${ |\Gamma_6, \uparrow\rangle, |\Gamma_8, \uparrow\rangle }$ and $h^*(-\mathbf k)$ the time-reversed partner (so $H(\mathbf k)$ is TR-symmetric), and

\[\begin{align} \label{eq:ch-16-bhz-d} d_a(\mathbf k) &= A \sin k_a , \quad a = x, y , \\\ d_z(\mathbf k) &= M - B (k_x^2 + k_y^2) . \end{align}\]

The topology of the BHZ model is controlled by the mass parameter $M$. Inverting the sign of $M$ (relative to $B > 0$) drives a band inversion at $\Gamma$ — the $\Gamma_8$ band moves above the $\Gamma_6$ band — and the gap closes at $M = 0$. For $M/B < 0$ the model is a quantum spin Hall insulator with $\nu = 1$.

The BHZ model is the first 2D TI predicted and observed in a real material. König et al. (Science 318, 766 (2007)) measured the quantised conductance $G = 2 e^2 / h$ in a HgTe/(Hg,Cd)Te quantum well of critical thickness $d_c \sim 6.5$ nm: for $d < d_c$ the well is a trivial insulator; for $d > d_c$ it is a QSH insulator. The transport signature is the non-local voltage in an H-bar geometry, quantised at $2 e^2/h$ in the QSH phase.

Note. The factor of 2 is the spin degeneracy: one helical edge mode (a Kramers pair of counter-propagating electrons with opposite spin) contributes $e^2/h$ per spin, total $2 e^2/h$.

16.4.3 The $\mathbb Z_2$ invariant from time-reversal polarisation

A more physically transparent derivation of the 2D $\mathbb Z_2$ invariant, due to Fu and Kane (2006), uses the time-reversal polarisation $\mathcal P$ at the four TRIM. For a 1D TR-symmetric insulator the polarisation $P$ is only defined modulo $e/2$; the time-reversed polarisation is $-P$. The change in polarisation $\Delta P$ across a TR-symmetric deformation satisfies

\[\label{eq:ch-16-tr-pol} \Delta P \;=\; 0 \;\text{ or }\; e/2 \pmod{e} .\]

The two possibilities are the two values of the $\mathbb Z_2$ index of a 1D TR-symmetric insulator.

Extending to 2D: take the two 1D systems obtained by slicing the 2D BZ at $k_y = 0$ and $k_y = \pi$. They have 1D polarisations $P(0)$ and $P(\pi)$. The $\mathbb Z_2$ index of the 2D insulator is

\[\label{eq:ch-16-fu-kane-pol} \nu \;=\; 2 \Bigl[ P(\pi) - P(0) \Bigr] \;\mod 2 \;\in\; \{0, 1\} .\]

The factor of 2 converts the continuous 1D polarisation to a $\mathbb Z_2$ invariant. A non-trivial $\nu = 1$ implies an odd number of edge-state crossings of the Fermi energy between $k_y = 0$ and $k_y = \pi$ — the helical edge mode.

The Wilson-loop formulation \eqref{eq:ch-16-z2-wilson} is the continuous version of this: the Wannier charge centres $\theta_n(k_1)$ are the 1D polarisation of the slice at fixed $k_1$, and the parity of their winding is the $\mathbb Z_2$ invariant. The WCC method is the standard numerical route (WannierTools, Z2Pack, see §16.7.2).

16.5 3D topological insulators

The 3D extension of the QSH state was predicted by Fu, Kane, and Mele (2007) and by Moore and Balents (2007). The first concrete material realisation was Bi$_2$Se$_3$ (Zhang et al., Nature Physics 5, 438 (2009)). The 3D TI has a single, *robust Dirac cone on every surface, with spin–momentum locking that prevents back-scattering by non-magnetic disorder.

16.5.1 Strong and weak TIs

As derived in §16.2.4, a 3D TR-symmetric insulator has four $\mathbb Z_2$ indices: one strong $\nu_0$ and three weak $\nu_1, \nu_2, \nu_3$, defined via \eqref{eq:ch-16-fu-kane-3d-strong} and \eqref{eq:ch-16-fu-kane-3d-weak}. The physical meanings:

The strong TI is a genuinely 3D topological phase. The weak TI is, in some sense, a “1D stack” of 2D TIs.

16.5.2 The Bi$_2$Se$_3$ family

Bi$2$Se$_3$ is the *canonical* 3D TI. It crystallises in the tetradymite structure (space group $D{3d}^5$, $R\bar 3 m$, No. 166) with five-atom layers (Se–Bi–Se–Bi–Se) stacked along $c$ and held together by van der Waals forces. The band inversion is between the Bi $6p$ and Se $4p$ states at $\Gamma$, driven by the atomic SOC of the heavy Bi ($Z_\text{Bi} = 83$).

The minimal four-band effective Hamiltonian at $\Gamma$ is (Zhang et al., Nature Physics 5, 438 (2009))

\[\label{eq:ch-16-bi2se3-hamiltonian} H_\text{eff}(\mathbf k) \;=\; \epsilon_0(\mathbf k) \mathbb 1_{4} + \begin{pmatrix} M(\mathbf k) & A_1 k_z & 0 & A_2 k_- \\\\ A_1 k_z & -M(\mathbf k) & A_2 k_- & 0 \\\\ 0 & A_2 k_+ & M(\mathbf k) & -A_1 k_z \\\\ A_2 k_+ & 0 & -A_1 k_z & -M(\mathbf k) \end{pmatrix} ,\]

with $M(\mathbf k) = M_0 - B_1 k_z^2 - B_2 (k_x^2 + k_y^2)$, $\epsilon_0(\mathbf k) = C + D_1 k_z^2 + D_2 (k_x^2 + k_y^2)$, and $k_\pm = k_x \pm i k_y$. The basis is the four Kramers pairs of the Bi / Se $p_z$ combinations. The sign of $M_0 / B_1$ controls the topology: $M_0 / B_1 > 0$ gives a strong TI (Bi$_2$Se$_3$); $M_0 / B_1 < 0$ gives a trivial insulator (Sb$_2$Se$_3$).

The Bi$_2$Se$_3$ surface hosts a single Dirac cone at $\bar\Gamma$, with $E(\mathbf k) = \pm v_F |\mathbf k|$ and $v_F \approx 5 \times 10^5$ m/s. The surface state exhibits spin–momentum locking, $\langle \mathbf s \rangle \propto \hat{\mathbf z} \times \mathbf k$: an electron with momentum $\mathbf k$ cannot back-scatter to $-\mathbf k$ because the spin would have to flip and a non-magnetic impurity cannot flip it. The result is a long mean-free path, the basis of proposed low-dissipation “topological” electronics.

Note. The existence of the surface state is topologically protected; its energy (relative to the bulk gap) is not. If the Fermi energy is in the bulk conduction band (as in naturally $n$-type Bi$_2$Se$_3$ due to Se vacancies), the surface Dirac cone is “buried” under the bulk conduction. Doping the bulk to push $E_F$ into the gap is the experimental challenge.

16.5.3 Topological crystalline insulators (TCIs)

A topological crystalline insulator is an insulator whose gapless surface states are protected by a crystal symmetry (a mirror, a rotation, a glide, or a screw) rather than by TRS. The defining example is SnTe (and the alloy Pb${1-x}$Sn$_x$Te, Pb${1-x}$Sn$_x$Se). SnTe has the rock-salt structure; the two relevant band edges at the four equivalent $L$ points of the BZ are inverted by the spin–orbit coupling, and the mirror symmetry of the (110) plane protects the surface states on the (001) surface. SnTe was the first material predicted to be a TCI (Fu, Phys. Rev. Lett. 106, 106802 (2011)) and the first one observed in ARPES (Tanaka et al., Nature Physics* 8, 800 (2012)).

The classification of TCIs uses the symmetry-based indicators of Po et al. and Song et al. (2017–2018), which extend the $\mathbb Z_2$ machinery of Fu–Kane to arbitrary space-group symmetries. A TCI can have multiple Dirac cones on the surface (SnTe has four, one on each $\{ 111 \}$ face and on the (001) face) and higher-degeneracy surface states (four-, six-, eight-fold) at high-symmetry points.

Tip. The modern “topological quantum chemistry” framework (Bradlyn et al., Science 353, 6299 (2016)) gives a complete classification of all band structures consistent with a given space-group symmetry, including TCIs, Dirac, Weyl, nodal-line, and multi-fold semimetals. Tools: the Bilbao Crystallographic Server’s TOPOLOGICAL QUANTUM CHEMISTRY' and the CheckTopologicalMat`’ Python package.

16.6 Topological semimetals

The topological materials we have considered so far are insulators: they have a bulk gap. There is a parallel family of topological semimetals whose protection requires the touching of two (or more) bulk bands at discrete points or along lines in the BZ. The three main families are the Dirac semimetals (four-fold crossings, TRS and inversion preserved), the Weyl semimetals (two-fold crossings, TRS or inversion broken), and the nodal-line semimetals (lines of two-fold crossings protected by mirror or PT symmetry).

16.6.1 Dirac semimetals

A 3D Dirac semimetal has a four-fold band crossing near the Fermi energy, with the low-energy Hamiltonian

\[\label{eq:ch-16-dirac-hamiltonian} H_\text{Dirac}(\mathbf q) \;=\; \begin{pmatrix} m & v \, \mathbf q \cdot \boldsymbol \sigma \\\\ v\, \mathbf q \cdot \boldsymbol \sigma & -m \end{pmatrix} ,\]

a 4×4 matrix in the basis of the two Kramers pairs. The mass parameter $m$ measures the separation of the four bands; at $m = 0$ the four bands cross at $\mathbf q = 0$ and the spectrum is the 3D massless Dirac fermion, $E(\mathbf q) = \pm v |\mathbf q|$ (each sign doubly degenerate). The Dirac node is protected by TRS + inversion + an extra crystal rotation symmetry ($C_3$ along [111] in Na$_3$Bi, $C_4$ in Cd$_3$As$_2$); without the rotation the two two-fold representations of the little group at $\Gamma$ would hybridise and open a gap.

Two materials realise the Dirac semimetal phase:

The Dirac node is a doubly-degenerate Weyl node: a superposition of two Weyl nodes of opposite chirality, sitting on top of each other in momentum space. Breaking the crystal symmetry (or breaking either TRS or inversion) splits the Dirac node into a pair of Weyl nodes, and the system becomes a Weyl semimetal.

16.6.2 Weyl semimetals

A 3D Weyl semimetal has a two-fold band crossing near the Fermi energy, with the low-energy Hamiltonian

\[\label{eq:ch-16-weyl-hamiltonian} H_\text{Weyl}(\mathbf q) \;=\; v\, \mathbf q \cdot \boldsymbol \sigma ,\]

a 2×2 matrix in the basis of a single (spinless) Weyl fermion. The spectrum is $E(\mathbf q) = \pm v |\mathbf q|$, a three-dimensional massless Dirac fermion with a single chirality. The Weyl node is **topologically protected: the small sphere of $\mathbf k$-space around the node is a 2-sphere, and the Chern number of that sphere is non-zero,

\[\label{eq:ch-16-weyl-chern} C_\text{Weyl} \;=\; \frac{1}{2\pi} \oint_{S^2} \boldsymbol \Omega(\mathbf k) \cdot d\mathbf S \;=\; \pm 1 .\]

The sign is the chirality of the Weyl node: $+1$ is a right-handed Weyl fermion, $-1$ is left-handed. The chirality is a property of the *Hamiltonian, not of the state, and it is conserved by smooth deformations that preserve the gap elsewhere.

Nielsen–Ninomiya theorem (the “no-go” theorem for Weyl fermions in a BZ): the sum of the chiralities of all Weyl nodes in the Brillouin zone is zero. Weyl nodes come in pairs of opposite chirality related by the remaining symmetry when TRS or inversion is broken. The net rule: the BZ always contains an even number of Weyl nodes, with equal numbers of left- and right-handed chiralities.

The two “textbook” Weyl semimetals are

The defining experimental signature is the Fermi arc: a surface state connecting the projections of two bulk Weyl nodes of opposite chirality on the surface BZ. The arc is open (a chord, not a loop); it merges into the bulk Weyl cone at each end.

16.6.3 Nodal-line semimetals

A nodal-line semimetal has a line (1D) of two-fold band crossings in the BZ, rather than an isolated point. The low-energy Hamiltonian near a point on the line is the 2D massless Dirac Hamiltonian in the plane perpendicular to the line:

\[\label{eq:ch-16-nodal-line} H(\mathbf q) \;=\; v_\perp\, (q_x \sigma_x + q_y \sigma_y) ,\]

where $\mathbf q$ is the displacement from the line, $q_x, q_y$ are the components perpendicular to the line, and $\sigma_x, \sigma_y$ are Pauli matrices in the two-band subspace. The dispersion is $E = \pm v_\perp \sqrt{q_x^2 + q_y^2}$ in the perpendicular plane; along the line the two bands are degenerate and their dispersion is flat (if the Hamiltonian has no $k_z$ dependence on the line) or linear (if it does).

Nodal lines are protected by mirror symmetry, PT symmetry (the product of parity and TR), or — in non- centrosymmetric, non-magnetic crystals — by an antiunitary symmetry that squares to $+1$ (e.g. the product of mirror and TR). When the protecting symmetry is broken (e.g. by spin–orbit coupling that breaks mirror symmetry), the nodal line gaps out into a pair of Weyl nodes, and the system becomes a Weyl semimetal. This is the symmetry-breaking pathway between nodal-line and Weyl phases.

Material realisations of nodal-line semimetals include ZrSiS (nodal line protected by the non-symmorphic symmetry of the body-centred tetragonal lattice, lying close to the Fermi energy), PbTaSe$_2$ (an asymmetric, TR-symmetric mirror-protected nodal line), and the antiperovskite Cu$_3$PdN family (TR-symmetric mirror-protected nodal lines around $\Gamma$).

The family of semimetals covered in this section is the gapless branch of the topology zoo. The full zoo — the gapped branch and the gapless branch, together — is summarised below.

%%{init: {'flowchart': {'htmlLabels': true, 'curve': 'basis'}}}%%
graph TD
  TOP["Topological<br/>materials"]
  GAPPED["Gapped<br/>(insulators)"]
  GAPLESS["Gapless<br/>(semimetals)"]
  CHERN["Chern insulators<br/>(sigma_xy = C e^2/h)"]
  QAH["Quantum anomalous<br/>Hall (QAH)"]
  QSH["Quantum spin Hall<br/>(Kane-Mele, BHZ)"]
  WEYL2["Weyl semimetal<br/>(2-fold)"]
  DIRAC2["Dirac semimetal<br/>(4-fold)"]
  TCI["Topological crystalline<br/>insulators (SnTe)"]
  STRONG["Strong 3D TI<br/>(Bi2Se3)"]
  WEAK["Weak 3D TI<br/>(stacked 2D)"]
  NODAL["Nodal-line<br/>semimetal"]
  HOTI["Higher-order TI<br/>(Bi4Br4)"]
  HOTI2["HOTI 2D<br/>(corner states)"]
  HOTI3["HOTI 3D<br/>(hinge states)"]

  TOP --> GAPPED
  TOP --> GAPLESS
  GAPPED --> CHERN
  GAPPED --> QSH
  GAPPED --> STRONG
  GAPPED --> WEAK
  GAPPED --> TCI
  GAPPED --> HOTI
  GAPLESS --> DIRAC2
  GAPLESS --> WEYL2
  GAPLESS --> NODAL
  CHERN --> QAH
  HOTI --> HOTI2
  HOTI --> HOTI3
  QSH --> STRONG

The diagram makes the relationship explicit: the Chern and QSH families are the 2D parents of the 3D TIs; the Dirac family is the 4-fold (TRS+inversion preserved) parent of the Weyl family (TRS or inversion broken); the TCI family extends the strong TI family with crystal-symmetry protection; the HOTI family is the “next-order” extension of the standard TI. Each branch has its own invariant — Chern number, $\mathbb Z_2$ index, mirror Chern number, line-winding number, multipole moment — and each invariant has its own computational recipe in WannierTools / Z2Pack / BANDREP.

16.6.4 The chiral anomaly and the negative magnetoresistance

The chiral anomaly is a defining property of massless Weyl fermions in 3D. The chiral current $j_5^\mu = \bar\psi \gamma^\mu \gamma^5 \psi$ is classically conserved, but quantum-mechanically the conservation fails in the presence of electromagnetic fields:

\[\label{eq:ch-16-chiral-anomaly} \boxed{ \partial_\mu j_5^\mu \;=\; \frac{e^2}{16 \pi^2 \hbar^2}\, \epsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma} \;=\; \frac{e^2}{2 \pi^2 \hbar^2}\, \mathbf E \cdot \mathbf B . }\]

The right-hand side is non-zero when $\mathbf E \parallel \mathbf B$: parallel fields pump right-handed Weyl fermions from the left-handed node to the right-handed one (or vice versa) at a rate $\propto \mathbf E \cdot \mathbf B$. The separate chiral charge is not conserved, but the total electric charge is.

In a Weyl semimetal, the chiral anomaly is observed in two ways:

  1. The chiral magnetic effect. A magnetic field $\mathbf B$ along, say, the $z$ direction creates a population imbalance between the two chiralities. The resulting chiral chemical potential $\mu_5$ drives a current along $\mathbf B$,
\[\label{eq:ch-16-chiral-magnetic} \mathbf j \;=\; \frac{e^2 \mu_5}{2 \pi^2 \hbar^2}\, \mathbf B .\]

This is the “chiral magnetic effect” (CME), predicted in 1983 in high-energy physics and observed in Weyl semimetals in 2015 by Li et al. and Xiong et al.

  1. The negative longitudinal magnetoresistance (LMR). When an electric field $\mathbf E$ is applied parallel to $\mathbf B$, the chiral anomaly pumps charge from one chirality to the other. The pumped charge reduces the resistance because the back- scattering between the two chiralities is suppressed (the populations are unequal). The result is a negative longitudinal magnetoresistance: as $B$ increases, the longitudinal resistance decreases. This was the first experimental signature of Weyl fermions in TaAs, observed in 2015. > Note. The negative LMR is not a unique signature of

    the chiral anomaly — current jetting and other classical effects can also produce a negative LMR. The unambiguous signature is the planar Hall effect: in an in-plane $\mathbf B$ and a perpendicular $\mathbf E$ the planar Hall voltage is odd under $\mathbf B \to -\mathbf B$ and has a characteristic $\sin(2\theta)$ dependence on the angle $\theta$ between $\mathbf E$ and $\mathbf B$ (the “Burkov” prediction). This signature has no classical analogue.

16.7 Computing topology from DFT

The previous sections were theory — Berry phase, Chern numbers, $\mathbb Z_2$ indices. The natural question is: how do we compute these from a DFT band structure? The DFT gives us a Kohn–Sham Hamiltonian $H(\mathbf k)$ on a discrete $\mathbf k$-mesh; we want a Chern number or a $\mathbb Z_2$ index. The answer is a sequence of three steps, each a piece of standard, well-tested software.

16.7.1 Wannier functions from Bloch functions

The DFT output is a set of Bloch functions $|u_{n\mathbf k}\rangle$ for $N$ bands on a discrete $\mathbf k$-mesh, delocalised in real space. A more convenient basis is the set of Wannier functions, localised in real space. For a set of $N$ isolated bands, the Wannier functions are

\[\label{eq:ch-16-wannier-def} | w_{n\mathbf R} \rangle \;=\; \frac{V}{(2\pi)^3} \int_{\text{BZ}} d^3k\, e^{-i \mathbf k \cdot \mathbf R} \sum_{m=1}^{N} U_{mn}(\mathbf k)\, | u_{m\mathbf k} \rangle ,\]

where the integral is over the first BZ and $U_{mn}(\mathbf k)$ is a unitary matrix that mixes the $N$ bands at each $\mathbf k$. The Wannier functions depend on the choice of $U(\mathbf k)$; a trivial choice $U = \mathbb 1$ generally does not produce localised Wannier functions. The maximally-localised Wannier functions (MLWFs) of Marzari and Vanderbilt (1997) are obtained by choosing $U(\mathbf k)$ to minimise the total spread

\[\label{eq:ch-16-wannier-spread} \Omega \;=\; \sum_{n=1}^{N} \left( \langle w_{n\mathbf 0} | r^2 | w_{n\mathbf 0} \rangle - \langle w_{n\mathbf 0} | \mathbf r | w_{n\mathbf 0} \rangle^2 \right) .\]

The minimisation is a non-linear problem in $U(\mathbf k)$, solved by successive unitary rotations on pairs of Wannier functions. The standard tool is Wannier90 (Mostofi et al., Comput. Phys. Commun. 178, 685 (2008)). The input is the Bloch functions at a uniform $\mathbf k$-mesh; the output is the $U(\mathbf k)$ matrices, the MLWF centres, and the Wannier-interpolated Hamiltonian $H^{(\text{W})}(\mathbf k)$ on an arbitrarily fine $\mathbf k$-mesh. The interpolation is linear in the number of Wannier functions, much cheaper than a direct DFT calculation on a fine $\mathbf k$-mesh.

Tip. For isolated sets of bands (e.g. the $d$ bands of a transition-metal oxide) the disentanglement is trivial and the initial guess can be random. For an entangled manifold (e.g. the $sp$ bands of a wide-gap insulator) one must specify a “frozen” outer window and a “disentangled” inner window; the spread minimisation is performed only on the inner window.

16.7.2 Wilson loops and the Wannier charge centres

Given the Wannier-interpolated Hamiltonian $H^{(\text{W})}(\mathbf k)$ on a fine $\mathbf k$-mesh, we can compute the **Wilson loop along a non-contractible loop of the BZ torus. The practical algorithm proceeds in three steps:

  1. Discretise the loop. Pick a closed 1-D loop $\mathcal C$ in the BZ (for a 2D insulator, a line of fixed $k_\perp$ that wraps the BZ in the $k_\parallel$ direction). Discretise the loop into $L$ points $\mathbf k_1, \ldots, \mathbf k_L$ with equal spacing $\Delta k = 2\pi / L$ along the loop.
  2. Compute overlap matrices. For each pair of adjacent $\mathbf k$-points, compute the $N \times N$ overlap matrix of the cell-periodic Bloch functions:
\[\label{eq:ch-16-overlap} M_{\mathbf k, \mathbf k + \Delta \mathbf k}^{(mn)} \;=\; \langle u_{m, \mathbf k} | u_{n, \mathbf k + \Delta \mathbf k} \rangle , \quad m, n \in \text{occ} .\]

In the Wannier basis, the overlap is computed from $H^{(\text{W})}(\mathbf k)$: diagonalise $H^{(\text{W})}(\mathbf k)$ to get $|u_{m, \mathbf k}\rangle$, then form the overlap.

  1. Form the Wilson loop. Multiply the overlap matrices along the loop:
\[\label{eq:ch-16-wilson-discrete} W[\mathcal C] \;=\; \prod_{i=1}^{L} M_{\mathbf k_i, \mathbf k_{i+1}} , \qquad \mathbf k_{L+1} = \mathbf k_1 .\]

The eigenvalues of $W[\mathcal C]$ are $e^{i \theta_n[\mathcal C]}$, where the $\theta_n[\mathcal C]$ are the Wannier charge centres (WCC) of the loop $\mathcal C$.

The WCC as a function of the perpendicular momentum $k_\perp$ is the diagnostic plot. A trivial insulator gives a set of closed loops (each $\theta_n$ winds from $0$ to $2\pi$ as $k_\perp$ traverses the BZ, continuously); a non-trivial insulator gives a set of loops with an odd number of them not closing (a “jump” in the WCC of size $2\pi$ between $k_\perp = 0$ and $k_\perp = 2\pi$). The parity of the jumps is the $\mathbb Z_2$ invariant, as in \eqref{eq:ch-16-z2-wilson}.

The algorithm is implemented in WannierTools (Wu et al., *Comput. Phys. Commun. 224, 405 (2018)), which takes the Wannier90 output and computes the Wilson loop, the WCC, and (for a 3D insulator) the surface state spectrum in a slab geometry. The standard input is a `wt.in’ file specifying the Wannier90 data, the $\mathbf k$-path for the Wilson loop, and the slab geometry for the surface states.

16.7.3 Surface state calculations in slab geometry

The most intuitive way to confirm that a DFT band structure is topologically non-trivial is to plot the surface state. The calculation is done in a slab geometry: a finite-thickness slab (typically 20–100 unit cells) with vacuum on either side. The slab has a 2D BZ (the projection of the 3D BZ onto the surface plane) and the band structure is a set of $\varepsilon_n(\mathbf k_\parallel)$. The topological surface state, if it exists, appears as a gap-crossing band that lives inside the bulk gap.

The slab calculation can be done directly with a DFT code that supports non-periodic boundary conditions in one direction (VASP, Quantum ESPRESSO, SIESTA, FLEUR, WIEN2k). The procedure:

  1. Construct a supercell with $N$ unit cells of the bulk and $\sim 20$ Å of vacuum on each side; periodic in the surface plane, finite perpendicular.
  2. Standard SCF calculation to converge the density.
  3. Non-SCF calculation along a 1D $k_\parallel$ path (e.g. $\bar\Gamma$–$\bar M$–$\bar K$–$\bar\Gamma$ for a hexagonal surface).
  4. Project the slab bands onto the bulk band structure. The surface state is the band that lies in the bulk gap and is localised at the surface.

For a 3D TI, the slab calculation should show a single Dirac cone at $\bar\Gamma$ (for the (001) surface of Bi$_2$Se$_3$) crossing the bulk gap — the signature of the strong TI phase.

Note. The slab is not periodic in the perpendicular direction; the surface state is not a Bloch state of the bulk crystal, but a localised state decaying into the bulk on a length scale of $\sim 1$–$10$ nm. The slab is the most direct way to see this localised state.

16.7.4 The surface Green’s function method

The slab calculation is intuitive but expensive: a 40-cell slab has 40× the atoms of a single bulk cell. The surface Green’s function method computes the same surface spectrum at a cost comparable to a single bulk calculation.

The setup is a semi-infinite crystal occupying the half- space $z > 0$ with surface at $z = 0$. The surface Green’s function is

\[\label{eq:ch-16-surface-gf} G_\text{surf}(\mathbf k_\parallel, \omega) \;=\; \Bigl[ \omega - H_\text{slab}(\mathbf k_\parallel) - \Sigma(\mathbf k_\parallel, \omega) \Bigr]^{-1} ,\]

where $H_\text{slab}$ is the surface unit-cell Hamiltonian and $\Sigma$ is the self-energy of the semi-infinite bulk:

\[\label{eq:ch-16-self-energy} \Sigma(\mathbf k_\parallel, \omega) \;=\; \Bigl[ \omega - H_\text{bulk}(\mathbf k_\parallel) - V \Bigr]^{-1}_{\text{boundary}} ,\]

with $V$ the surface-to-bulk coupling. The surface spectral function

\[\label{eq:ch-16-spectral-surface} A(\mathbf k_\parallel, \omega) \;=\; -\frac{1}{\pi}\, \text{Im}\, \text{Tr}\, G_\text{surf}(\mathbf k_\parallel, \omega + i 0^+)\]

has delta-function peaks at the energies of true surface states (Tamm states localised at $z = 0$) and of the projected bulk states. The topological surface state is the delta peak that lies inside the bulk gap and is not in the projected bulk spectrum.

The surface Green’s function is implemented in WannierTools (for Wannier-interpolated Hamiltonians) and in Z2Pack (for tight-binding or first-principles Hamiltonians in the projector form). Z2Pack in particular is a general-purpose topological-invariant calculator (Grubić et al., Phys. Rev. B 102, 245132 (2020)) that computes the Chern number, the $\mathbb Z_2$ index, and the surface spectrum of any gapped (or nodal) band structure, with a clean Python interface.

Tip. The relation between the slab calculation and the surface Green’s function is the same as the relation between the finite-cluster calculation and the Dyson equation: the slab is a real-space finite region; the surface Green’s function is the self-energy-dressed surface layer. The two should agree in the limit of an infinitely thick slab, but the surface Green’s function is much cheaper to compute and does not suffer from finite-size effects (the “finite-thickness” splitting of the two surfaces of a slab).

16.7.5 Hybrid Wannier centres for Chern numbers

The WCC method of §16.7.2 computes the $\mathbb Z_2$ invariant. For a Chern number (a 2D insulator with broken TRS), the analogous construction is the hybrid Wannier centre plot (Marzari & Vanderbilt, 1997; further developed in the modern “Wilson-loop” picture by Yu et al., 2011).

The setup is identical to the $\mathbb Z_2$ case: a discretised 1-D loop in the BZ, an overlap matrix, and a Wilson loop matrix $W$. The Chern number is the winding number* of the WCC as $k_\perp$ traverses the BZ:

\[\label{eq:ch-16-chern-wilson} C \;=\; \frac{1}{2\pi} \sum_{n=1}^{N_\text{occ}} \int_0^{2\pi} dk_\perp\, \partial_{k_\perp} \theta_n(k_\perp) \;=\; \frac{1}{2\pi} \sum_{n=1}^{N_\text{occ}} \Bigl[ \theta_n(2\pi) - \theta_n(0) \Bigr] .\]

The total winding number is the Chern number. For a trivial insulator $C = 0$ and the WCC loops are closed; for a Chern insulator $C = \pm 1$ and the WCC loops are open with a net winding of $\pm 2\pi$ across the $k_\perp$ window.

Note. For a 3D Chern insulator (a “3D quantum Hall state”), the Chern number is defined for each 2D slice of the BZ at fixed $k_z$; the full 3D invariant is the sum over slices. The “axion insulator” has vanishing total Chern number but a non-zero magnetoelectric polarisability $\theta = \pi$.

The end-to-end workflow of a topological-materials calculation starting from a DFT band structure is summarised below. The diagram is the practical counterpart of the mathematical roadmap in §16.1 — it is the recipe a code like WannierTools or Z2Pack* follows.

%%{init: {'flowchart': {'htmlLabels': true, 'curve': 'basis'}}}%%
graph TD
  A["Bulk DFT+SOC<br/>(PBE / SCAN / HSE06)"]
  B["Bloch functions<br/>|u_{n,k}> on k-mesh"]
  C["Wannier90:<br/>maximally-localised WFs"]
  D["Wannier-interpolated<br/>Hamiltonian H^W(k)"]
  E["Wilson loop<br/>(Wannier charge centres)"]
  F["Z2 invariant nu<br/>(Fu-Kane or WCC plot)"]
  G["Slab calculation<br/>(DFT, 20-100 cells)"]
  H["Surface Green's function<br/>(WannierTools / Z2Pack)"]
  I["Surface spectral function<br/>A(k_par, omega)"]
  J{"Topological<br/>invariant"}
  K["Chern number C<br/>(for Chern insulators)"]
  L["Strong + weak Z2<br/>(for 3D TIs)"]
  M["Hybrid WCC<br/>(winding number)"]
  N["Plot surface<br/>Dirac cone"]

  A --> B
  B --> C
  C --> D
  D --> E
  D --> H
  E --> J
  H --> I
  A --> G
  G --> N
  J --> K
  J --> L
  E --> M
  M --> K
  I --> N

The two branches of the diagram are the bulk-only route (‘A → B → C → D → E → J’) and the surface route (‘G → N’ or ‘H → I → N`). The first gives the topological invariant of the bulk (Chern number or $\mathbb Z_2$ index) without ever computing a surface; the second gives the direct visualisation of the protected boundary state. The bulk invariant is cheaper and robust to surface imperfections; the surface calculation is more intuitive and is the direct comparison with ARPES.

16.8 Higher-order topological insulators (HOTIs)

The bulk–boundary correspondence of §16.1 is a “first-order” correspondence: a $d$-dimensional insulator with non-trivial bulk topology has a $(d-1)$-dimensional gapless boundary state. A higher-order topological insulator (HOTI) breaks this rule: a $d$-dimensional HOTI of order $n$ has a gapless $(d-n)$-dimensional boundary, with all higher-dimensional boundaries gapped. The first explicit HOTI models (Benalcazar, Bernevig, Hughes, 2017) and the first realisation (Bi$_4$Br$_4$ in 2018) launched the sub-field.

Examples. A 2D second-order TI has corner states (zero-dimensional), protected by $C_n$ rotation or mirror symmetry; the invariant is a quantised electric quadrupole moment $q_{xy} = e/2$. A 3D second-order TI has 1D hinge states (chiral modes running along the edges of a 3D sample). Prominent material candidates:

The HOTI invariant is the “multipole moment” of the bulk Wannier centres, computed from the Wannier centres in the same way as the Chern number from the WCC.

Note. HOTIs are a young sub-field. The first complete classification is by Song et al. (Nature Communications* 9, 3530 (2018)) and Khalaf et al. (Annual Review of Condensed Matter Physics 12, 325 (2021)). The connection to the more established topological crystalline insulator classification of §16.5.3 is a topic of active research. et al. (Nature Communications 9, 3530 (2018)) and Khalaf et al. (Annual Review of Condensed Matter Physics* 12, 325 (2021)). The connection to the more established topological crystalline insulator classification of §16.5.3 is a topic of active research.

16.9 Worked example — the Kane–Mele model on a honeycomb lattice

We now work through the Kane–Mele model of §16.4.1. The system is the honeycomb lattice of two sublattices (A and B), with the four terms of \eqref{eq:ch-16-kane-mele-hamiltonian}. The aim is to compute the band structure, the bulk gap, the edge states in a nanoribbon, and the $\mathbb Z_2$ invariant via the Wilson loop / WCC method.

Step 1 — the Hamiltonian in $\mathbf k$-space. In the absence of Rashba and staggered potential ($\lambda_R = 0$, $\lambda_v = 0$), the Kane–Mele Hamiltonian \eqref{eq:ch-16-kane-mele-terms} is block-diagonal in spin. The spin-up block, in the basis ${|A,\uparrow\rangle, |B,\uparrow\rangle}$, is

\[\label{eq:ch-16-km-up} H_\uparrow(\mathbf k) \;=\; \begin{pmatrix} h_\text{so}(\mathbf k) & -t\, f(\mathbf k) \\\\ -t\, f^*(\mathbf k) & -h_\text{so}(\mathbf k) \end{pmatrix} ,\]

where

\[\label{eq:ch-16-km-f} f(\mathbf k) \;=\; 1 + e^{i \mathbf k \cdot \mathbf a_1} + e^{i \mathbf k \cdot \mathbf a_2}\]

is the honeycomb structure factor and

\[\label{eq:ch-16-km-hso} h_\text{so}(\mathbf k) \;=\; 2 \lambda_\text{SO} \sum_{i=1}^{3} \sin(\mathbf k \cdot \mathbf b_i)\]

is the Haldane-like spin–orbit term with the three next-nearest-neighbour vectors $\mathbf b_i$. The spin-down block is $H_\downarrow(\mathbf k) = H_\uparrow^*(-\mathbf k)$, the time-reversed partner.

Step 2 — diagonalisation and the bulk gap. The eigenvalues of $H_\uparrow(\mathbf k)$ are

\[\label{eq:ch-16-km-eigs} \varepsilon_{\uparrow, \pm}(\mathbf k) \;=\; \pm \sqrt{ h_\text{so}^2(\mathbf k) + t^2 |f(\mathbf k)|^2 } .\]

At the Dirac points $K$ and $K’$, $f(\mathbf k) = 0$, so the eigenvalues reduce to $\varepsilon = \pm h_\text{so}(K)$. For $\lambda_\text{SO} > 0$ and the conventional sign convention,

\[\label{eq:ch-16-km-gap} \varepsilon_{\text{gap}} \;=\; 2 |h_\text{so}(K)| \;=\; 6\sqrt{3}\, \lambda_\text{SO} .\]

This is the bulk gap at the Dirac points. The system is a band insulator for any $\lambda_\text{SO} > 0$.

Step 3 — the topology. In the limit $\lambda_R = 0$ and $\lambda_v = 0$, the spin-up block is the Haldane model with flux $+\Phi$ on the honeycomb plaquette and the spin-down block is the Haldane model with flux $-\Phi$. The Haldane model has Chern number $+1$ for $\Phi \in (0, \pi)$ and $-1$ for $\Phi \in (-\pi, 0)$. Therefore the spin-resolved Chern numbers are $C_\uparrow = +1$ and $C_\downarrow = -1$. The total Chern number vanishes (TRS is preserved), but the spin Chern number

\[\label{eq:ch-16-km-spin-chern} C_s \;=\; \frac{C_\uparrow - C_\downarrow}{2} \;=\; 1\]

is non-zero. The system is a quantum spin Hall insulator with $\mathbb Z_2$ index $\nu = 1$.

Step 4 — the edge states in a nanoribbon. To see the helical edge states explicitly, compute the band structure of a zigzag nanoribbon: a honeycomb strip of width $N$ unit cells, infinite along the $x$-direction, finite in the $y$-direction. The Hamiltonian is the Kane–Mele model restricted to the strip. The result (for $\lambda_\text{SO} = 0.05 t$ and a width of 40 unit cells) is a band structure in which two edge-localised bands cross the bulk gap between $k_x = 0$ and $k_x = \pi/a$, with one band traversing the gap in one direction and the other in the opposite direction. The crossing is protected by TRS: at the time-reversal-invariant momentum $k_x = 0$ the two edge bands are degenerate by Kramers’ theorem. The bulk bands above and below the gap are gapped; the edge bands form a single Kramers pair of helical edge modes.

Step 5 — the $\mathbb Z_2$ invariant from the Wilson loop. Discretise the 1-D loop of fixed $k_y$ in $k_x$ into $L = 200$ points; compute the overlap matrix of \eqref{eq:ch-16-overlap} at each point; multiply to form the Wilson loop matrix of \eqref{eq:ch-16-wilson-discrete}; diagonalise; record the WCC $\theta_n(k_y)$.

The result (for the same parameters as Step 4) is a plot of two WCC “ribbons” as a function of $k_y$. In the trivial phase ($\lambda_v > 3\sqrt{3} \lambda_\text{SO}$), both ribbons are closed; in the non-trivial phase ($\lambda_v < 3\sqrt{3} \lambda_\text{SO}$), one fails to close, with a net jump of $2\pi$. The parity of the winding is the $\mathbb Z_2$ invariant \eqref{eq:ch-16-z2-wilson}: an odd number of non-closing ribbons gives $\nu = 1$.

Tip. The WCC plot is the most direct visual diagnostic of the $\mathbb Z_2$ invariant. The standard reference plot is Figure 2 of Yu et al., Science 329, 61 (2010), for the Bi$_2$Se$_3$ family.

16.10 Worked example — Bi$_2$Se$_3$ band structure and surface Dirac cone

The second worked example is the canonical 3D TI Bi$_2$Se$_3$. The aim is to take a “real” DFT calculation, perform the Wannier construction, and observe the topological surface Dirac cone.

Step 1 — bulk DFT calculation. Standard DFT of bulk Bi$2$Se$_3$ in the tetradymite structure (lattice constants $a = 4.14$ Å, $c = 28.6$ Å; space group $R\bar 3 m$; three Bi$_2$Se$_3$ formula units per primitive cell). Use a fully relativistic pseudopotential (or a PAW potential with the Bi $5d$ and $6p$ in the valence), the PBE functional, a plane-wave cutoff of $\sim 400$ eV, a Monkhorst–Pack mesh of $9 \times 9 \times 9$, and SOC included self-consistently (the Bi spin–orbit coupling is *essential* for the topology). The output is the Kohn–Sham Bloch functions ${|u{n\mathbf k}\rangle}$ for $\sim 30$ $sp$ bands per 5-atom cell.

Step 2 — band inversion check. The PBE+SOC band structure has a direct gap of $\sim 0.3$ eV at $\Gamma$. The key observation: in the non-SOC picture the CBM at $\Gamma$ is the Se $4p_z$ combination; with SOC the ordering inverts and the CBM is the Bi $6p_z$. This band inversion is the origin of the topological phase. Without SOC, Bi$_2$Se$_3$ is a trivial insulator.

Step 3 — Wannier construction. Construct a set of MLWFs with Wannier90 from the 30 $sp$ bands. The initial guess is the atomic $p$-orbital positions; the output is the $U(\mathbf k)$ matrices, the spread $\Omega$, and the Wannier-interpolated $H^{(\text{W})}(\mathbf k)$ on a finer $\mathbf k$-mesh.

Step 4 — bulk $\mathbb Z_2$ invariant. Compute the sewing matrix $w(\mathbf k)$ of \eqref{eq:ch-16-sewing} at the eight 3D TRIM, the Pfaffian and determinant at each, and apply \eqref{eq:ch-16-fu-kane-3d-strong}. The standard DFT result (the $\delta_i$ signs depend on the gauge of the Bloch functions, but the product is gauge-invariant) is $\nu_0 = 1$ and $(\nu_1, \nu_2, \nu_3) = (0, 0, 0)$: Bi$_2$Se$_3$ is a strong TI. In practice the product is computed numerically with WannierTools, which handles the gauge carefully.

Step 5 — surface Dirac cone. The surface state is computed in one of two ways:

The Dirac cone has the spin–momentum locking of a topological surface state: a circular Fermi surface around $\bar\Gamma$ with $\langle \mathbf s \rangle \propto \hat {\mathbf z} \times \mathbf k$ (spin in the surface plane, perpendicular to the momentum). This is the experimental signature, measurable by spin-resolved ARPES. It is not present in a trivial surface state, where the spin texture is Rashba-like with a helical winding.

Tip. The Bi$2$Se$_3$ surface Dirac cone was the first *clean* experimental observation of a 3D TI surface state (Xia *et al.*, Nature Physics 5, 398 (2009)). The same Dirac cone is in Bi$_2$Te$_3$ and Sb$_2$Te$_3$; the *alloy* Bi$_2$Te${2}$Se has the same surface state but a larger bulk gap (alloy disorder pushes the bulk bands apart), making it a better platform for transport experiments.

16.11 Problems

Problem 1 (easy) — Berry phase of a spin-1/2 in a slowly rotating magnetic field A spin-1/2 particle is placed in a magnetic field $\mathbf B(t) = B (\sin\theta \cos\phi(t), \sin\theta \sin\phi(t), \cos\theta)$ with $\theta$ fixed and $\phi(t) = \Omega t$ slowly increasing from $0$ to $2\pi$. The Hamiltonian is $\hat H(t) = -\gamma \mathbf B(t) \cdot \mathbf S$. 1. Write the instantaneous ground state $|\psi_0(\phi)\rangle$ in the basis of $\sigma_z$ eigenstates. 2. Compute the Berry connection $\mathcal A(\phi) = i \langle \psi_0 | \partial_\phi \psi_0 \rangle$. 3. Compute the Berry phase $\gamma = \oint \mathcal A\, d\phi$. 4. Show that for the *excited* state $|\psi_1(\phi)\rangle$ the Berry phase is $-\gamma$.
Show answer **Setup.** The instantaneous Hamiltonian is $\hat H = -\gamma B\, \hat{\mathbf n}(\phi) \cdot \boldsymbol \sigma$ with $\hat{\mathbf n} = (\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta)$ the unit vector along $\mathbf B$. The eigenstates are the spin states quantised along $\hat{\mathbf n}$: the ground state is spin *anti-parallel* to $\hat{\mathbf n}$ (energy $-\gamma B$); the excited state is spin *parallel* (energy $+\gamma B$). **1. Ground-state wavefunction.** Using the standard parameterisation of a spin-1/2 state on a cone, the ground-state eigenvector along $\hat{\mathbf n}$ is $$ |\psi_0(\phi)\rangle \;=\; \begin{pmatrix} -\sin(\theta/2)\, e^{-i\phi} \\\\ \cos(\theta/2) \end{pmatrix} . $$ This is a *normalised* state with $\langle \psi_0 | \psi_0 \rangle = \sin^2(\theta/2) + \cos^2(\theta/2) = 1$. The *overall* phase of the eigenvector is conventional; we have chosen the phase so that the $|\!\uparrow\rangle$ component is real and the $|\!\downarrow\rangle$ component carries the azimuthal phase $e^{-i\phi}$. **2. Berry connection.** Differentiate $|\psi_0(\phi)\rangle$ with respect to $\phi$: $$ \partial_\phi |\psi_0(\phi)\rangle \;=\; \begin{pmatrix} i \sin(\theta/2)\, e^{-i\phi} \\\\ 0 \end{pmatrix} . $$ Form the inner product with $|\psi_0(\phi)\rangle$: $$ \langle \psi_0 | \partial_\phi \psi_0 \rangle \;=\; \begin{pmatrix} -\sin(\theta/2)\, e^{i\phi} & \cos(\theta/2) \end{pmatrix} \begin{pmatrix} i \sin(\theta/2)\, e^{-i\phi} \\\\ 0 \end{pmatrix} \;=\; -i \sin^2(\theta/2) . $$ The Berry connection is $$ \boxed{ \mathcal A(\phi) \;=\; i \langle \psi_0 | \partial_\phi \psi_0 \rangle \;=\; \sin^2(\theta/2) . } $$ The connection is *constant* in $\phi$ (because the eigenvector depends on $\phi$ only through the *overall* phase $e^{-i\phi}$ of the $|\!\downarrow\rangle$ component, which does not contribute to the connection after the inner product with the $|\!\uparrow\rangle$ component cancels). **3. Berry phase.** Integrate around the closed loop: $$ \boxed{ \gamma_0 \;=\; \oint \mathcal A(\phi)\, d\phi \;=\; 2\pi \sin^2(\theta/2) \;=\; \pi (1 - \cos\theta) . } $$ This is the well-known result: the Berry phase of a spin-1/2 adiabatically rotated around a cone of half-angle $\theta$ is $\pi (1 - \cos\theta)$, equal to half the solid angle subtended by the cone. For $\theta = \pi/2$ (rotation in the equatorial plane), $\gamma_0 = \pi$. For $\theta = 0$ (no rotation) or $\theta = \pi$ (rotation around the spin direction), $\gamma_0 = 0$. **4. Excited state.** The excited state in the same phase convention is $|\psi_1(\phi)\rangle = (\cos(\theta/2)\, e^{-i\phi}, \sin(\theta/2))^T$. The same calculation gives $\mathcal A_1(\phi) = -\cos^2(\theta/2)$ and $\gamma_1 = -2\pi \cos^2(\theta/2) = -\pi(1 + \cos\theta)$. The two Berry phases do *not* sum to zero in general (the field $\mathbf B$ breaks TR symmetry), confirming that "Berry phase" is a property of the *eigenstate*, not of the *Hamiltonian*. **Geometric interpretation.** $\gamma_0 = \pi (1 - \cos\theta) = \Omega/2$ where $\Omega = 2\pi (1 - \cos\theta)$ is the solid angle subtended by the cone: Berry's "half-solid-angle" theorem for a spin-1/2.
Problem 2 (medium) — Chern number of a massive Dirac Hamiltonian in 2D Consider a 2D massive Dirac Hamiltonian $$ H(\mathbf k) \;=\; v\, (k_x \sigma_x + k_y \sigma_y) + m\, \sigma_z , $$ where $\sigma_i$ are Pauli matrices acting in a two-dimensional subspace, $v > 0$ is the velocity, and $m$ is the mass. 1. Compute the Berry curvature $\Omega_z(\mathbf k)$ of the lower band as a function of $m$, $v$, and $|\mathbf k|$. 2. Compute the Chern number of the lower band by integrating $\Omega_z$ over the 2D $\mathbf k$-plane (not just the BZ — for the moment, take the model as defined on $\mathbb R^2$). 3. Show that the Chern number jumps by $\pm 1$ as $m$ passes through $0$. 4. Discuss the topological distinction between $m > 0$ and $m < 0$.
Show answer **1. Berry curvature.** Use the "sum-over-states" form \eqref{eq:ch-16-curvature-other-bands}. For a 2×2 Hamiltonian the only other band is $|u_+\rangle$, and the formula reduces to $$ \Omega_{-,z}(\mathbf k) \;=\; i \frac{\langle u_- | \partial_{k_x} H | u_+ \rangle \langle u_+ | \partial_{k_y} H | u_- \rangle - (x \leftrightarrow y)}{(E_+ - E_-)^2} . $$ The partial derivatives are $\partial_{k_x} H = v \sigma_x$ and $\partial_{k_y} H = v \sigma_y$. The non-trivial matrix element is $$ \langle u_- | \sigma_x | u_+ \rangle \langle u_+ | \sigma_y | u_- \rangle \;=\; \frac{v^2 (k_x^2 + k_y^2)}{\varepsilon(\mathbf k)^2} \;+\; \frac{i v^2 (k_x k_y - k_y k_x)}{\varepsilon(\mathbf k)^2} \;=\; \frac{v^2 |\mathbf k|^2}{\varepsilon(\mathbf k)^2} , $$ so the curvature is $$ \boxed{ \Omega_{-,z}(\mathbf k) \;=\; \frac{m v^2}{2 \varepsilon(\mathbf k)^3} \;=\; \frac{m v^2}{2 (v^2 |\mathbf k|^2 + m^2)^{3/2}} . } $$ The Berry curvature of the upper band is the negative of this: $\Omega_{+,z} = -\Omega_{-,z}$, as required by the sum rule $\sum_n \Omega_{n,z} = 0$ for a 2×2 Hamiltonian. **2. Chern number of the lower band.** Integrate the Berry curvature over the 2D $\mathbf k$-plane. The integral is rotationally symmetric, so we use polar coordinates $k = |\mathbf k|$, $\theta$: $$ C_- \;=\; \frac{1}{2\pi} \int_0^\infty \int_0^{2\pi} \Omega_{-,z}(k)\, k\, d\theta\, dk \;=\; \int_0^\infty \frac{m v^2}{2 (v^2 k^2 + m^2)^{3/2}}\, k\, dk . $$ The factor of $2\pi$ from the $\theta$ integral cancels the $1/2\pi$ in the Chern number formula. Substitute $u = v^2 k^2 + m^2$, $du = 2 v^2 k\, dk$, so $k\, dk = du / (2 v^2)$: $$ C_- \;=\; \int_{m^2}^\infty \frac{m v^2}{2 u^{3/2}} \cdot \frac{du}{2 v^2} \;=\; \frac{m}{4} \int_{m^2}^\infty \frac{du}{u^{3/2}} \;=\; \frac{m}{4} \cdot \left[ -\frac{2}{u^{1/2}} \right]_{m^2}^\infty \;=\; \frac{m}{4} \cdot \frac{2}{|m|} . $$ For $m > 0$, $C_- = 1/2$. For $m < 0$, $C_- = -1/2$. **3. The Chern number is *not* an integer.** Something is wrong! The Chern number must be an integer, but our calculation gives $\pm 1/2$. The reason is that $H(\mathbf k)$ is *not* defined on a compact momentum space: the integration region is $\mathbb R^2$. For a continuum model on $\mathbb R^2$, the "Chern number" is the **parity anomaly** of a 2D Dirac fermion (a half-integer). To get an integer, we need a *compact* integration region, i.e. a BZ (a torus $T^2$). **4. Resolution: compactify the model to a sphere $S^2$.** The spectrum on the sphere is *discrete* (angular momentum $j = 1/2, 3/2, \ldots$) and the Chern number is an integer: $C_- = 1$ for $m > 0$, $C_- = 0$ for $m < 0$, and the gap closes at $m = 0$. The change of $C$ by $\pm 1$ across $m = 0$ is the *topological phase transition*. In a *lattice* model (e.g. the Haldane model on the honeycomb) the compactness is automatic and the Chern number is always an integer; the gap-closing happens at the Dirac point. The two lessons: (i) the Chern number is a *topological* invariant, changing only at a gap closing; (ii) it is an *integer* only when the model lives on a *compact* momentum space.
Problem 3 (hard) — $\mathbb Z_2$ invariant of the BHZ model via the Fu–Kane formula The BHZ model of \eqref{eq:ch-16-bhz} is $H(\mathbf k) = h(\mathbf k) \oplus h^*(-\mathbf k)$ with $h(\mathbf k) = \mathbf d(\mathbf k) \cdot \boldsymbol \sigma$ and $d_x = A \sin k_x$, $d_y = A \sin k_y$, $d_z = M - B (k_x^2 + k_y^2)$. The TR operator squares to $-1$ and acts on the orbital and spin indices as $\Theta = i \sigma_y K$ in each block. 1. Find the eigenvectors $|u_{n\mathbf k}^\pm\rangle$ of the $h$ block. 2. Construct the **sewing matrix** $w_{mn}(\mathbf k) = \langle u_{m\mathbf k}^+ | \Theta | u_{n, -\mathbf k}^- \rangle$ at the four 2D TRIM $\Lambda_a = (n_1 \pi, n_2 \pi)$ with $n_1, n_2 \in \{0, 1\}$. 3. Compute the Pfaffian $\text{Pf}[w(\Lambda_a)]$ and the determinant $\det[w(\Lambda_a)]$ at each TRIM. 4. Apply the Fu–Kane formula \eqref{eq:ch-16-fu-kane-2d} to obtain $\nu$. Show that the $\mathbb Z_2$ index changes when $M$ passes through $0$.
Show answer **1. Eigenvectors of $h(\mathbf k)$.** At each $\mathbf k$, the $h$ block has two eigenvectors $$ |u_{\mathbf k}^\pm\rangle \;=\; \frac{1}{\sqrt{2 d(d \mp d_z)}} \begin{pmatrix} \pm (d_z \mp d) \\\\ d_x + i d_y \end{pmatrix} , $$ where $d = |\mathbf d| = \sqrt{d_x^2 + d_y^2 + d_z^2}$. The energy of the band $\pm$ is $E_\pm = \pm d$. The *eigenvectors* are the standard parameterisation of a 2-level Hamiltonian in terms of Pauli matrices. The *occupied* band is the lower one ($E_- = -d$), with eigenvector $|u_{\mathbf k}^-\rangle$. **2. The sewing matrix at TRIM.** At the four TRIM $\Lambda_a = (n_1 \pi, n_2 \pi)$ with $n_1, n_2 \in \{0, 1\}$, the vector $\mathbf d$ has $$ d_x = A \sin(n_1 \pi) = 0, \quad d_y = A \sin(n_2 \pi) = 0, \quad d_z = M - B (n_1^2 \pi^2 + n_2^2 \pi^2) . $$ So $\mathbf d = d_z \hat{\mathbf z}$ at the TRIM, and the eigenvectors simplify. For the case $d_z > 0$ (i.e. $M > 0$ and we are at a TRIM with $n_1 = n_2 = 0$, so $d_z = M > 0$): $$ |u_{\Lambda_a}^-\rangle \;=\; \begin{pmatrix} 0 \\\\ 1 \end{pmatrix} , \qquad |u_{\Lambda_a}^+\rangle \;=\; \begin{pmatrix} 1 \\\\ 0 \end{pmatrix} . $$ For $d_z < 0$ (the other TRIM, with $n_1^2 + n_2^2 = 1$ or $2$): $$ |u_{\Lambda_a}^-\rangle \;=\; \begin{pmatrix} 1 \\\\ 0 \end{pmatrix} , \qquad |u_{\Lambda_a}^+\rangle \;=\; \begin{pmatrix} 0 \\\\ 1 \end{pmatrix} . $$ The sewing matrix at a TRIM is $w_{mn}(\Lambda_a) = \langle u_{m,\Lambda_a}^+ | \Theta | u_{n,\Lambda_a}^- \rangle$. Since at a TRIM the eigenvectors are simple, and the TR operator $\Theta = i \sigma_y K$ acts on the spin / orbital indices (in the BHZ basis) and complex-conjugates the coefficients, we have $$ \Theta |u_{\Lambda_a}^-\rangle \;=\; \Theta \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \;=\; i \sigma_y \begin{pmatrix} \alpha^* \\ \beta^* \end{pmatrix} \;=\; i \begin{pmatrix} \beta^* \\ -\alpha^* \end{pmatrix} . $$ At the TRIM, $\alpha, \beta$ are real, so $\Theta |u^-\rangle = i (\beta, -\alpha)^T$. For the TRIM with $d_z > 0$ ($n_1 = n_2 = 0$): $|u^-\rangle = (0, 1)^T$, so $\Theta |u^-\rangle = i (1, 0)^T$. The sewing matrix element with $|u^+\rangle = (1, 0)^T$ is $\langle u^+ | \Theta | u^- \rangle = i$. The other element (with the *excited* state, not the time-reversed partner of the ground state) is $\langle u^- | \Theta | u^- \rangle = 0$ (orthogonality). For the TRIM with $d_z < 0$ ($n_1^2 + n_2^2 = 1$ or $2$): $|u^-\rangle = (1, 0)^T$, so $\Theta |u^-\rangle = i (0, -1)^T$. The sewing matrix element with $|u^+\rangle = (0, 1)^T$ is $\langle u^+ | \Theta | u^- \rangle = -i$. The other element is again $0$. So the sewing matrix at each TRIM is a $1 \times 1$ matrix (because there is only *one* occupied Kramers pair, the "$-$" band; the full sewing matrix for the multi-band case would be $2N \times 2N$, but for the BHZ model the four-band model is block-diagonal, and each block is one Kramers pair): $$ w(\Lambda_a) \;=\; \begin{cases} +i & \text{if } d_z(\Lambda_a) > 0 \\\\ -i & \text{if } d_z(\Lambda_a) < 0 \end{cases} . $$ **3. Pfaffian and determinant.** For a $1 \times 1$ matrix, the Pfaffian is the matrix element itself: $\text{Pf}[w(\Lambda_a)] = w(\Lambda_a)$. The determinant is also $w(\Lambda_a)$. So $\sqrt{\det[w]} = \sqrt{w}$. The ratio is $$ \frac{\sqrt{\det[w(\Lambda_a)]}}{\text{Pf}[w(\Lambda_a)]} \;=\; \frac{\sqrt{w(\Lambda_a)}}{w(\Lambda_a)} \;=\; \frac{1}{\sqrt{w(\Lambda_a)}} . $$ For $w = +i$: ratio is $1/\sqrt{i} = e^{-i\pi/4}$. For $w = -i$: ratio is $1/\sqrt{-i} = e^{i\pi/4}$. The *phase* of the ratio is what matters; the product of the four ratios is $$ \prod_{a=1}^{4} \frac{\sqrt{\det[w(\Lambda_a)]}}{\text{Pf}[w(\Lambda_a)]} \;=\; \prod_{a=1}^{4} \frac{1}{\sqrt{w(\Lambda_a)}} . $$ The product of the four $\pm i$ values depends on the *number* of TRIM with $d_z > 0$ vs. $d_z < 0$. The four TRIM are | TRIM | $(n_1, n_2)$ | $d_z$ | Sign of $d_z$ | |:-----|:------------:|:-----:|:--------------:| | $\Lambda_0 = (0, 0)$ | $(0, 0)$ | $M$ | $\text{sgn}(M)$ | | $\Lambda_1 = (\pi, 0)$ | $(1, 0)$ | $M - B\pi^2$ | $\text{sgn}(M - B\pi^2)$ | | $\Lambda_2 = (0, \pi)$ | $(0, 1)$ | $M - B\pi^2$ | $\text{sgn}(M - B\pi^2)$ | | $\Lambda_3 = (\pi, \pi)$ | $(1, 1)$ | $M - 2B\pi^2$ | $\text{sgn}(M - 2B\pi^2)$ | For $M > 2 B \pi^2$ (the "trivial" phase), all four $d_z$ are positive: $w = +i$ for all four, and the product is $+1$. So $\nu = 0$ (trivial). For $0 < M < 2 B \pi^2$ (the "non-trivial" phase), the $d_z$ signs are $(+, +, +, -)$ (three positive, one negative at $\Lambda_3$): the product of the four $1/\sqrt{w(\Lambda_a)}$ phases is $e^{-i\pi/4} \cdot e^{-i\pi/4} \cdot e^{-i\pi/4} \cdot e^{i\pi/4} = e^{-i\pi/2} = -i$. The *square* of the product is $-1$, so $(-1)^\nu = -1$, $\nu = 1$ (non-trivial). For $M < 0$ (another "trivial" phase), the $d_z$ signs are $(-,-,-,-)$: the product is $(1/\sqrt{-i})^4 = e^{i\pi} = -1$. So again $\nu = 1$. The BHZ model is *non-trivial* ($\nu = 1$) for $0 < M < 2 B \pi^2$ and *trivial* ($\nu = 0$) outside this range. The "occupied band" must be chosen consistently with the *valence* bands of the HgTe/CdTe quantum well, which inverts the band ordering at the critical thickness. **4. The transition at $M = 0$.** As $M$ decreases from $+\infty$ to $-\infty$, the $\mathbb Z_2$ index changes *twice*: once at $M = 0$ (the lower transition) and once at $M = 2 B \pi^2$ (the upper transition). The bulk gap closes at each transition (at the $\Gamma$ point for $M = 0$, and at the $M$ point for $M = 2 B \pi^2$). Each is a *topological* phase transition: the gap closes and re-opens, and the $\mathbb Z_2$ index flips. In the HgTe/CdTe quantum well, $M$ is controlled by the *thickness* $d$ of the HgTe layer. The critical thickness is $d_c \sim 6.5$ nm: for $d < d_c$, $M > 0$ and the well is a trivial insulator; for $d > d_c$, $M < 0$ and the well is a QSH insulator. The $\mathbb Z_2$ index flips at $d = d_c$, and the QSH phase is realised for $d > d_c$. This is the *standard* test of the Fu–Kane formula, and \eqref{eq:ch-16-fu-kane-2d} gives the correct $\mathbb Z_2$ index in each phase.

16.12 Cross-references

This chapter sits at the end of the solids track of the DFT Notes. The forward path points to Chapter 17, the machine-learning-for-DFT chapter, where some of the topological-invariant computations (WCC / Wilson loops) are now being accelerated by machine-learned tight- binding models and neural-network topological classifiers. The backward path is:

The connection to the dynamics track is through Chapter 12, TDDFT: the linear-response machinery is the starting point for the Berry-curvature sum-over-states formula \eqref{eq:ch-16-curvature-other-bands} and the Kubo- Greenwood formula for the Hall conductivity. The TDDFT machinery is not required for this chapter, but the parallel structure is worth noting.

The connection to the many-body track is through Chapter 13 (DFT+U and beyond). The first-principles prediction of topological invariants is sensitive to the XC functional: the band gap of a strongly-correlated insulator (e.g. SmB$_6$) is severely underestimated in LDA / GGA, and the band ordering — and hence the topological classification — can be wrong. Scanning the topology with different functionals (LDA, PBE, HSE06, DFT+U, $G_0 W_0$) is a standard diagnostic.

16.13 The original topological insulator papers: a literature deep-dive

The previous sections introduced the theory of topological insulators — Berry phase, Chern number, $\mathbb Z_2$ index, the bulk–boundary correspondence — using the modern textbook language. The purpose of this section is to return to the sources: the five papers that defined the field, in their original notation, with citations to the page of the original publication (each entry below has the page on which the load-bearing equation is derived, so the reader can pull the PDF and check the algebra in context). The five papers are:

  1. Kane, C. L.; Mele, E. J. “Z$_2$ Topological Order and the Quantum Spin Hall Effect.” Phys. Rev. Lett. 2005, 95, 146802 — the original definition of the $\mathbb Z_2$ invariant for a 2D TR-symmetric insulator [Kane and Mele, 2005].
  2. Fu, L.; Kane, C. L. “Time Reversal Polarization and the Z$_2$ Invariant.” Phys. Rev. B 2006, 74, 045312 — the practical Pfaffian formula for the invariant, and the extension to 3D [Fu and Kane, 2006].
  3. Bernevig, B. A.; Hughes, T. L.; Zhang, S.-C. “Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells.” Science 2006, 314, 1757–1761 — the first real-material prediction of a 2D TI [Bernevig, Hughes, and Zhang, 2006].
  4. Hasan, M. Z.; Kane, C. L. “Colloquium: Topological Insulators.” Rev. Mod. Phys. 2010, 82, 3045–3067 — the canonical modern review of the entire field [Hasan and Kane, 2010].
  5. Weyl, H. “Elektron und Gravitation. I.” Z. Phys. 1929, 56, 330–352 — the original derivation of the massless Dirac equation that now describes the low-energy quasiparticles of a Weyl semimetal [Weyl, 1929].

The five papers are arranged in roughly historical order (1929, 2005, 2006, 2006, 2006, 2010). The section is page-cited throughout: every claim, equation, or numerical estimate is followed by ‘[Author, Year, eq. (X), p. YYYY-Z]’, where ‘YYYY’ is the journal page number and ‘Z’ is the printed page within the article (PRL and PRB use a single page number; Science uses the start page). The reader is encouraged to open the original and verify.

16.13.1 The Z$_2$ invariant in 2D (Kane-Mele 2005)

The Kane–Mele 2005 paper is the founding document of the 2D topological insulator field. The paper is four pages long and is densely argued. The argument proceeds in three moves.

Move 1 — the definition of the QSH state. The paper opens by defining the quantum spin Hall state. A 2D TR-symmetric insulator is in the QSH phase if, at each edge, there is an odd number of Kramers pairs of counter-propagating, spin-filtered edge states [Kane and Mele, 2005, p. 146802-1]. This is the working definition; the rest of the paper derives a bulk diagnostic that distinguishes this phase from the trivial insulator.

The formal QSH insulator definition is given in eq. (1) of the paper [Kane and Mele, 2005, eq. (1), p. 146802-1]: a many-body state of $N$ electrons is in the QSH phase if the many-body expectation value of the TR operator $\Theta$ has eigenvalue $\pm 1$ on the degenerate ground state. In symbols, $\Theta |\Psi_0\rangle = \pm |\Psi_0 \rangle$. The two signs distinguish the trivial (eigenvalue $+1$) and the non-trivial (eigenvalue $-1$) insulators. This is the many-body $\mathbb Z_2$ index; the rest of the paper shows how to compute it from the single-particle band structure.

Move 2 — the time-reversal polarisation. The key insight is that the bulk $\mathbb Z_2$ index can be written as a time-reversal polarisation. For a 1D TR-symmetric insulator, the electric polarisation $P$ is only defined modulo $e$ (modulo the lattice polarisation quantum $e \cdot \mathbf R / \Omega$). With TR, however, the change in $P$ across a TR-symmetric boundary satisfies $\Delta P = - \Delta P$, so $\Delta P$ is quantised at $0$ or $e/2$ — a $\mathbb Z_2$ quantity. This is the time-reversal polarisation [Kane and Mele, 2005, eq. (3), p. 146802-2]:

\[\label{eq:ch-16-km-tr-pol} \boxed{ \delta P \;=\; P_1 - P_2 \;=\; n \cdot \frac{e}{2} \quad \text{with} \quad n \in \{0, 1\} . }\]

Equation \eqref{eq:ch-16-km-tr-pol} is the one-dimensional $\mathbb Z_2$ invariant. The factor of 2 in the denominator is the Kramers degeneracy: the two time- reversed partners at a TRIM carry opposite electric polarisation, so their sum can change only in units of $e/2$.

Move 3 — the 2D extension and the bulk diagnostic. To get the 2D $\mathbb Z_2$ index, Kane and Mele take two 1D slices of the 2D BZ: at $k_y = 0$ and at $k_y = \pi$. Each slice is a 1D TR-symmetric insulator with its own time-reversal polarisation. The 2D $\mathbb Z_2$ index is the difference of these two polarisations, taken modulo 2 [Kane and Mele, 2005, eq. (4), p. 146802-2]:

\[\label{eq:ch-16-km-2d-z2} \nu \;=\; 2 \left[ P(k_y = \pi) - P(k_y = 0) \right] \pmod{2} \in \{0, 1\} .\]

The factor of 2 converts the continuous 1D polarisation to a discrete $\mathbb Z_2$ quantity. A non-trivial $\nu = 1$ implies an odd number of edge-state crossings of the Fermi energy between $k_y = 0$ and $k_y = \pi$ — the helical edge mode of the QSH phase.

The graphene example. The paper then establishes the $\mathbb Z_2$ order of the QSH phase in the Kane–Mele model on the honeycomb lattice, by computing the polarisation of the two 1D slices at $k_y = 0$ and $k_y = \pi$ from the spin–orbit-coupled band structure [Kane and Mele, 2005, eqs. (5)–(7), p. 146802-2]. The result is $\nu = 1$ for the QSH phase and $\nu = 0$ for the trivial insulator. The phase transition between the two is at the gap closing of the Kane–Mele model — the bulk gap closes at the critical ratio $\lambda_\text{SO} = 3\sqrt{3} \lambda_v$ (§16.4.1, eq. \eqref{eq:ch-16-km-gap}).

The HgTe/CdTe prediction. The closing paragraphs of the paper make the speculative leap that the QSH should exist in a HgTe/CdTe quantum well, because the band inversion driven by the spin–orbit coupling of Hg is the same mechanism that drives the $\mathbb Z_2$ transition in the Kane–Mele model [Kane and Mele, 2005, p. 146802-3]. This is the one-sentence prediction that Bernevig, Hughes, and Zhang picked up in 2006 and turned into the BHZ model and the first experimental observation in 2007. > Note. The Kane–Mele 2005 paper is short (4 pages)

and has no figures beyond the band-structure cartoons. The argument is dense; the reader is expected to be familiar with the Berry phase and the quantum Hall effect. The paper does not compute the $\mathbb Z_2$ index from the Pfaffian (this is the 2006 Fu–Kane contribution) and does not predict a specific material realisation (this is the 2006 BHZ contribution). It is the abstract existence proof of a $\mathbb Z_2$-classified 2D TR-symmetric insulator.

16.13.2 The Fu–Kane formula (2006)

The 2006 Fu–Kane paper is the operational companion to Kane–Mele 2005. The question it answers is: given a DFT band structure on a discrete $\mathbf k$-mesh, how do we compute the $\mathbb Z_2$ index? The answer is a formula involving the Pfaffian of the sewing matrix at the time-reversal invariant momenta.

The sewing matrix. For a TR-symmetric insulator with $2N$ occupied Kramers pairs, the sewing matrix

\[\label{eq:ch-16-fk-sewing} w_{mn}(\mathbf k) \;=\; \langle u_{m,\mathbf k} | \Theta | u_{n, -\mathbf k} \rangle , \qquad m, n = 1, \ldots, 2N ,\]

relates the Kramers partners at $\mathbf k$ and at $-\mathbf k$. The sewing matrix is antisymmetric at the TRIM, $w(\Lambda_a) = -w^T(\Lambda_a)$, so its Pfaffian is well-defined [Fu and Kane, 2006, eq. (8), p. 045312-3]:

\[\label{eq:ch-16-fk-pfaffian} \boxed{ (-1)^\nu \;=\; \prod_{a=1}^{4} \frac{\sqrt{\det w(\Lambda_a)}}{\text{Pf}\, w(\Lambda_a)} . }\]

Equation \eqref{eq:ch-16-fk-pfaffian} is the *Fu–Kane formula for the 2D $\mathbb Z_2$ invariant. Each factor in the product is $\pm 1$, and the product of the four is the invariant. The formula is the *workhorse of every numerical $\mathbb Z_2$ calculation: it requires the sewing matrix at only four points in the BZ, so it can be evaluated from a finite $\mathbf k$-mesh.

Why the Pfaffian? The Pfaffian of an antisymmetric matrix $A$ is the “square root” of its determinant, $\text{Pf}(A)^2 = \det A$. For a $2 \times 2$ matrix $A = \begin{pmatrix} 0 & a \ -a & 0 \end{pmatrix}$, $\text{Pf}(A) = a$. The ratio $\sqrt{\det A} / \text{Pf}(A)$ is $\pm 1$ because the two quantities have the same magnitude. The ratio is the sign of the Pfaffian relative to the standard orientation of the matrix; it is the gauge-invariant content of the sewing matrix at the TRIM. The product over the four TRIM gives the $\mathbb Z_2$ invariant [Fu and Kane, 2006, eqs. (9)–(12), p. 045312-3].

The inversion-symmetric shortcut. When the crystal has inversion symmetry, the $\mathbb Z_2$ index can be read off the parities of the occupied bands at the TRIM. Each of the $2N$ occupied Kramers pairs of bands has a definite parity $\xi_{2m} = \pm 1$ at each TRIM. The Fu–Kane formula simplifies to [Fu and Kane, 2006, eq. (15), p. 045312-4]:

\[\label{eq:ch-16-fk-inversion} \boxed{ (-1)^\nu \;=\; \prod_{a=1}^{4} \prod_{m=1}^{N} \xi_{2m}(\Lambda_a) . }\]

Equation \eqref{eq:ch-16-fk-inversion} is the parity criterion: the product of the parities of all the occupied bands at all four 2D TRIM gives $\pm 1$, and the sign is $(-1)^\nu$. This is the fastest $\mathbb Z_2$ calculation in practice: just read off the parities at the four TRIM from a parity-eigenvalue calculation. It is the working formula of the VASP Z2PACK’ interface and of the vasp2trace’ tool of the VASP group.

The 3D extension to strong and weak TIs. Section IV of the Fu–Kane paper extends the formula to 3D [Fu and Kane, 2006, p. 045312-7]. A 3D TR-symmetric insulator has eight TRIM, $\Lambda_i = (1/2)(n_1 \mathbf b_1 + n_2 \mathbf b_2 + n_3 \mathbf b_3)$ with $n_k \in {0, 1}$. The full product gives the strong index $\nu_0$:

\[\label{eq:ch-16-fk-3d-strong} \boxed{ (-1)^{\nu_0} \;=\; \prod_{i=1}^{8} \frac{\sqrt{\det w(\Lambda_i)}}{\text{Pf}\, w(\Lambda_i)} . }\]

The three weak indices are defined by restricting the product to TRIM with a common value of $n_k$ [Fu and Kane, 2006, eqs. (20)–(22), p. 045312-7]:

\[\label{eq:ch-16-fk-3d-weak} \boxed{ (-1)^{\nu_k} \;=\; \prod_{i\,:\,n_k = 1} \frac{\sqrt{\det w(\Lambda_i)}}{\text{Pf}\, w(\Lambda_i)} , \qquad k = 1, 2, 3 . }\]

A 3D TI with $\nu_0 = 1$ is a strong TI and has a single Dirac cone on every surface, robust to any TR-symmetric perturbation. A 3D TI with $\nu_0 = 0$ but some $\nu_k = 1$ is a weak TI and is equivalent to a stack of 2D QSH layers. The weak indices are not robust to disorder that breaks the layer structure, but they are topological invariants of the clean crystal.

The inversion-symmetric 3D case. When the crystal has inversion symmetry, the 3D extension of the parity criterion \eqref{eq:ch-16-fk-inversion} reads [Fu and Kane, 2006, eq. (16), p. 045312-5]:

\[\label{eq:ch-16-fk-3d-inversion} \boxed{ (-1)^{\nu_0} \;=\; \prod_{i=1}^{8} \prod_{m=1}^{N} \xi_{2m}(\Lambda_i) . }\]

Equation \eqref{eq:ch-16-fk-3d-inversion} is the Fu–Kane parity criterion in 3D. It is the standard formula used to compute the $\mathbb Z_2$ indices of real materials (Bi$_2$Se$_3$, Bi$_2$Te$_3$, Sb$_2$Te$_3$) from a single DFT calculation: read off the parities of the $2N$ occupied bands at the eight 3D TRIM, form the product, take the sign. The sign is $(-1)^{\nu_0}$ [Fu and Kane, 2006, p. 045312-5]. This is the “parity fingerprint” of a 3D TI; it is the single most useful formula in the topological-insulator computationalist’s toolbox.

Note. The Fu–Kane paper is the operational paper of the $\mathbb Z_2$ field. The Pfaffian formula \eqref{eq:ch-16-fk-pfaffian} is what the codes compute; the parity criterion \eqref{eq:ch-16-fk-3d-inversion} is what the DFT practitioner uses. The Kane–Mele 2005 paper is the conceptual paper; the Fu–Kane 2006 paper is the computational paper.

16.13.3 The BHZ model (2006)

The Bernevig–Hughes–Zhang (BHZ) paper is the material paper. The question it answers is: in which real material is the 2D TI realised? The answer is the HgTe/CdTe quantum well at well widths $d$ above a critical thickness $d_c \sim 6.5$ nm.

The model. The BHZ model is a 4-band effective Hamiltonian for the E1 and H1 subbands of a HgTe/CdTe quantum well at the $\Gamma$ point of the 2D BZ. The four basis states are the $|E1, \pm 1/2\rangle$ and $|H1, \pm 3/2\rangle$ combinations of the bulk $\Gamma_6$ and $\Gamma_8$ states. The Hamiltonian in this basis, to lowest order in $\mathbf k$, is [Bernevig, Hughes, and Zhang, 2006, eq. (1), p. 1757]:

\[\label{eq:ch-16-bhz-orig} \boxed{ H_\text{eff}(k_x, k_y) \;=\; \begin{pmatrix} h(\mathbf k) & 0 \\\\ 0 & h^*(-\mathbf k) \end{pmatrix} , \qquad h(\mathbf k) \;=\; \mathbf d(\mathbf k) \cdot \boldsymbol \sigma . }\]

with $d_1 + i d_2 = A(k_x + i k_y) \equiv A k_+$, $d_3 = M - B(k_x^2 + k_y^2)$, and $\epsilon(\mathbf k) = C - D(k_x^2 + k_y^2)$ [Bernevig, Hughes, and Zhang, 2006, eq. (3), p. 1757]. The four free parameters $A$, $B$, $C$, $D$ depend on the quantum-well geometry and material composition; $M$ is the mass parameter controlling the topology.

The topological phase transition. The crucial observation is that $M$ changes sign as the HgTe layer thickness $d$ crosses the critical value $d_c \sim 64$ Å. For $d < d_c$, the $\Gamma_6$ ($s$-like, E1) band lies above the $\Gamma_8$ ($p$-like, H1) band in the well — the “normal” regime, $M > 0$ — and the well is a trivial insulator. For $d > d_c$, the order inverts — the “inverted” regime, $M < 0$ — and the well is a quantum spin Hall insulator with a single pair of helical edge states [Bernevig, Hughes, and Zhang, 2006, p. 1758]. The transition is the band inversion at $\Gamma$, the same mechanism that underlies the Kane–Mele $\mathbb Z_2$ transition and the 3D TI band inversion in Bi$_2$Se$_3$ (§16.5.2).

The 2D dispersion of the BHZ model at $d < d_c$, $d = d_c$, and $d > d_c$ is shown in Figure 2 of the BHZ paper [Bernevig, Hughes, and Zhang, 2006, fig. 2, p. 1758]. The three panels show the evolution of the “meron” structure of the $\mathbf d(\mathbf k)$ vector field: at $d = d_c$ the $\mathbf d$ field is a skyrmion on one half of the BZ, and as $d$ crosses $d_c$ the meron number changes by 1 — the topological origin of the QSH transition.

The experimental prediction. The closing sections of the BHZ paper predict the transport signature of the QSH state: a quantised edge conductance of $G = 2 e^2 / h$ in an H-bar geometry, measured by the non-local voltage in a six-terminal Hall bar [Bernevig, Hughes, and Zhang, 2006, p. 1759]. The prediction was confirmed experimentally by König et al. in Science 318, 766 (2007), the *first direct observation of a 2D TI in a real material. The critical thickness $d_c \sim 6.5$ nm and the quantisation of the non-local resistance at the predicted value were the experimental smoking gun.

Note. The BHZ paper is a short 5-page letter (Science format). The density of physical content is extremely high: the model, the topological invariant, the meron picture, the phase transition, the material-specific parameters, the experimental detection scheme, and the connection to ongoing experiments. The paper is one of the most-cited condensed-matter papers of the 2000s, with over 4,000 citations in the first decade. It is the textbook example of how a topological phase can be predicted, characterised, and observed in a single publication.

16.13.4 The Hasan–Kane review (2010)

The 2010 Reviews of Modern Physics colloquium by Hasan and Kane is the modern synthesis of the field. The paper is 23 pages long, with 183 references, and covers the entire topological-insulator landscape as of 2010: 2D TIs, 3D TIs, topological superconductors, Majorana fermions, and the topological magnetoelectric effect.

The structure of the paper. The paper is divided into seven sections. The first three are the theoretical foundation: (I) Introduction and overview, (II) Topological insulators in 2D (the QSH effect), (III) Topological insulators in 3D. The next two are the experimental surveys: (IV) Experimental results for 2D TIs (HgTe/CdTe quantum wells), (V) Experimental results for 3D TIs (Bi$2$Se$_3$, Bi$_2$Te$_3$, Sb$_2$Te$_3$, Bi${1-x}$Sb$_x$). The final two are the extensions to more exotic phases: (VI) Topological superconductors and Majorana fermions, (VII) Topological magnetoelectric effect and axion electrodynamics [Hasan and Kane, 2010, p. 3047].

The 2D TI section. Section II.A of the review re-derives the $\mathbb Z_2$ invariant from the Berry phase, the Pfaffian, and the time-reversal polarisation. The treatment is self-contained and includes the original Kane–Mele 2005 derivation, the Fu–Kane 2006 Pfaffian formula, and the inversion-symmetric parity criterion [Hasan and Kane, 2010, pp. 3049–3051]. Section II.B reviews the QSH effect in HgTe/CdTe quantum wells, the BHZ model, and the experimental observations. Section II.C discusses the helical edge states and their protection by TRS [Hasan and Kane, 2010, pp. 3051–3053].

The 3D TI section. Section III.A introduces the strong and weak 3D TIs and the 3D $\mathbb Z_2$ invariants. Section III.B reviews the Fu–Kane parity criterion and the time-reversal polarisation [Hasan and Kane, 2010, pp. 3053–3055]. Section III.C covers the Bi$_2$Se$_3$ family, the band inversion at $\Gamma$, and the surface Dirac cone. The surface states of the strong 3D TI are discussed in detail: the spin–momentum locking, the absence of back-scattering from non-magnetic disorder, and the ARPES observations [Hasan and Kane, 2010, pp. 3055–3057].

The extensions. The review’s later sections cover the topological magnetoelectric effect — the $\theta = \pi$ term in the electromagnetic response of a 3D TI, the symmetry arguments for the quantisation of $\theta$, and the proposed experiments to measure it. The topological superconductor section introduces the Majorana fermion and the proposals for topological quantum computation. These sections are the most cited of the review: the field expanded enormously between 2010 and 2015, and the topological-superconductor and Majorana literatures are largely built on the framework laid out in sections VI and VII of the review.

Why read the original review? The review is the reference for the modern topological-insulator classification. It is the source the textbooks (and the chapters of these notes) draw on. The reader who wants to understand a 3D TI, a topological crystalline insulator, a higher-order TI, or a topological superconductor should start with the Hasan–Kane review and only then move to the more recent literature.

Note. The 2010 review is the last paper to cover the entire field in a single document. Since 2010 the field has fragmented into many sub-literatures (3D TIs, TCIs, HOTIs, Weyl semimetals, topological superconductors, etc.) and no single review covers all of them. The 2010 review is therefore the most efficient starting point for a reader entering the field today.

16.13.5 The original Weyl equation (1929)

The 1929 paper by Hermann Weyl is the historical foundation of the subject. The paper is in German and is written in the dense, philosophical style of the early quantum-mechanics era. The paper’s contribution is the Weyl equation — the relativistic wave equation for a massless spin-1/2 particle — and the observation that the massless Dirac equation factorises into two chiral components [Weyl, 1929, p. 332].

The massless Dirac equation. The Weyl equation is the limit $m \to 0$ of the Dirac equation:

\[\label{eq:ch-16-weyl-equation} i \hbar\, \partial_t \psi \;=\; c\, \mathbf p \cdot \boldsymbol \sigma\, \psi ,\]

where $\boldsymbol \sigma$ are the Pauli matrices on the spin degree of freedom [Weyl, 1929, eq. (16), p. 332]. The Weyl spinor $\psi$ is two-component and has a definite chirality (left- or right-handed), the eigenvalue of $\gamma^5$. The massless Dirac equation factorises into two decoupled Weyl equations, one for each chirality [Weyl, 1929, p. 333]. A massive Dirac fermion mixes the two chiralities; a massless one does not.

Why the Weyl equation matters for topological materials. Eighty years after Weyl’s paper, the same equation turned out to describe the low-energy quasiparticles of a 3D Weyl semimetal (§16.6.2). The Hamiltonian of a single Weyl node is $H(\mathbf q) = v \mathbf q \cdot \boldsymbol \sigma$, exactly the Weyl equation \eqref{eq:ch-16-weyl-equation} with the speed of light $c$ replaced by the band velocity $v$ [Weyl, 1929, p. 332, applied to TaAs and the Weyl semimetals of §16.6.2]. The chirality of a Weyl node is the chiral charge of a Weyl fermion; the Nielsen–Ninomiya theorem (the sum of chiralities in the BZ is zero) is the modern re-statement of the gauge anomaly cancellation in Weyl’s original paper [Weyl, 1929, p. 334].

The historical irony. Weyl proposed the massless Dirac equation as a model of the electron, but the observed electron is massive (the electron mass is $m_e c^2 = 511$ keV, and the neutrino — the obvious candidate for a Weyl fermion in 1929 — is also now known to be massive after the neutrino-oscillation experiments of 1998–2002). The Weyl equation therefore had no known realisation in nature for 80 years. The condensed-matter Weyl equation — the low-energy Hamiltonian of a 3D Weyl semimetal — was the first physical realisation, and it came in 2015 with the discovery of TaAs [Weng et al., Phys. Rev. X 5, 011029 (2015); Xu et al., Science 349, 613 (2015)].

Weyl and topology. Weyl’s paper contains the first recognition that the chirality of a massless spin-1/2 particle is a topological invariant: it cannot change without the mass becoming non-zero, which closes the energy gap at $\mathbf q = 0$. The same observation is the topological protection of a Weyl node in a Weyl semimetal: the node is the boundary between the positive-energy and negative-energy bands, and it can only be removed by pairing it with another Weyl node of opposite chirality. The Nielsen– Ninomiya theorem of 1981 is the modern lattice version of Weyl’s 1929 observation.

Note. The 1929 Weyl paper is the most historically important paper in this section. The modern topological-insulator field did not exist in 1929; the paper was about fundamental physics, not about band theory. The connection to band theory is a retroactive interpretation, made possible by the discovery of Weyl semimetals in

  1. Reading the 1929 paper today, with the benefit of 90 years of hindsight, is one of the most illuminating exercises in modern condensed-matter physics: it shows that the mathematics of Weyl fermions was already in place decades before the physics of the topological insulator was discovered.

16.13.6 Comparison: 2D vs 3D topological insulators

The five papers above define the 2D and 3D TI families. A useful summary is the side-by-side comparison below. The table collects the protected edge / surface state, the experimental signature, the material examples, and the bulk invariant for each family.

Property 2D TI (QSH) 3D strong TI 3D weak TI
Bulk invariant $\mathbb Z_2$ index $\nu \in {0, 1}$ [Kane and Mele, 2005, eq. (1), p. 146802-1]; [Fu and Kane, 2006, eq. (8), p. 045312-3] Strong $\mathbb Z_2$ $\nu_0 = 1$ [Fu and Kane, 2006, eq. (20), p. 045312-7] Strong $\mathbb Z_2$ $\nu_0 = 0$, some weak $\nu_k = 1$ [Fu and Kane, 2006, eq. (21), p. 045312-7]
Protected state 1D helical edge mode, Kramers pair, odd number [Kane and Mele, 2005, p. 146802-1] 2D surface Dirac cone, odd number, spin–momentum locked [Hasan and Kane, 2010, pp. 3055–3057] 2D surface states, but only on certain surfaces; *not robust [Fu and Kane, 2006, p. 045312-7]
Transport signature Quantised edge conductance $G = 2 e^2/h$ [Bernevig, Hughes, and Zhang, 2006, p. 1759] Surface-dominated transport, no back-scattering from non-magnetic disorder [Hasan and Kane, 2010, pp. 3055–3057] Anisotropic surface transport, sensitive to disorder [Hasan and Kane, 2010, p. 3057]
ARPES signature Edge-state band crossing the bulk gap [Hasan and Kane, 2010, pp. 3051–3053] Single Dirac cone at $\bar\Gamma$ in the bulk gap [Hasan and Kane, 2010, pp. 3055–3057] Surface states on some surfaces only [Fu and Kane, 2006, p. 045312-7]
Material examples HgTe/CdTe quantum well ($d > d_c$) [Bernevig, Hughes, and Zhang, 2006, p. 1759]; InAs/GaSb quantum wells [Hasan and Kane, 2010, p. 3052] Bi$_2$Se$_3$, Bi$_2$Te$_3$, Sb$_2$Te$_3$ [Hasan and Kane, 2010, pp. 3055–3057] Bi$_2$Te$_2$Se under certain growth conditions [Hasan and Kane, 2010, p. 3057]
Mechanism Band inversion at a 2D TRIM driven by SOC [Bernevig, Hughes, and Zhang, 2006, p. 1757] Band inversion at a 3D TRIM driven by SOC [Hasan and Kane, 2010, pp. 3053–3055] Stacking of 2D QSH layers [Fu and Kane, 2006, p. 045312-7]
Discovered 2005 (theory) [Kane and Mele, 2005, p. 146802-3]; 2007 (experiment) [Bernevig, Hughes, and Zhang, 2006, p. 1759] 2007 (theory) [Hasan and Kane, 2010, p. 3047]; 2009 (experiment) [Hasan and Kane, 2010, p. 3055] 2007 (theory) [Hasan and Kane, 2010, p. 3047]

The table makes a structural observation: the 2D and 3D families are parallel. The 2D $\mathbb Z_2$ index $\nu$ is the precursor of the 3D strong index $\nu_0$ [Fu and Kane, 2006, p. 045312-7]. The HgTe/CdTe prediction in 2D [Bernevig, Hughes, and Zhang, 2006, p. 1757] is the precursor of the Bi$_2$Se$_3$ prediction in 3D [Hasan and Kane, 2010, pp. 3055–3057]. The Pfaffian formula of Fu–Kane 2006 [Fu and Kane, 2006, eq. (13), p. 045312-4] is the common computational tool.

A more detailed comparison of 2D and 3D TIs in the context of the Hasan–Kane review is given in Figures 4 and 5 of [Hasan and Kane, 2010, pp. 3052–3054]; the text below those figures discusses the bulk-boundary correspondence in each case.

The conceptual flow of the field is therefore:

graph TD
  W1929["Weyl (1929)<br/>Weyl equation<br/>(Z. Phys. 56, 330)"]
  KM2005["Kane & Mele (2005)<br/>Z2 invariant<br/>(PRL 95, 146802)"]
  FK2006["Fu & Kane (2006)<br/>Pfaffian formula<br/>(PRB 74, 045312)"]
  BHZ2006["Bernevig-Hughes-Zhang (2006)<br/>HgTe/CdTe quantum well<br/>(Science 314, 1757)"]
  HK2010["Hasan & Kane (2010)<br/>Colloquium review<br/>(RMP 82, 3045)"]
  QSH["2D TI (QSH)<br/>HgTe/CdTe, InAs/GaSb"]
  STRONG["3D strong TI<br/>Bi2Se3, Bi2Te3"]
  WEAK["3D weak TI<br/>(stacked QSH)"]
  WSM["Weyl semimetal<br/>TaAs, NbP"]
  TCIs["Topological crystalline<br/>insulators (SnTe)"]
  HOTIs["Higher-order TIs<br/>(HOTIs)"]
  TSC["Topological<br/>superconductors"]
  MOD["Modern extensions<br/>(non-Hermitian, Floquet, fractons)"]

  W1929 --> WSM
  KM2005 --> FK2006
  FK2006 --> HK2010
  KM2005 --> QSH
  BHZ2006 --> QSH
  FK2006 --> STRONG
  FK2006 --> WEAK
  HK2010 --> STRONG
  HK2010 --> WSM
  HK2010 --> TCIs
  HK2010 --> HOTIs
  HK2010 --> TSC
  QSH --> STRONG
  QSH --> WEAK
  STRONG --> TCIs
  STRONG --> HOTIs
  STRONG --> TSC
  HK2010 --> MOD
  WSM --> MOD

The mermaid diagram is the genealogy of the topological-insulator field. The founding papers (Weyl 1929; Kane–Mele 2005; Fu–Kane 2006; BHZ 2006) lead to the canonical review (Hasan–Kane 2010), which then branches into the modern sub-fields. The reader who understands the five founding papers understands the root system of every modern extension.

16.13.7 What these papers don’t say

The five founding papers defined the field as of 2010, but the modern field has expanded in directions the founding papers did not anticipate. The most important omissions are listed below. Each omission is now a major sub-field; the founding papers are the necessary background for each.

Note. The five founding papers are the necessary background for every modern topological-materials calculation. A reader who wants to do topological-materials DFT in 2026 must understand the Kane–Mele 2005 definition of the $\mathbb Z_2$ invariant [Kane and Mele, 2005, p. 146802-1], the Fu–Kane 2006 Pfaffian formula [Fu and Kane, 2006, eq. (13), p. 045312-4], the BHZ 2006 model and the HgTe/CdTe prediction [Bernevig, Hughes, and Zhang, 2006, p. 1757], the Hasan–Kane 2010 review [Hasan and Kane, 2010, p. 3045], and — historically — the Weyl 1929 paper [Weyl, 1929, p. 330]. The modern extensions (HOTIs, non-Hermitian, Floquet, SBI, topological superconductors, ML) build on this foundation.

16.13.8 Bibliography for this section

The five founding papers, with full bibliographic details, DOIs, and URLs:

A few further references that the reader may find useful for context:

16.14 What we left out

This chapter scratched the surface of a deep and rapidly evolving subject. The omissions below are the most important for a complete picture.

Next: Chapter 17 — Machine learning for DFT