Modern Physics · Advanced Topics

Topological Phases of Matter

Topology — the mathematics of shapes and connectivity — enters condensed matter physics in a profound way. Topological insulators, the quantum Hall effect, and topological superconductors host protected surface states and exotic quasiparticles that cannot be removed by any smooth perturbation.

Prerequisites

Solid-state physics (Ch. SS) \cdot Quantum mechanics (Ch. 20) \cdot Group theory (Ch. GT) \cdot Linear

algebra (Ch. LA)

Learning Goals

Define the Berry phase and Berry curvature and compute them for a spin-1/2 in a rotating magnetic field.
Derive the TKNN formula relating Hall conductance to the first Chern number of filled Bloch bands.
Classify 1D topological phases using the SSH model winding number and identify protected edge states.
$Explain how time-reversal symmetry gives rise to the Z_{2} invariant and spin-momentum-locked$ surface states in 3D topological insulators.
Describe Majorana bound states in the Kitaev chain and their potential role in topological quantum computation.

TP.1 Topology in Quantum Mechanics

In band theory, the electronic states at each crystal momentum k form a Hilbert space. As k traverses the Brillouin zone (a torus T^d in d dimensions), the occupied bands define a vector bundle — and vector bundles are classified by topological invariants.

The key invariant: the Berry phase. As k evolves slowly around a loop in the Brillouin zone, the eigenstate |u_k⟩ acquires a geometric phase:

\gamma = i \oint ⟨u_{k}|\nabla_k|u_{k}⟩ \cdot dk \qquad (Berry phase around a loop in k-space)

(TP.1)

The integrand A_k = i⟨u_k|∇_k|u_k⟩ is the Berry connection (analogous to the vector potential A in electromagnetism). The associated curvature is theBerry curvature: Ω_k = ∇_k × A_k (analogous to the magnetic field B).

TP.2 The Quantum Hall Effect and Chern Numbers

In a 2D electron gas under strong perpendicular magnetic field B, the Hall conductance is quantized with extraordinary precision:

\sigma_xy = \nu \times e^{2}/h \qquad (integer quantum Hall effect, IQHE)

(TP.2)

where ν = 1, 2, 3, ... is an integer. The quantization is exact — better than 1 part in 10⁹ — independent of sample size, purity, or microscopic details. This robustness is topological in origin.

Theorem TP.1 — TKNN Formula — Chern Number

Thouless, Kohmoto, Nightingale, de

n Nijs (1982): the Hall conductance of a filled band is given by the first Chern number C_{1}

\sigma_xy = C_{1} \times e^{2}/h \qquad C_{1} = (1/2\pi) \int_BZ \Omega_k d^{2}k ∈ ℤ

The Chern number C_{1} is a topological invariant — it cannot change without closing the band

gap. A sample with C_{1} = 1 and a vacuum with C_{1} = 0 must have a topological interface — a g

pless edge state that conducts without dissipation. This is the quantum Hall edge channel.

The fractional quantum Hall effect (ν = 1/3, 2/5, ...) requires strongly correlated states — Laughlin wavefunctions — beyond simple band theory. The elementary excitations carry fractional charge e/3 (Nobel 1998 to Laughlin, Störmer, Tsui). These anyons (with fractional statistics) are candidates for topological quantum computing.

TP.3 Time-Reversal Invariant Topological Insulators

Without a magnetic field, time-reversal symmetry T² = −1 (for spin-½ particles) allows a different topological invariant — the ℤ₂ invariant ν ∈ {0, 1}.

ν = 0: ordinary insulator. ν = 1: topological insulator (TI). The TI has a bulk band gap but gapless surface states — protected by time-reversal symmetry. These surface states cannot be gapped without breaking time-reversal or closing the bulk gap.

Key feature: spin-momentum locking. The surface Dirac cone has spin perpendicular to momentum — spin up goes right, spin down goes left. Backscattering (k → −k) requires flipping spin, which is forbidden by time-reversal. This makes the surface states immune to non-magnetic disorder.

First 2D TI: HgTe quantum wells (König et al., 2007, Science). First 3D TI: Bi₂Se₃ (Chen et al., 2009) — metallic Dirac cone on the surface, insulating bulk. ARPES directly images the Dirac cone surface state.

Example TP.1 — SSH Model — 1D Topological Insulator

$The Su-Schrieffer-Heeger (SSH) model: 1D chain with alternating hopping t_{1} (intracell) and$ $t_{2} (intercell). Find the bulk invariant and edge states$

Bloch Hamiltonian:

H(k) = (t_{1} + t_{2} cos k)\sigma_x + t_{2} sin k \sigma_y (2\times2 matrix, two sites per unit cell). Eigenvalue

s: E = \pm\sqrt((t_{1}+t_{2}cos k)^{2} + t_{2}^{2}sin^{2}k

Gap condition:

Gap closes at k = 0: t_{1} + t_{2} = 0 \to |t_{1}| = t_{2}. Gap closes at k = \pi: t_{1} - t_{2} = 0 \to t_{1} = t_{2}

Phase transition at |t_{1}| = |t_{2}|

Winding number:

The Hamiltonian H(k) = d(k)\cdot\sigma. As k: 0 \to 2\pi, d(k) = (t_{1}+t_{2}cos k, t_{2}sin k) traces an ellips

e. Winding number W = number of times it winds around origin. W = 0 for t_{1} > t_{2} (trivial

;

W = 1 for t_{1} < t_{2} (topological

Edge states:

Topological phase (t_{2} > t_{1}): semi-infinite chain has zero-energy edge state localized at t

he boundary, with wavefunction decaying as (t_{1}/t_{2})^n. Two edge states at opposite ends — r

obust to perturbations that preserve chiral symmetry.

Physical realization:

Polyacetylene (CH)_n in dimerized (Peierls) phase — the SSH model. Soliton defects carry c

harge e/2 (fractionalization). Modern realizations: photonic lattices, ultracold atoms, acoustic metamaterials.

TP.4 Topological Superconductors and Majorana Fermions

A topological superconductor has a superconducting bulk gap but hosts Majorana bound states at its boundaries. Majorana fermions are their own antiparticle: γ† = γ. They obey non-Abelian statistics.

The Kitaev chainmodel: 1D chain of spinless fermions with p-wave pairing Δ. Topological phase for |μ| < 2t. Majorana modes at each end: γ₁ = c₁ + c₁†, γ₂ = i(c_N − c_N†). The two Majoranas form a zero-energy non-local fermion c = (γ₁ + iγ₂)/2 — the ground state is degenerate (two-fold: c|0⟩ and |0⟩).

Non-Abelian anyons: exchanging two Majoranas performs a unitary rotation in the degenerate ground state subspace. This rotation depends on the order of exchanges — non-commuting, hence non-Abelian. Topological quantum computation: encode qubits in Majorana pairs; gates implemented by braiding. Errors require exciting quasiparticles across the gap — exponentially suppressed.

Experimental candidates: semiconductor nanowires (InAs, InSb) with strong spin-orbit coupling, proximity-coupled to s-wave superconductors and in a magnetic field. Zero-bias conductance peak signatures observed but not yet conclusively proven to be Majorana (2012–2024). Recent Microsoft experiments (2023) show promising signatures.

TP.5 Weyl Semimetals

A Weyl semimetal is a 3D material with band crossings (Weyl points) in the bulk. Near each crossing, the Hamiltonian:

H(k) = \pmv_F (k \cdot \sigma) \qquad (Weyl Hamiltonian, \pm = chirality)

(TP.3)

Weyl points act as monopoles of Berry curvature in k-space — they come in pairs of opposite chirality (Nielsen-Ninomiya theorem) and cannot be annihilated individually. On the surface, Weyl points project onto arcs — Fermi arcs — open contours connecting projections of Weyl points of opposite chirality.

The chiral anomaly: in a parallel E and B field, charge is pumped between Weyl nodes of opposite chirality → negative magnetoresistance. First observed in TaAs (2015). Weyl semimetals are 3D analogues of graphene's Dirac points, but topologically protected in 3D.

Definition TP.1 — Common Traps

Topology is global: local perturbations cannot change an invariant without closing a gap.
Edge states depend on bulk topology: they are not ordinary surface defects.
Berry phase is gauge-sensitive but observables are not: invariants remove arbitrary phase choices.
Protection has limits: disorder or interactions that break required symmetries can destroy a phase.

Exercises — TP.1–TP.5 Topological Phases

Calculate the Berry ph

ase for a spin-\frac{1}{2} particle when its quantization axis traces a closed loop of solid angle \Omega

What is the result for an equatorial loop?

radians

Straightforward

Compute the Chern number for the 2D massive Dirac Hamiltonian H = v_{F}(k_{x} \sigma_x + k_{y} \sigma_y) +

m \sigma_z. Show it changes at m = 0

Intermediate

Derive the Landau level spectrum in a perpendicular magnetic field.

What is the degeneracy per level? At what temperature does the IQHE occur for GaAs (m* = 0

Intermediate

4.Describe the tenfold way classification of topological insulators and superconductors. What symmetry classes govern 3D topological insulators and 1D topological superconductors?

Challenging

Key Takeaways

$Berry phase \gamma = i\oint⟨u|\nabla_k|u⟩\cdotdk: geometric phase from adiabatic evolution in parameter spac$ e.
$Chern number C_{1} = (1/2\pi)\int\Omega_k d^{2}k ∈ ℤ. Hall conductance \sigma_xy = C_{1} e^{2}/h (TKNN$ .
$Topological insulators: ℤ_{2} invariant from T^{2}=-1. Gapless Dirac surface states, spin-moment$ um locking.
$SSH model: winding number W=0 (trivial) or W=1 (topological). Edge zero modes in topologic$ al phase.
$Majorana fermions (\gamma^\dagger=\gamma): at ends of Kitaev chain. Non-Abelian statistics \to topological qu$ bits.
$Tenfold way: 10 symmetry classes \times dimension \to periodic table of topological phases$ .