Chern number

In plain words

A whole number that adds up the twist of the electron waves in a band – the twist describes – over every way the electrons can move. Like the number of holes in a doughnut, it cannot change a little, so the effects it controls, such as a sideways resistance locked to an exact value, are extremely robust.

Going deeper

The Chern number counts how many times the electron states turn over as momentum sweeps the whole Brillouin zone. It can only take whole-number values, which is why the Hall conductance it fixes is quantised exactly, whatever the sample’s shape or purity.

An integer from the wavefunctions

Each Bloch state has a phase whose change around a closed path in momentum space is the Berry phase. The Berry curvature is the local density of that phase, and integrating it over a whole band gives 2π times an integer: the Chern number. It is a invariant – a property of how the states are arranged rather than of any energy – and it cannot change continuously. To change it, the gap must close somewhere.

The result, derived in 1982 for electrons in a periodic potential and a magnetic field, is that a filled band with Chern number C contributes exactly C·e²/h to the Hall conductance. This is what makes the plateaus so precise: a small deformation of the sample cannot change an integer.

When it can be non-zero

forces the Berry curvature at opposite momenta to be opposite, so the total over a band vanishes: an ordinary non-magnetic crystal has C = 0. A non-zero Chern number therefore requires time reversal to be broken – by an applied magnetic field, as in the quantum Hall effect, or by magnetism inside the material, which gives the quantum anomalous Hall effect with no magnet at all.

That was realised first in thin films of chromium-doped (Bi,Sb)₂Te₃ and later in MnBi₂Te₄ and in systems, where with non-zero Chern numbers appear in twisted graphene and twisted . Where time reversal is preserved but inverts the bands, the relevant invariant is instead the Z₂ index of the state.

Why it is more than bookkeeping

The bulk–boundary correspondence ties the number to something measurable: a boundary between regions of different Chern number must carry exactly that many one-way . Those channels are what carry the current in the quantum Hall effect, and they are robust for the same reason the integer is.

The idea generalises. Chern numbers classify photonic and acoustic crystals, cold-atom lattices and bands; with interactions, partially filled flat bands with non-zero Chern number can host , the zero-field analogue of the fractional quantum Hall effect. In each case the working test is the same: look for a quantised response and for edge modes that survive .

For specialists

The integral of Berry curvature over a band, an integer that fixes the quantised Hall conductance contributed by that band.

Where this comes from

Quantized Hall conductance in a two-dimensional periodic potential Thouless et al. · Physical Review Letters 49, 405 (1982) cited by 6,954

Topological phase

A state of matter told apart not by how its atoms are arranged but by a whole number that describes how its electrons’ waves twist across the crystal – the way a doughnut differs from a ball by its one hole. A whole number cannot change a little, so what it guarantees, such as current running along an edge without loss, survives defects and dirt until the band gap itself closes.

A gapped phase characterised by a topological invariant of its occupied bands – a Chern number, a Z₂ index, a winding number – that cannot change under deformations that keep the gap open and any protecting symmetry intact. Where regions with different invariants meet, the gap must close, which forces boundary states: chiral edge channels in Chern insulators, helical ones in quantum spin Hall insulators, Fermi arcs in Weyl semimetals, Majorana modes in topological superconductors. Topological order in the strict sense – the long-range entanglement of fractional quantum Hall states and spin liquids – is a distinct, stronger notion.

Full explanation

Quantum Hall effect

In a strong magnetic field at low temperature, electrons flowing along a flat sheet are pushed to one side, so a voltage builds up across it. That sideways voltage, divided by the current, stops changing smoothly and locks onto exact steps set only by fundamental constants – the same in every sample, and so precise that they are used to define the ohm.

Quantisation of the Hall conductance in units of e²/h when a 2D electron system sits in a strong perpendicular field and the Fermi level lies between Landau levels, with the longitudinal resistance vanishing. Graphene shows a half-integer sequence with a level pinned at zero energy, a direct consequence of its Dirac dispersion and π Berry phase, and the effect survives to room temperature there. The zero-field analogues are the quantum anomalous and quantum spin Hall effects.

Full explanation

Time-reversal symmetry

The idea that the laws governing electrons would look just as valid if you ran a film of them backwards. Reversing time flips every motion and every spin, so a material with no magnetism of its own usually keeps this symmetry, while a magnet – or a magnetic field – breaks it. Which case applies decides a lot: some protected edge currents need the symmetry, and others need it broken.

The antiunitary operation t → −t, which reverses momenta and spins; for spin-½ electrons T² = −1, giving Kramers degeneracy of every state at time-reversal-invariant momenta. It is broken by magnetic order or applied fields. Its presence protects the helical edge states of quantum spin Hall insulators and makes Berry curvature odd in k, so valley Hall but not anomalous Hall effects survive; its breaking allows Chern insulators, the anomalous Hall effect and Kerr rotation, the standard probes of spontaneous breaking in correlated phases.

Full explanation

Edge state

Electrons that can only travel along the border of a sheet while its interior stays insulating. In a topological material these border lanes are guaranteed to exist, and in the cleanest cases they carry current without losing energy to scattering.

States localised at a sample boundary and dispersing inside the bulk gap. In quantum Hall and Chern insulators they are chiral, one-way channels; in a quantum spin Hall insulator they are helical, counter-propagating with opposite spins and protected by time reversal, giving e²/h per edge in the ideal case – monolayer 1T′-WTe₂ keeps that edge conduction up to about 100 K. Trivial edges carry states too, from dangling bonds, reconstructions and graphene’s zigzag edge, so the evidence for topology is quantised conductance and its magnetic-field dependence, not conduction at the edge alone.

Full explanation

Disorder and localisation

Every real crystal has some disorder: missing or misplaced atoms, impurities, stray charges in the substrate, a random mixture of elements in an alloy. Electrons moving through it scatter off these imperfections, which raises the resistance. If the disorder is strong enough, electron waves become trapped in small regions – localised – and the material stops conducting, even if it would be a metal when perfect. In a sheet one atom thick, disorder in its surroundings matters especially.

Deviations from perfect periodicity – point defects, substitutional or alloy disorder, interface roughness, charged impurities and strain variations in the substrate – that scatter carriers and, beyond a threshold, localise their wavefunctions through interference (Anderson localisation). Scaling theory predicts that non-interacting electrons in two dimensions without spin–orbit coupling are localised by any disorder; its precursor, weak localisation, appears as a logarithmic rise of resistance at low temperature that a magnetic field suppresses, while strong spin–orbit coupling gives weak antilocalisation. In graphene on SiO₂, charged impurities break the sheet near charge neutrality into electron–hole puddles, which hBN encapsulation reduces. Strong disorder leads to hopping conduction, broadens optical lines and smears phase transitions; in frustrated magnets and correlated oxides, structural or cation disorder can mimic exotic states.

Full explanation