Z₂ invariant

In plain words

A yes-or-no label, 0 or 1, that says whether an has protected conducting edges. A value of 1 marks a particular kind of .

Going deeper

The Z₂ invariant asks a parity question: how many pairs of edge states cross the Fermi level between the two time-reversal-invariant momenta. An even number can be pushed out of the gap; an odd number cannot, and those edges must conduct.

A different kind of index

The is labelled by the , an integer that counts channels and requires a magnetic field to break . The quantum spin Hall phase is different: it is time-reversal invariant, which forces the Chern number to vanish, yet it still has a gapped bulk and conducting edges, with counter-propagating states of opposite .

Kane and Mele showed that this phase carries its own label – not an integer, but a two-valued one. An insulator with time-reversal symmetry is either trivial or not, with nothing in between, and that Z₂ classification plays the role for the quantum spin Hall effect that the Chern number plays for the quantum Hall effect. They established it first in a two-band model of graphene with , then generalised it to multiband and interacting systems.

Why only a parity survives

Time reversal acts twice over on electrons: it relates a state at momentum k and spin up to one at −k and spin down, and Kramers’ theorem guarantees these come in degenerate pairs at the special momenta where k and −k coincide. Edge bands therefore connect between those special points in pairs.

Now count how many times edge bands cross the in half the Brillouin zone. If that number is even, the bands can be pushed around and out of the gap in pairs without violating any symmetry – the edge can be gapped, and the insulator is ordinary. If it is odd, one crossing is always left over, because removing it would require pairing a state with itself. That leftover crossing is a conducting edge channel that no amount of deformation or non-magnetic can remove. Only the parity is robust, which is exactly why the invariant takes two values.

Computing it, and breaking it

Evaluating the invariant from the definition is awkward. The practical breakthrough was the observation that in a crystal with a centre of inversion it reduces to a product of parity eigenvalues of the occupied bands at a handful of time-reversal-invariant momenta – a calculation so cheap that it turned topology into something one can screen whole materials databases for, which is largely how the current catalogue of candidate topological materials was assembled. Without the modern approach tracks Wannier centres, or Wilson loops, around the Brillouin zone.

The protection has an obvious weak point: it rests entirely on time-reversal symmetry. Magnetic impurities, an applied magnetic field, or at the edge will gap the crossing and destroy the quantisation, while ordinary non-magnetic disorder will not. In two dimensions the canonical candidate is -WTe₂, where edge conduction consistent with the quantum spin Hall phase has been measured up to temperatures far above where anyone expected it – which is why the invariant appears in a glossary about layered materials rather than only in a topology course.

For specialists

The topological index of time-reversal-symmetric insulators that distinguishes a quantum spin Hall insulator from a trivial one.

Where this comes from

Z₂ topological order and the quantum spin Hall effect Kane and Mele · Physical Review Letters 95, 146802 (2005) cited by 6,293
Topological insulators with inversion symmetry Fu and Kane · Physical Review B 76, 045302 (2007) cited by 4,251
Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal Wu et al. · Science 359, 76 (2018) cited by 823

Metal, semimetal and insulator

The three basic ways a solid treats electricity, set by how its electrons fill the energies open to them. In a metal they fill a band of energies only partly, so the smallest push sets them moving, and it conducts at any temperature. In an insulator they fill a band completely and a wide gap separates it from the next, so nothing can move. A semimetal sits in between: two bands just touch or overlap a little, leaving a small number of mobile electrons and holes. NbSe₂ is a metal, graphene a semimetal, hBN an insulator.

A classification by where the Fermi level falls in the band structure. In a metal it crosses a partly filled band, leaving a Fermi surface, a carrier density of order one per atom (about 10²² cm⁻³) and a resistivity that falls on cooling. In a band insulator it lies in a gap of several electronvolts; a semiconductor is an insulator whose gap is small enough to be doped or gated. A semimetal has a slight overlap between conduction and valence bands (graphite, bismuth, bulk WTe₂) or bands that touch at isolated points (graphene), giving low and nearly equal densities of electrons and holes. Interactions can open gaps that band theory does not predict: Mott insulators from half-filled bands, excitonic insulators from semimetals.

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

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

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

Spin

A built-in property of every electron that makes it a tiny magnet. Measure it along any direction and you only ever find one of two answers, ‘up’ or ‘down’. Despite the name, nothing is actually spinning – the word stuck from an early picture – but the magnetism is real: countless electron spins lined up are what make a fridge magnet stick. Spintronics tries to carry information in spin rather than in charge.

The intrinsic angular momentum of the electron, ħ/2, with a magnetic moment of almost exactly one Bohr magneton; a measurement along any axis yields one of two values. In solids spin couples to orbital motion through spin–orbit coupling and to other spins through exchange; in 2D materials it sets magnetic order, spin–valley locking and how long a spin survives as a carrier of information.

Full explanation

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

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