Fractional Chern insulator

In plain words

A state in which electrons act together so strongly that a disturbance in them behaves like a particle carrying a fraction – a third, say – of an electron’s charge, although no electron has been split. Such states were long thought to need enormous magnetic fields; in some twisted and stacked they appear without any.

Going deeper

A fractional Chern insulator is the lattice version of a fractional quantum Hall state: the same fractionally charged excitations, but produced by the band structure of a twisted stack rather than by a laboratory magnet.

Fractions without a magnet

In a strong magnetic field a two-dimensional electron gas has no kinetic energy left to speak of: the states collapse into , all at the same energy. Fill one of those levels only partly and the electrons have nothing to do but arrange themselves around each other. The result is the fractional quantum Hall effect, whose excitations carry a fraction of an electron’s charge.

A is a lattice band that carries the same index as a Landau level. If such a band is also flat, the argument above applies word for word, with no magnetic field anywhere in it. That possibility was written down in 2011; the question was which crystal would supply a flat Chern band.

What a twisted stack supplies

Twisting two layers makes a with a very small Brillouin zone and correspondingly narrow minibands. In rhombohedrally stacked homobilayers, strong and the layer degree of freedom give those minibands a non-zero Chern number, with opposite signs for the two . Interactions then pick one valley – the layer becomes on its own – and what is left is a single partly filled Chern band.

In 3.7° twisted MoTe₂ that is what was seen. Magnetic showed ferromagnetic states at fractional hole fillings, and the way those states moved with magnetic field followed the Streda formula for Hall conductances of −2/3 and −3/5 e²/h, while the ν = −1 state behaved as a Chern number −1 quantum anomalous Hall . States on the electron side did not shift at all, marking them as ordinary .

What still has to be shown

The first reports read the fractions optically, from how the states dispersed with field rather than from a measured conductance; quantised transport in the same material followed soon afterwards. That order matters, because a Landau-fan slope is consistent with a fractional Chern insulator but does not by itself exclude other states that shift for other reasons.

The open questions are about what else lives in these bands. Charge-ordered and Wigner-like states compete at nearby fillings, smears the plateaux, and the fillings where non-Abelian excitations would appear – the reason anyone is interested in the first place – remain harder to reach than the ones already seen. , pressure and are the knobs; sample-to-sample reproducibility is the constraint.

For specialists

A lattice analogue of a fractional quantum Hall state that forms in a partially filled Chern band, potentially without any external magnetic field.

Where this comes from

Fractional Chern insulator Regnault and Bernevig · Physical Review X 1, 021014 (2011) cited by 605
Signatures of fractional quantum anomalous Hall states in twisted MoTe₂ Cai et al. · Nature 622, 63 (2023) cited by 718

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

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

Linear and circular dichroism

When a material absorbs light differently depending on how the light is polarised. In linear dichroism the difference is between light vibrating along one direction of the crystal and across it, as in a polarising filter; in circular dichroism, between light whose vibration turns one way and the other. Shining polarised light on a flake this way reveals hidden directions in it – its crystal axes, its magnetic order, which valley its electrons sit in – without touching it.

Polarisation-dependent absorption: linear dichroism between orthogonal linear polarisations, from in-plane anisotropy of the lattice or of an order that breaks rotational symmetry; circular dichroism between left and right circular polarisations, from broken mirror or time-reversal symmetry. In black phosphorus, ReS₂ and the transition metal trichalcogenides linear dichroism follows the crystal axes; in FePS₃ it appears with zigzag antiferromagnetic order and tracks it with temperature. In monolayer TMDCs valley-selective circular dichroism at K and K′ underlies optical valley polarisation, and magnetic circular dichroism – reflective in the visible, XMCD at X-ray absorption edges – measures magnetisation, with sum rules separating spin and orbital moments.

Full explanation

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

Strong electron correlation

What happens when electrons in a material repel each other so strongly that they can no longer be treated as moving independently. Each electron’s motion then depends on where all the others are, and the result can be an insulator where simple theory predicts a metal, or magnetism, superconductivity and other collective states.

The regime in which the Coulomb repulsion U between electrons is comparable to or larger than their kinetic energy, set by the bandwidth W, so single-particle band theory fails. At half filling with U well above W the Hubbard model gives a Mott insulator with local moments and antiferromagnetic exchange of order t²/U; doping or tuning U/W gives correlated metals, heavy-fermion behaviour and unconventional superconductivity. In 2D, weak screening keeps U large and moiré superlattices shrink W, so twisted bilayers make U/W tunable by gate and twist. Semi-local DFT misses the physics; DFT+U, DMFT, exact diagonalisation and quantum Monte Carlo are the usual tools.

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

Gating

Changing a material by putting a voltage on a nearby electrode – a gate – separated from it by an insulator. The voltage pulls electrons into the material or pushes them out, as in every transistor. Because a 2D material is so thin, the gate reaches all of it, so the number of electrons can be dialled up and down continuously: a single device can be turned from insulator to metal, its magnetism strengthened, or superconductivity switched on. Two gates, above and below, can also apply an electric field across the layer.

Electrostatic control of carrier density and electric field through a gate coupled capacitively across a dielectric: the induced density equals the gate capacitance times the voltage beyond threshold, divided by the electron charge, reaching a few 10¹³ cm⁻² with oxide or hBN gates before breakdown. Dual gating sets density and perpendicular displacement field independently, opening the gap of bilayer graphene and tuning moiré flat bands. Ionic-liquid and solid-electrolyte gating form an electric double layer about a nanometre thick and reach 10¹⁴–10¹⁵ cm⁻², enough to induce superconductivity in MoS₂ or raise the Curie temperature of Fe₃GeTe₂, but can intercalate or react electrochemically and work only while the ions are mobile. Hysteresis from traps, contact effects and quantum capacitance are the usual caveats.

Full explanation