The two families every particle belongs to. Fermions – electrons, protons, neutrons – will not share: no two can be in the same state, the Pauli exclusion principle. That is why electrons fill an atom’s shells and a crystal’s bands one state at a time, up to the , and why matter takes up room. Bosons – , , and pairs of fermions bound together – do the opposite and crowd into the same state, which gives lasers, superfluids and . In two dimensions a third kind is possible: the anyon.
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Heavy-fermion physics, where localised f electrons hybridise with conduction electrons and make quasiparticles hundreds of times heavier than a free electron, used to require bulk intermetallic crystals.
The best-studied candidate for a Kitaev quantum spin liquid: a state in which magnetic moments never settle into an ordered pattern, even at absolute zero, and which theory predicts should carry exotic excitations called Majorana fermions.
Fermions cannot share a state, so electrons fill levels two at a time, spins opposite, up to the Fermi level; bosons can, and when cold they crowd into the lowest. In gated graphene the exclusion shows up in light: photons with less than twice the Fermi energy would lift electrons into states that are already full, so they are not absorbed.
Two kinds of particle
Identical particles cannot be told apart, and quantum mechanics takes that literally: swap two of them and nothing measurable may change. That leaves two possibilities. For bosons the stays exactly as it was; for fermions it changes sign. Which one a particle is follows from its – half-integer for fermions, such as the electron with spin ½, whole numbers for bosons, such as the photon.
Particles made of others follow the count of fermions inside them: an even number makes a boson, an odd number a fermion. A Cooper pair of two electrons is a boson, and so is an , an electron bound to a hole. Vibrations of a crystal, phonons, and its , magnons, behave as bosons too.
Taking turns
The sign change has a striking consequence: two fermions in the same state would have to cancel themselves out, so no two can share a state – the Pauli exclusion principle. Electrons therefore fill an atom’s and a crystal’s bands from the bottom, at most two to an orbital with opposite spins, up to the Fermi level. Chemistry’s shells, the difference between metals and and the stiffness of matter all follow from it, and so, together with electrostatic repulsion, does the behind magnetism.
It also decides what light can do. An electron can only be lifted into an empty state. In graphene a gate raises or lowers the Fermi level, and photons with less than twice the Fermi energy then find the states they would lift electrons into already full, so they pass without being absorbed – Pauli blocking, which follows as the changes (Wang and colleagues, 2008).
Crowding together, and anyons
Bosons have no such restriction, and at low temperature they prefer to share: photons pile into one mode in a laser, a cold gas of atoms gathers in a single quantum state – a Bose–Einstein condensate – and superconductivity is a condensate of Cooper pairs. Excitons, far lighter than atoms, should condense at much higher temperatures. In double layers of MoSe2 and WSe2 carrying at densities up to 1012 per square centimetre, light emission switches on sharply above a threshold density, in a way that persists above 100 K – evidence for exciton condensation (Wang and colleagues, 2019).
In three dimensions those are the only two options. Confined to a plane, particles can be exchanged in a way that remembers how they went round each other, and the wavefunction can pick up any phase: such particles are called anyons. The charged excitations of the fractional are anyons – a phase of 2π/3 has been measured for the state at one third filling (Nakamura and colleagues, 2020), and colliding them at a beam splitter shows their in-between statistics (Bartolomei and colleagues, 2020).
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Particles with half-integer spin obey Fermi–Dirac statistics and have wavefunctions antisymmetric under exchange; those with integer spin obey Bose–Einstein statistics and symmetric wavefunctions. Exclusion fills bands up to the Fermi level and causes Pauli blocking: gating graphene switches off interband absorption below twice the Fermi energy. Bound pairs of fermions act as bosons – Cooper pairs, excitons – and interlayer excitons in MoSe2/WSe2 double layers, at densities up to 1012 cm−2, show signs of condensation that persist above 100 K. In two dimensions exchange can multiply the wavefunction by any phase: the of the fractional quantum Hall state at ν = 1/3 are anyons, with a measured braiding phase of 2π/3.
The effective interaction between composite fermions, set entirely by the Coulomb potential and the underlying electronic Landau level orbitals, can stabilize exotic fractional quantum Hall states. In particular, half-filled Landau levels with different orbital character can host either metallic or … [Phys. Rev. Lett.…
The square-hexagon-dodecagon (SHD) lattice hosts an exactly solvable , while further-neighbor couplings frustrate its magnet into classically degenerate lines where quantum fluctuations can melt the Néel order. To place this state among the lattice's spin liquids, we classify fermionic…
Binding anyons to optically active excitons offers a route to manipulate fractional excitations and reveal the internal spatial structure of anyons. We establish the formation of anyon bound states between a mobile dipolar exciton and an e/3 quasihole in a MoTe2 . Using exact…