⭕ Landau Levels — Electrons in a Magnetic Field

Landau levels are quantised energy states of electrons in a 2D plane subjected to a perpendicular magnetic field. Classically, electrons undergo circular cyclotron orbits; quantum mechanics quantises the orbital energy into discrete levels Eₙ = ℏωc(n + 1/2). Between levels are energy gaps where no states exist — the system behaves like a 1D harmonic oscillator in the direction perpendicular to its motion.

When a 2D electron system in a strong perpendicular magnetic field is cooled to low temperatures, the Hall conductance (the ratio of current to transverse voltage) is quantised to integer multiples of e²/h (approximately 3.87×10⁻⁵ siemens). This quantisation is topologically protected, extraordinarily precise, and independent of sample impurities — a consequence of filled Landau levels and the topology of electronic band structure.

The von Klitzing constant RK = h/e² ≈ 25,813 ohms is a fundamental constant depending only on Planck's constant and the electron charge. The quantised Hall resistance can be measured with a precision of 10⁻¹⁰ and is independent of temperature, material, and sample geometry, making it an ideal primary resistance standard. Since 2019, the International System of Units (SI) defines the ohm in terms of h and e using this effect.

The FQHE occurs at very high magnetic fields when a Landau level is partially filled. Strong Coulomb repulsion between electrons within the level creates a correlated ground state (Laughlin state) with a gap to excitations. Quasiparticle excitations carry fractional charge (1/3 of the electron charge for the ν=1/3 state) and obey fractional (anyonic) statistics, behaviour impossible for free electrons and with potential applications in topological quantum computing.

As the magnetic field is swept, Landau levels pass through the Fermi energy successively. Each time a level empties, there is a discontinuity in the number of electrons at the Fermi level, causing oscillations in magnetisation (de Haas-van Alphen) and resistance (Shubnikov-de Haas) with period 1/B. The frequency of these oscillations is proportional to the area of the Fermi surface cross-section perpendicular to B, allowing its detailed mapping.