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Robert Laughlin's 1983 variational method uses pairwise zeros to suppress close electron encounters within the lowest Landau level. For aligned spin, odd $m$ gives fermionic exchange symmetry.^[ref]

Here $\mathcal{N}$ enforces wavefunction normalization, ${\ell }_B$ is the magnetic length, and complex coordinates $z_j=x_j-i{y}_j$ describe electrons in a magnetic field $B\hat{\mathbf{z}}$, $B>0$, with the symmetric gauge. Reversing the orientation conjugates the polynomial without changing probability densities. The Gaussian function confines the wavefunction.

The exchange check is especially simple for two electrons. Writing $w=z_1-z_2$, exchange sends $w\mapsto -w$, so $w^m\mapsto (-1{)}^m{w}^m$. An aligned two-electron spin state is symmetric, requiring the spatial wavefunction to change sign. Hence $m=1,3,5,\dots$ works, whereas $m=2$ would give the wrong sign. At a close encounter, the probability density vanishes as $|w{|}^{2m}$: $m=3$ imposes a sixth-order zero, stronger than the quadratic zero at $m=1$. This is the local geometric mechanism for avoiding a large repulsive Coulomb potential energy.

A useful large-$N$ counting outline follows directly from the polynomial. Each complex coordinate has maximum degree $m(N-1)$. A lowest Landau level orbital proportional to $z^k{e}^{-|z{|}^2/(4{\ell }_B^2)}$ has its areal probability density maximum at $r^2=2k{\ell }_B^2$, by differentiating its logarithm. This suggests droplet area $A\simeq 2\pi mN{\ell }_B^2$, neglecting the boundary correction. Dividing $N/A$ by the degeneracy of a Landau level per area, $1/(2\pi {\ell }_B^2)$, gives filling factor $\nu \simeq 1/m$. This counting is a motivation for the fraction, not an independent proof that the wavefunction minimizes the interacting Hamiltonian.

For an electron in a uniform perpendicular magnetic field of magnitude $B>0$, the magnetic length sets the orbital length scale of a Landau level; $e$ is the positive elementary charge.

For a particle of charge $q$ and mass $m$ in a uniform magnetic field $\mathbf{B}=B\hat{z}$, minimal coupling replaces momentum by $\mathbf{p}-q\mathbf{A}$. Choose Landau gauge $\mathbf{A}=(0,Bx,0)$ and first restrict motion to the $xy$ plane. Since $p_y=\hslash k_y$ is conserved,These Landau levels quantize the classical cyclotron motion. The wavefunctions are harmonic-oscillator functions centred at $x_c$, multiplied by $e^{i{k}_{y}y}$. Changing $k_y$ shifts the guiding centre without changing the energy. Gauge-dependent labels such as $k_y$ are not themselves observables; the level spacing and degeneracy are gauge independent. The magnetic length ${\ell }_B=\sqrt{\hslash /|qB|}$ is the transverse orbital scale.

For a large rectangle of sides $L_x,L_y$, periodicity along $y$ gives $\Delta k_y=2\pi /L_y$. Requiring centres inside the sample gives, for each orbital level and each specified spin state,Thus each magnetic flux quantum supplies one guiding-centre state. The continuum bulk count neglects boundary corrections; magnetic periodic conditions on a torus require integer flux count. A finite confining edge lifts the simple flat guiding-centre spectrum near that edge.

spin adds a Zeeman term. For electrons with gyromagnetic factor $g$, its contribution is $g{\mu }_{B}B{m}_s$, with $m_s=\pm 1/2$ and ${\mu }_B=e\hslash /(2m_e)$. If $g=2$ and the orbital mass is $m_e$, the branches become $\hslash {\omega }_{c}n$ and $\hslash {\omega }_c(n+1)$: the lowest combined level has one spin branch, while positive levels coincide in pairs. In a material, effective mass and effective $g$ need not give that coincidence.

In three dimensions, free motion parallel to $B$ adds ${\hslash }^2{k}_z^2/(2m)$, so each orbital index labels a one-dimensional continuum rather than an isolated three-dimensional energy. Resolved magnetic quantization requires the spacing to exceed substantial thermal and disorder broadening. The changing occupation of these levels underlies magnetic quantum oscillations; in two dimensions their degeneracy and filling are central to the integer quantum Hall effect. These statements assume nonrelativistic, approximately independent particles, rather than arbitrary interacting or disordered spectra.

A uniform magnetic field quantizes the transverse cyclotron motion of an electron. With electron charge $-e$ and the Landau gauge for a uniform magnetic field $\mathbf{A}=(0,Bx,0)$, minimal coupling givesFor an eigenstate of $p_y$ with eigenvalue $\hslash k_y$, this is a quantum harmonic oscillator centered at $x_c=-\hslash k_y/(eB)$, with cyclotron frequency ${\omega }_c=eB/m$. Its Landau levels areThe oscillator width is set by the magnetic length ${\ell }_B=\sqrt{\hslash /(eB)}$. In three dimensions, the free motion parallel to the magnetic field adds ${\hslash }^2{k}_z^2/(2m)$. The electron spin adds a Zeeman splitting $\pm g_s{\mu }_{B}B/2$. In a crystal the appropriate band effective mass and effective spin factor replace the free-electron values.

Changing $k_y$ moves the orbit center without changing the energy. In a rectangle $L_x{L}_y$, adjacent allowed centers differ by $2\pi {\ell }_B^2/L_y$. Requiring the center to lie in the sample gives the degeneracy of a Landau level, per spin,This large state degeneracy, together with the increasing level spacing, rearranges the occupied states as the magnetic field changes. To resolve the Landau levels, thermal broadening $k_{B}T$ and disorder broadening should both be small compared with $\hslash {\omega }_c$.

The de Haas-van Alphen effect is an oscillation of magnetization as successive Landau levels pass the Fermi energy. The Shubnikov-de Haas effect is the corresponding oscillation in longitudinal magnetoresistance. For a simple orbit, successive crossings occur periodically in $1/B$; the general relation to the extremal orbit area is Onsager quantization. These measurements reveal the geometry and effective masses of a Fermi surface.

In a two-dimensional electron system, filling an integer number $\nu$ of resolved levels leads to the integer quantum Hall effect, withand very small longitudinal resistivity on a plateau. Disorder localization helps produce finite plateau widths. Cyclotron resonance provides another direct signature: radiation is absorbed at the adjacent-level spacing $\hslash {\omega }_c$.

The conduction and valence dispersions are two cones touching at the Dirac point, with slopes $\pm \hslash v_F$. Here $\mathbf{k}$ is measured relative to one valley centre. Each cone has a density of states proportional to $|E|$, which vanishes at the touching energy. Gate-induced carriers satisfy approximately $n=C_g(V_g-V_D)/e$ per unit area. Negative and positive offsets from $V_D$ produce holes and electrons, respectively. Away from neutrality a simple charge carrier mobility model gives $\sigma \simeq e\mu |n|$, so resistance decreases on either side of a maximum near $V_D$. Disorder, electron–hole puddles, contacts and quantum transport round the peak; the cone dispersion alone fixes neither its height nor its width, nor does it require exact electron–hole symmetry in a real device.

A direct derivation of the magnetic spectrum starts from the single-valley Dirac Hamiltonian operator $H=v_F({\sigma }_x{\pi }_x+{\sigma }_y{\pi }_y)$. Introduce oscillator operators using the magnetic length ${\ell }_B=\sqrt{\hslash /(e|B|)}$. Up to exchanging sublattices or valleys,Squaring gives adjacent oscillator number operators, henceThe positive expression printed in the source describes the electron branch only; the negative-energy partners must also be included. The same square-root dependence follows from quantizing circular orbit area, with the Dirac Berry phase cancelling the usual half-integer offset.

Each spin and valley contributes $Ae|B|/h$ states in area $A$. Including two spins and two valleys gives $N_{\mathrm{LL}}=4Ae|B|/h$, or areal degeneracy $4e|B|/h$. The zero-energy level is half-filled at charge neutrality: the carrier density counts departures from that state.

For $n=10^{16}{\mathrm{m}}^{-2}$, the filling factor and fourfold graphene Hall sequence areIn a plateau, ${\rho }_{xx}\simeq 0$ and $|{\rho }_{xy}|=h/({\nu }_N{e}^2)$. The Hall sign reverses between electron and hole doping; the sketch uses its magnitude to avoid tensor-index sign conventions. Representative plateau-centre fields are $20.68$, $6.893$, $4.136$ and $2.954\mathrm{T}$ for $\nu =2,6,10,14$. Between plateaux, crossing extended states in broadened levels gives longitudinal peaks and Hall steps. Their ideal filling-centre positions have $\nu =4j$, namelyThus peaks near $10.34$, $5.17$ and $3.45\mathrm{T}$ separate the first prominent plateaux as field increases. Many weak oscillations accumulate toward zero field and are damped there. Finite-width plateaux require localized states between extended-state energies; an exactly clean Landau spectrum does not by itself prescribe plateau widths. The illustration uses an explicitly schematic broadening, not inferred experimental widths.

At $6.5\mathrm{T}$, $\nu =6.36$, close to the $\nu =6$ plateau. With $v_F=10^6\mathrm{m}{\mathrm{s}}^{-1}$, $E_1=92.5\mathrm{meV}$ and $E_2=130.8\mathrm{meV}$. The relevant adjacent gap near the chemical potential is therefore $\Delta =E_2-E_1\simeq 38.3\mathrm{meV}$, rather than the gap from the zero level. Resolving levels requires quantum-lifetime broadening $\hslash /{\tau }_q\lesssim \Delta$ and thermal broadening appreciably smaller than $\Delta$, giving the order-of-magnitude limitsThe upper temperature is an energy-scale estimate, not a guaranteed sharp-feature temperature. For oscillation amplitudes the thermal damping factor is $X/\sinh X$, $X=2{\pi }^2{k}_{B}T/(\hslash {\omega }_c^{\ast })$, with $\hslash {\omega }_c^{\ast }=\hslash eB{v}_F^2/E_F\simeq 36.7\mathrm{meV}$ and $E_F=\hslash v_F\sqrt{\pi n}$. Requiring $X\lesssim 1$ gives the more conservative scale $T\lesssim 22\mathrm{K}$ for little thermal damping. Visible weaker features may persist above that scale; their actual cutoff depends on resolution, disorder and temperature-dependent scattering. The transport lifetime and quantum lifetime also need not coincide.

The fractional quantum Hall effect occurs when interactions produce an incompressible correlated state within a partially occupied Landau level. The orbital kinetic energy is already quantized; the important degeneracy is among many possible arrangements of electrons within the level. Repulsion can lift that degeneracy and create a many-body excitation gap. The interaction scale is of order $e^2/(4\pi ϵ{\ell }_B)$, where magnetic length ${\ell }_B=(\hslash /(e|B|){)}^{1/2}$; it is not a gap predicted by independent-electron fractional occupation alone.

A composite fermion description incorporates strong correlations by attaching $2p$ statistical flux quanta to each electron, with positive integer $p$. The attachment is a representation of correlated wavefunction phases and zeros, not a literal solenoid attached to a particle. Attaching an even number preserves fermionic exchange character. In a mean-field approximation for positive applied field,The particles consequently experience a reduced effective field. Define the signed effective filling ${\nu }^{\ast }=n_s{\varphi }_0/B^{\ast }$ and electronic filling $\nu =n_s{\varphi }_0/B$. Their relation isWhen $q$ effective levels are filled with the same field orientation, ${\nu }^{\ast }=q$, giving $\nu =q/(2pq+1)$. Reversed effective-field orientation has ${\nu }^{\ast }=-q$, giving $\nu =q/(2pq-1)$. Thus the Jain sequence isFor $p=1$, examples are $1/3$, $2/5$, $3/7$ on one branch and $2/3$, $3/5$ on the reversed branch. Fractional electronic plateaus can be understood as integer effective Hall states of these composite objects. The original proposal explicitly makes this connection.^[ref]

At a gapped fractional state, longitudinal dissipation becomes very small and the Hall magnitude approachesDisorder can localize quasiparticles and allow the plateau to persist over a filling interval, provided it does not overwhelm the interaction gap. The Laughlin wavefunction at $\nu =1/(2p+1)$ is a particularly simple correlated example; its charged excitations carry fractions of the electron charge. More generally the basic quasiparticle charge in the simple Jain sequence has magnitude $e/(2pq\pm 1)$. Such excitations are distinct from noninteracting bare electrons occupying a fraction of a level.

At $\nu =1/(2p)$, the mean effective field $B^{\ast }$ vanishes. This permits a compressible composite-fermion Fermi-sea description, rather than an ordinary integer effective-level gap. The external field remains large: cancellation is in the effective description. Not every fractional state is explained by the simplest integer composite fermion sequence, and the effective masses and gaps reflect interactions rather than simply the GaAs band mass. The utility of the construction is that it organizes many observed fractions and their nearby excitations into one understandable pattern.