--- title: "The Fractional Quantum Hall Effect and Topological Order" module: Nanostructures moduleNumber: 11 lessonNumber: 3 order: 1103 summary: > When the lowest Landau level is only partly filled, the non-interacting theory predicts no gap, yet a plateau appears at filling one-third. It is a many-body effect: Coulomb repulsion selects a correlated ground state, the Laughlin wavefunction, whose excitations carry a fraction of the electron charge. This lesson builds the Laughlin state, introduces composite fermions that map the fractional effect onto an integer one, and explains how the quantum Hall effect brought the Chern number and topology into condensed-matter physics. topics: [Nanostructures] draft: false sources: - book: Kittel ref: "Ch. 18 — Nanostructures; fractional quantum Hall effect" - book: Simon ref: "topological invariants and band structure background" --- The [integer quantum Hall effect](/condensed-matter/nanostructures/integer-quantum-hall-effect) rests on filling an integer number of Landau levels, each a set of non-interacting single-particle states. When the lowest Landau level is only **partly** filled, that theory offers no gap and predicts a compressible metal. In 1982 Tsui, Störmer, and Gossard found instead a quantized Hall plateau at filling factor $\nu = 1/3$, with $\rho_{xx}$ vanishing exactly as in the integer case. A partly filled level has an enormous degeneracy and no kinetic energy scale to lift it, so the state is decided entirely by the Coulomb interaction among the electrons. The fractional quantum Hall effect is the first laboratory realization of a genuinely many-body incompressible liquid, and its excitations carry a fraction of an electron's charge. ## A macroscopically degenerate level Within one Landau level every single-particle state has the same energy $\hbar\omega_c(n+\tfrac12)$, so the kinetic energy is a fixed constant and drops out of the problem. At filling $\nu = 1/3$ there are three times as many available states as electrons, and the number of ways to place the electrons is astronomically large. Any infinitesimal interaction lifts this degeneracy and selects a ground state. The relevant scale is the Coulomb energy at the magnetic length, $$ \ell_B = \sqrt{\frac{\hbar}{eB}}, \qquad E_{\text{Coul}} = \frac{e^2}{4\pi\varepsilon\,\ell_B}, $$ where $\ell_B$ is the characteristic radius of a single-particle orbit in the lowest level. For $B = 10\ \text{T}$, $\ell_B = 8.1\ \text{nm}$ and $E_{\text{Coul}}\approx 10\ \text{meV}$. The many-body ground state is separated from its excitations by a gap of this order, which is what makes the liquid incompressible and the plateau flat. ## The Laughlin wavefunction Lowest-Landau-level states are built from analytic functions of the complex coordinate $z = x - iy$ (in units of $\ell_B$) times a common Gaussian. Robert Laughlin proposed that at $\nu = 1/m$ the ground state of $N$ electrons is $$ \psi_m(z_1,\dots,z_N) = \prod_{i **Definition (Incompressible quantum liquid).** An incompressible quantum > liquid is a many-body ground state separated from all excitations by an energy > gap, so that adding or removing a small amount of charge or compressing the > system costs a finite energy. The Laughlin state is incompressible; the gap is > the origin of the vanishing $\rho_{xx}$ on a fractional plateau. $$ % caption: Below the integer plateaus the Hall resistance develops finer % fractional plateaus at filling 1/3, 2/5, 2/3, each a many-body incompressible % state selected by the Coulomb interaction. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black, ->] (0,0) -- (7.4,0) node[below] {magnetic f\/ield $B$}; \draw[black, ->] (0,0) -- (0,4.0) node[left] {$R_{xy}$}; % integer plateau nu=1 at right, then fractional structure to the left \draw[very thick] (5.6,2.4) -- (7.0,2.4); \node[anchor=south] at (6.3,2.42) {integer}; % transition \draw[very thick] (5.6,2.4) .. controls (5.2,2.7) and (5.0,2.9) .. (4.6,3.1); % nu = 2/3 plateau \draw[acc, very thick] (3.4,3.1) -- (4.6,3.1); \node[acc, anchor=south] at (4.0,3.12) {$\frac{2}{3}$}; \draw[acc, very thick] (3.4,3.1) .. controls (3.1,3.25) .. (2.8,3.4); % nu = 2/5 plateau \draw[acc, very thick] (2.0,3.4) -- (2.8,3.4); \node[acc, anchor=south] at (2.4,3.42) {$\frac{2}{5}$}; \draw[acc, very thick] (2.0,3.4) .. controls (1.8,3.5) .. (1.6,3.6); % nu = 1/3 plateau \draw[acc, very thick] (0.5,3.6) -- (1.6,3.6); \node[acc, anchor=south] at (1.0,3.62) {$\frac{1}{3}$}; \end{tikzpicture} $$ ## Fractional charge The elementary excitation of the Laughlin state is created by piercing the liquid with one flux quantum, which pushes charge outward and leaves a **quasihole**. Laughlin's argument uses the plasma analogy: $|\psi_m|^2$ is the Boltzmann weight of a two-dimensional one-component plasma, and inserting a flux quantum adiabatically transfers a deficit of exactly charge $e/m$. The quasihole and its partner the quasielectron carry $$ q^\ast = \pm\frac{e}{m}, $$ a fraction of the electron charge. At $\nu = 1/3$ the excitations carry $e/3$. Shot-noise experiments confirm this fractional charge directly: the granularity of the current on a fractional plateau is $e/3$, not $e$. Fractional charge does not contradict the indivisibility of the electron; it is a collective property of the correlated liquid, in which a localized disturbance behaves as if a third of an electron had been removed. $$ % caption: A Laughlin quasihole is a local charge deficit created by inserting % one flux quantum into the incompressible liquid; the missing charge is exactly % e over m, a fraction of the electron charge. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % uniform liquid as a filled disk \draw[thick] (2.4,1.5) circle (1.9); \node[black, anchor=south] at (2.4,3.5) {Laughlin liquid}; % electrons as small dots on a grid inside \foreach \x in {1.3,2.0,2.7,3.4}{ \foreach \y in {0.8,1.5,2.2}{ \fill[black] (\x,\y) circle (1.4pt); } } % quasihole: cleared region with inserted flux \fill[white] (2.4,1.5) circle (0.42); \draw[acc, thick, dashed] (2.4,1.5) circle (0.42); \draw[acc, thick, ->] (2.4,1.0) arc (-90:200:0.28); \node[acc, anchor=west] at (4.6,1.5) {charge def\/icit $q=\frac{e}{m}$}; \draw[black, ->] (4.55,1.5) -- (2.85,1.5); \end{tikzpicture} $$ ## Composite fermions Jainendra Jain unified the fractional plateaus with a change of variables. Attach $2p$ flux quanta to each electron; the bound object is a **composite fermion**. In mean field the attached flux partly cancels the applied field, so a composite fermion moves in a reduced effective field $$ B^\ast = B - 2p\,n_s\Phi_0. $$ When the composite fermions fill an integer number $q$ of their own Landau-like levels in this residual field, the electrons sit on a fractional plateau. Setting the composite-fermion integer effect at $B^\ast$ equal to the electron filling gives the **Jain sequence** $$ \nu = \frac{q}{2pq \pm 1}, \qquad q = 1, 2, 3, \dots $$ which reproduces the observed fractions $\tfrac13, \tfrac25, \tfrac37,\dots$ ($p=1$) and $\tfrac23,\tfrac35,\dots$ as the strongest plateaus. The fractional quantum Hall effect of electrons is the integer quantum Hall effect of composite fermions: the interaction has been absorbed into the flux attachment, leaving weakly interacting quasiparticles in a reduced field. $$ % caption: A composite fermion is an electron bound to two flux quanta; the % attached flux cancels part of the applied field, so composite fermions see a % reduced field B-star and fill integer Landau levels there. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % electron \fill[black] (1.2,1.5) circle (5pt); \node[black, anchor=south] at (1.2,1.85) {electron}; % two attached flux quanta as loops \draw[acc, thick, ->] (1.2,1.5) ++(0.7,0) arc (0:320:0.42); \draw[acc, thick, ->] (1.2,1.5) ++(1.4,0) arc (0:320:0.42); \node[acc, anchor=north] at (1.9,0.85) {two f\/lux quanta}; % equals sign \node[black] at (4.0,1.5) {$=$}; % composite fermion \fill[black] (5.6,1.5) circle (5pt); \draw[acc, thick] (5.6,1.5) circle (0.5); \draw[acc, thick] (5.6,1.5) circle (0.75); \node[black, anchor=south] at (5.9,2.35) {composite fermion}; \node[black, anchor=north] at (5.6,0.6) {reduced f\/ield}; \end{tikzpicture} $$ ## Anyons Exchanging two identical particles in three dimensions can only multiply the wavefunction by $+1$ (bosons) or $-1$ (fermions), because a double exchange is a loop that can be contracted to a point. In two dimensions the exchange path cannot be undone, and the phase acquired can be any value $e^{i\theta}$. Laughlin quasiparticles are such **anyons**: braiding one quasihole around another multiplies the state by $e^{i\pi/m}$, a statistical phase intermediate between bosonic and fermionic. Anyonic statistics is a direct consequence of the fractional charge and the flux it carries, and it is the property that motivates proposals for fault-tolerant quantum computation using quantum Hall liquids. ## Topology and the Chern number The exactness of the quantized Hall conductance, integer and fractional alike, reflects a topological invariant rather than any fine-tuning. Thouless, Kohmoto, Nightingale, and den Nijs showed that for a filled band the Hall conductance is $$ \sigma_{xy} = \frac{e^2}{h}\,C, \qquad C = \frac{1}{2\pi}\int_{\text{BZ}} \mathcal{F}(\vec k)\,\d^2 k, $$ where $\mathcal{F} = \nabla_{\vec k}\times \vec{\mathcal{A}}$ is the Berry curvature of the Bloch states and $\vec{\mathcal{A}}$ the Berry connection. The integral of the curvature over the Brillouin zone, a closed surface (a torus), is quantized to an integer $C$ — the **Chern number** — by the same theorem that quantizes the total curvature of a closed surface in the Gauss–Bonnet theorem. A Chern number is a topological invariant: it cannot change under any smooth deformation of the Hamiltonian that keeps the gap open, so disorder, geometry, and interactions leave $\sigma_{xy}$ exactly quantized. This insight, that a transport coefficient equals a topological integer, opened the field of topological phases of matter. $$ % caption: The Hall conductance is a topological invariant: it counts the Berry % curvature wrapped over the Brillouin-zone torus, so smooth changes to the % Hamiltonian leave the integer plateau untouched. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % Brillouin zone torus (schematic) \draw[thick] (1.8,1.5) circle (1.3); \draw[thick] (1.8,1.5) ellipse (0.55 and 0.22); \draw[thick] (1.15,1.6) .. controls (1.5,1.35) and (2.1,1.35) .. (2.45,1.6); \node[black, anchor=south] at (1.8,2.95) {BZ torus}; \node[black, anchor=north] at (1.8,0.15) {Berry curvature}; % arrow to plateau \draw[black, ->] (3.3,1.5) -- (4.4,1.5); % quantized plateau robust to deformation \draw[black, ->] (4.8,0.3) -- (4.8,3.0) node[left] {conductance}; \draw[black, ->] (4.8,0.3) -- (7.4,0.3) node[below] {disorder strength}; \draw[acc, very thick] (5.0,2.1) -- (7.2,2.1); \node[acc, anchor=south] at (6.1,2.12) {$\frac{Ce^2}{h}$ f\/ixed}; \end{tikzpicture} $$ The topology of the two-dimensional electron gas required a magnetic field to break time-reversal symmetry and lift the Landau levels. A crystal can carry the same Berry curvature in its band structure without any external field. The [final lesson](/condensed-matter/nanostructures/graphene-and-dirac-materials) turns to graphene, whose honeycomb lattice produces massless Dirac electrons, a Berry phase of $\pi$, and an anomalous quantum Hall effect that carries the topological ideas of this lesson into a real material.