Quantum Hall Effect

Definition

Take a two-dimensional electron system whose bulk is an insulator: gapped, with vanishing longitudinal conductivity. A weak in-plane electric field then produces a perpendicular current, $J_y={\sigma }_{xy}{E}_x$, with

$${\sigma }_{xy}=\nu \frac{e^2}{h},$$

and $\nu$ is quantized. It can be an integer (von Klitzing–Dorda–Pepper 1980; measured to about $10^{-9}$, now a resistance standard) or a rational number with a small, usually odd, denominator (Tsui–Störmer–Gossard 1982, $\nu =1/3$). The original setting is a strong perpendicular magnetic field. Zero-field “anomalous” versions also exist: the quantum anomalous Hall effect (Haldane’s 1988 model; observed by Chang et al. 2013) and the fractional quantum anomalous Hall effect (Park et al. 2023, twisted MoTe₂).

Three complementary explanations:

• Effective action (Witten 2026).
  • In the effective action of a 2+1d insulator, the only term not built from $\vec{E},\vec{B}$ is $2\pi k\mathrm{CS}(A)=\frac{k}{4\pi }\int AdA$.
  • Well-definedness of $e^{iI}$ forces $k\in \mathbb{Z}$, and odd $k$ requires a spin structure.
  • This is the only term whose current is proportional to the field, so ${\sigma }_{xy}=k{e}^2/h$.
  • In the fractional effect an emergent field $a$ couples through $(k,s,r)$ and $\nu =k-s^2/r$. See chern-simons-theory.
• Topology and index theory.
  • Laughlin’s flux-insertion argument (1981).
  • TKNN (1982): the Hall conductance is the Chern number of the filled bands. Avron–Seiler–Simon (1983).
  • Rigorous quantization for interacting gapped systems (Hastings–Michalakis 2015).
  • Rational indices with denominator bounded by the torus degeneracy (Bachmann–Bols–De Roeck–Fraas 2021).
• Wavefunctions. Laughlin (1983): $\Psi ={\prod }_{i<j}(z_i-z_j{)}^m{e}^{-\sum |z_i{|}^2/4}$ at $\nu =1/m$, with quasiparticle charge $e/m$. Moore–Read (1991): conformal blocks and the Pfaffian, the non-abelian candidate for $\nu =5/2$.

Bulk and edge.

• Bulk. The bulk is a topological-order described by abelian Chern–Simons theory, or by a $K$-matrix for multi-component states. The ground-state degeneracy is $|\det K{|}^g$ on genus $g$ and the excitations are anyons. The topological entanglement entropy is $\gamma =\ln \sqrt{|\det K|}$, i.e. $\ln \sqrt{3}$ at $\nu =1/3$.
• Edge. Chiral edge modes are forced by anomaly inflow. The edge charge density obeys $[\rho (x),\rho (y)]=\frac{i\nu }{2\pi }{\delta }^{\prime }(x-y)$: a $U(1)$ Kac–Moody algebra whose level is the Hall conductance.
• The Laughlin edge is a chiral boson (Wen 1990). The electron operator has scaling dimension $m/2$, so it is a fermion for odd $m$, and the quasiparticles carry charge $e/m$. In Witten’s conventions $r=-m$.

Role in Research

• confinement-duality, historical. The group’s Maxwell–Chern–Simons and Chern–Simons work lives here. The Chern–Simons level is the Hall conductance, and Maxwell–Chern–Simons is the fractional-effect effective theory with an irrelevant Maxwell term.
• condensed-matter-connections. The half-quantized surface Hall conductance $e^2/2h$ of a $\theta =\pi$ topological insulator is the boundary of the θ-term (axionic-electrodynamics). Witten’s integrality argument explains why that surface cannot be a standalone 2d insulator.
• Quantum-information program. The edge is a chiral CFT, a net of type III₁ algebras whose current has Pauli–Jordan distribution proportional to $\nu$. That is natural ground for the group’s Weyl- and vertex-operator Bell-CHSH constructions and for coherent-state Araki–Uhlmann computations; see the Questions Raised in 2026-witten-chern-simons-quantum-hall. In the bulk, abelian states are stabilizer states while non-abelian orders carry long-range magic (cf. magic-nonstabilizerness).
• Courses. The generalized-symmetries course treats the fractional effect in Semester II Week 10 (“one honest lecture”: Laughlin $\nu =1/3$, $K=3$, three ground states on the torus). The Hall response of a θ-wall is Semester I Week 12 §8.

Relations

• chern-simons-theory: the effective action; level = Hall conductance.
• topological-order: fractional quantum Hall fluids are the experimentally realized topological orders.
• axionic-electrodynamics: θ-walls carry Hall response $(\Delta \theta /2\pi )e^2/h$.
• t-hooft-anomaly: the chiral edge is the inflow partner of the bulk Chern–Simons term.
• spt-phases: the integer effect is invertible but not an SPT (see Notes).
• weyl-operators: the edge current generates a Weyl algebra with symplectic form proportional to $\nu$.
• julia-toulouse-mechanism: the composite-boson condensation picture (Zhang–Hansson–Kivelson; Read) is a defect-condensation derivation of the emergent Chern–Simons field.

Papers

• E. Hall (1879); K. von Klitzing, G. Dorda, M. Pepper, PRL 45 (1980) 494.
• R. B. Laughlin, PRB 23 (1981) 5632 (the gauge argument) and PRL 50 (1983) 1395 (the wavefunction).
• D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, PRL 49 (1982) 405. TKNN.
• D. C. Tsui, H. L. Störmer, A. C. Gossard, PRL 48 (1982) 1559.
• F. D. M. Haldane, PRL 61 (1988) 2015.
• S. C. Zhang, T. H. Hansson, S. Kivelson, PRL 62 (1989) 82; N. Read, PRL 62 (1989) 86.
• X.-G. Wen, PRB 41 (1990) 12838 (chiral Luttinger liquid); X.-G. Wen, A. Zee, PRB 46 (1992) 2290 ($K$-matrix).
• G. Moore, N. Read, NPB 360 (1991) 362.
• M. B. Hastings, S. Michalakis, CMP 334 (2015) 433.
• S. Bachmann, A. Bols, W. De Roeck, M. Fraas, J. Math. Phys. 62 (2021) 011901 [arXiv:2001.06458].
• H. Park et al., Nature 622 (2023) 74 [arXiv:2308.02657].
• 2026-witten-chern-simons-quantum-hall. The effective-action derivation.

Notes

• Integer quantum Hall states are invertible but not SPTs. Their chiral edge, with chiral central charge $c_{-}=k$, survives without any symmetry. The $U(1)$ response is the level-$k$ Chern–Simons term; the thermal Hall response is a gravitational Chern–Simons term (cf. Freed–Hopkins on invertible phases).
• “Insulator” in Witten’s sense means no gapless bulk degrees of freedom. ${\sigma }_{xx}=0$ with ${\sigma }_{xy}\ne 0$ also gives ${\rho }_{xx}=0$, which is why a Hall plateau shows zero longitudinal resistance.
• Orientation. The sign of ${\sigma }_{xy}$ depends on orientation; Witten fixes it by $dtdxdy$.