Topics#

Key Concepts#

• Cyclotron frequency \(\omega_c = qB/m\): energy-independent orbital frequency in a magnetic field

• Magnetic length \(\ell_B = \sqrt{\hbar/(qB)}\): the fundamental quantum length scale set by the field

• Landau levels: discrete energy levels \(E_n = (n + \tfrac{1}{2})\hbar\omega_c\) with macroscopic degeneracy \(N_\phi = BA/\Phi_0\)

• Guiding center noncommutativity: \([\hat{X}, \hat{Y}] = -\mathrm{i}\ell_B^2\) — position becomes fuzzy in a magnetic field

• Lowest Landau level: holomorphic wavefunctions \(\psi_m \propto z^m \mathrm{e}^{-|z|^2/(4\ell_B^2)}\) in complex coordinates

• Filling factor \(\nu = N_e/N_\phi\): ratio of electrons to flux quanta, determines the quantum Hall state

• Charge pumping: threading one flux quantum transfers exactly \(\nu\) electrons, giving \(\sigma_{xy} = \nu\, e^2/h\)

• Fractional quantum Hall effect: at fractional \(\nu\), interactions produce anyonic quasiparticles with fractional charge and statistics

Learning Objectives#

• Derive the cyclotron frequency and explain the classical and quantum Hall effects

• Solve the Landau level problem in the Landau gauge and identify the two sets of ladder operators (energy and guiding center)

• Calculate the degeneracy of a Landau level and relate it to the magnetic flux quantum

• Use the charge pumping argument to derive the quantized Hall conductance

• Connect the fractional quantum Hall effect to anyonic statistics from Chapter 2

Project: Quantum Hall Fractionalization and Composite Fermions#

Objective: Investigate the fractional quantum Hall effect (FQHE) in a two-dimensional electron gas using composite fermion theory and exact diagonalization, exploring how electron-electron interactions generate topologically ordered states.

Background: At integer filling factors \(\nu = n\) (where \(n\) is an integer), the quantum Hall effect is well-understood: electrons fill \(n\) Landau levels, and Hall conductance is quantized to \(\sigma = n e^2/h\). But at fractional fillings (e.g., \(\nu = 1/3, 2/5\)), remarkable new states emerge: the fractional quantum Hall effect, with fractionally quantized Hall conductance and exotic quasiparticle excitations (anyons). The composite fermion picture—viewing electrons as bound to an even number of flux quanta, forming emergent fermions—provides deep insight. This is a frontier in understanding topological order, and has implications for topological quantum computing.

Suggested Approach:

• Set up a small 2D lattice (or disk geometry) with \(N \approx 8\)–10 electrons in a perpendicular magnetic field, using the lowest Landau level (LLL) projection.

• Use exact diagonalization to compute ground and excited state wavefunctions at filling fraction \(\nu = 1/3\) and \(\nu = 2/3\).

• Analyze the ground state structure: compute entanglement entropy, topological entanglement entropy, and edge degeneracy (signatures of topological order).

• Implement the composite fermion picture: rewrite states in terms of flux-attached fermions and compare the CF ground state to exact results.

• Compute excitation spectrum and quasihole/quasiparticle properties: compare with Laughlin wavefunction predictions.

• Measure the fractional charge of quasiholes and investigate anyonic statistics (braiding properties).

Expected Deliverable: Research report with code. Include: (i) Hamiltonian in the LLL and lattice discretization, (ii) exact diagonalization algorithm and implementation, (iii) ground state properties and topological characterization, (iv) composite fermion analysis and comparison with exact results, (v) excitation spectrum and quasiparticle properties, (vi) physical significance for topological quantum information.

Key References: D. C. Tsui, H. L. Störmer, & A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982); J. K. Jain, Composite Fermions (Cambridge University Press); topological order reviews.