Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Condensed matter physics → Band structure and transport theory ↓


Quantum Hall Effect


The quantum Hall effect (QHE) is a quantum mechanical phenomenon observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. It is one of the most remarkable discoveries in condensed matter physics, which has not only expanded our understanding of electronic interactions in low-dimensional systems but also set the stage for exciting technological developments including topological insulators and quantum computing.

Understanding band structure

In solid state physics, the concept of band structure is quite fundamental. It describes the range of energy levels that electrons within a solid can have and the range of energy levels that they are forbidden to have. The band structure arises from the periodic potential experienced by electrons in the crystalline lattice.

Energy bands

When atoms are brought together to form a solid, their atomic orbitals overlap each other to form molecular orbitals, which spread throughout the solid as electron bands. The most important bands for understanding the QHE are the conduction band and the valence band. The energy difference between these, known as the band gap, determines whether the material is a conductor, insulator, or semiconductor.

    e(k) = ħ²k² / 2m

This formula describes the energy E of the electron wavevector k in the free electron model, where ħ is the reduced Planck constant and m is the electron mass. In a solid, the periodic potential modifies this relation, leading to a band structure.

Fermi surface

In metals, the Fermi surface is important for understanding electrical properties; it is a surface of constant energy in momentum space that separates filled states from unfilled states at absolute zero temperature.

Hall effect and the quantum leap

Classical Hall effect

Discovered by Edwin Hall in 1879, the classical Hall effect occurs when a magnetic field is applied perpendicular to the current flowing through a conductor. This magnetic field induces a Lorentz force, which deflects charge carriers towards one side of the conductor, creating a voltage difference in the transverse direction.

    V_H = (IB)/(NQ)

Where V_H is the Hall voltage, I is the current, B is the magnetic field, n is the charge carrier density, and q is the charge of the carriers.

Quantum Hall effect

Unlike the classical Hall effect, the quantum Hall effect occurs under extreme conditions: low temperatures and strong magnetic fields. Under these conditions, the Hall conductivity takes on quantized values, hence the term "quantum."

    (sigma_{xy} = frac{e^2}{h} cdot nu)

Here, ( nu ) is an integer or rational number known as the filling factor, e is the charge of an electron, and h is the Planck constant. This quantization is extremely precise, making it useful for defining standards of electrical resistance.

Landau levels

To understand the quantum Hall effect, one must delve deeper into the concept of Landau levels. When a two-dimensional electron gas is subjected to a perpendicular magnetic field, the electrons revolve in circular cyclotron orbits. The allowed energy levels of these orbits are Landau levels, which are described as:

    E_n = hbar omega_c (n + frac{1}{2})

where n is an integer representing the Landau level index, and (omega_c = frac{eB}{m^*}) is the cyclotron frequency, with m^* being the effective mass of the electron.

Visual example of Landau levels

energy index n

In this visual example, the vertical axis shows the energy of the Landau levels and the horizontal axis shows the index n. Each green circle corresponds to an available energy state at a particular Landau level.

Filling factors

The filling factor ( nu ) indicates how many Landau levels are filled and plays an important role in determining the quantum Hall conductivity.

    (nu = frac{n}{B/phi_0})

Where ( n ) is the electron density, ( B ) is the magnetic field, and (phi_0 = frac{h}{e}) is the magnetic flux quantum.

Integer quantum Hall effect (IQHE)

In the integer quantum Hall effect, the filling factor is an integer. The quantization of conductivity in such cases is a consequence of the topology of the electronic band structure, providing profound insights into the nature of electron flow in low-dimensional systems.

Fractional quantum Hall effect (FQHE)

In the fractional quantum Hall effect, the filling factor takes on fractional values. This phenomenon is particularly fascinating because it arises from electron-electron interactions that result in the formation of composite particles called "composite fermions."

Transportation theory

Transport theory describes how electrical conductivity behaves in two-dimensional electron systems subjected to a magnetic field. In the context of the QHE, transport properties provide important insights into the quantum states of matter.

The Hall resistance is quantified with extraordinary precision and is therefore used in resistance standards around the world. Measurements of this resistance allow characterizing the topological invariants of electronic states.

Edge states and topology

The quantum Hall effect is closely linked to the topological properties of materials. In particular, it has been found that the presence of strong edge states is required for the quantization of the Hall conductivity. These edge states are explained through topological invariants, known as Chern numbers, which are central to the field of topological insulators.

Applications and impact in the real world

Beyond its theoretical implications, the quantum Hall effect has also had practical consequences. It has contributed to the development of new tools in quantum computing and metrology. Quantization provides practical and precise definitions of electrical units such as resistance based on fundamental constants.

Challenges and future directions

While much work has been done to understand the QHE, challenges remain – particularly in fully understanding the fractional quantum Hall effect. New materials that can exhibit these effects more easily and at higher temperatures are of particular interest, providing potential routes to revolutionising electronic devices.

Thus, the continued exploration of the quantum Hall effect is a testament to the fascinating complexities and elegant beauty inherent in quantum mechanics and condensed matter physics.


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Quantum Hall Effect

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