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 ↓


Topological phases of matter


Topological phases of matter are a fascinating and emerging topic in the field of condensed matter physics. They represent a class of phases that go beyond the traditional symmetry-breaking description of matter, offering a new paradigm based on topological order. These phases are characterized by properties that depend not on local order parameters but on global topological aspects of the wave functions of matter.

Background and introduction

Traditionally, phases of matter are classified according to the principle of symmetry breaking. For example, in the transition from liquid to solid, the rotational symmetry of the liquid is broken. These classical phases can be understood within the framework of Landau's theory of phase transitions, where phases are distinguished by local order parameters.

In the 1980s, a new type of phase was discovered that could not be described by symmetry breaking alone. The first known topological phase of matter was the quantum Hall effect, observed in two-dimensional electron systems subjected to a strong magnetic field.

Key concepts of topological phases

Topological invariants

One of the defining features of topological phases is the presence of topological invariants. These are quantities that remain unchanged under a continuous deformation of the system. A well-known example of this is the Chern number, which is an integer indicating how many times the wave function of a system wraps around an imaginary space.

    Chern number, C = (1/2πi) ∫∫ F(kx, ky) d^2k

Here, F(kx, ky) is the Berry curvature, and the integration is over the Brillouin zone.

Edge States

Topological phases often host protected edge states at their boundaries. These edge states are robust against disturbances, meaning they cannot be easily destroyed by impurities or defects. This property leads to applications in creating devices that are resilient to external noise.

Examples of topological phases

To better understand topological phases, let's look at some examples.

Quantum Hall Effect

The quantum Hall effect is a prime example of a topological phase. When electrons in a two-dimensional electron gas are subjected to a strong perpendicular magnetic field, their motion becomes quantized into discrete levels known as Landau levels.

Landau Levels Magnetic field B

As the magnetic field strength increases, the Hall resistivity of the system becomes constant and the longitudinal resistivity tends to zero. These plateaus are quantized and can be described by integer Chern numbers:

    σ_xy = (e^2/h) * c

where σ_xy is the Hall conductivity, e is the electron charge, h is the Planck constant, and C is the Chern number.

Topological Insulators

Another example is topological insulators. These are materials that behave like insulators inside but conduct electricity on their surface. The state of the conducting surface is secured by the topological properties of the material.

Insulating Bulk Operating Edge

Theoretical framework

Berry phase and Berry curvature

An important mathematical concept in understanding topological phases is the Berry phase, which is a geometric phase obtained during a cycle when the system is subjected to adiabatic processes. The Berry curvature is the field strength corresponding to the Berry connection.

Topological band theory

In topological band theory, the properties of electrons in solids are described by energy bands. Topological band structures are characterized by non-trivial topology, which affects the motion and distribution of electrons within the bands.

Applications and implications

The study of topological phases of matter has many applications, especially in electronics and quantum computing. The stability of edge states makes topological materials ideal for use in devices where performance is critical, even amid environmental disturbances.

Quantum computing

Topological phases offer promising avenues for quantum computing, particularly in the design of qubits that are less prone to decoherence. Topological quantum computers take advantage of anyons, which are particles that emerge in certain two-dimensional systems, to create inherently stronger qubits.

Conclusion

Topological phases of matter represent a paradigm shift in understanding the behaviour of materials. With their unique properties rooted in their topological features, these phases offer exciting possibilities for technological innovations. As research progresses, the potential of topological phases in revolutionising technology becomes more tangible, bridging the gap between theoretical physics and practical applications.


Graduate → 6.3


U
username
0%
completed in Graduate

← Prev (6.2.3)
The Josephson Effect and SQUIDs
Next (6.3.1) →
Topological Insulators

Comments 



Topological phases of matter

https://www.physicsflow.com/g/6.3?pfal=en