"A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material."
Materials that conduct electricity on their surface but are insulators in their interior due to their unique electronic structure.
Topology: The study of the properties of an object that remain unchanged even when the object is continuously deformed. In the context of topological insulators, topology helps to define the unique properties of certain materials that give rise to their insulating behavior.
Band theory of solids: The theory that describes how electrons behave in a crystal lattice. This theory is an essential tool for understanding the electronic properties of materials, including topological insulators.
Quantum mechanics: The branch of physics that describes the behavior of matter and energy at the smallest scales. The principles of quantum mechanics are fundamental to understanding the behavior of electrons and other particles in materials, including topological insulators.
Symmetry: The study of the properties of objects or systems that remain unchanged under certain transformations. In the case of topological insulators, understanding the symmetry properties of materials is crucial for identifying and characterizing their topological properties.
Topological quantum field theory: A mathematical framework for understanding the behavior of particles and fields in a topologically nontrivial space. This theory is particularly useful for understanding the topological properties of materials such as topological insulators.
Berry phase: A phase factor that arises when a quantum system is adiabatically transported around a closed path. The Berry phase plays a critical role in defining the topological properties of materials, including topological insulators.
Spin-orbit coupling: The interaction between a particle's spin and its motion in an electric or magnetic field. This interaction is essential for understanding the electronic properties of materials, including topological insulators.
Dirac equation: A relativistic wave equation that describes the behavior of particles with spin 1/2, such as electrons. The Dirac equation is a key tool for understanding the electronic properties of materials, including topological insulators.
Edge states: States that occur at the boundary of a material and are localized to that boundary. In the case of topological insulators, edge states are typically topologically protected and propagate in only one direction.
Quantum Hall effect: A phenomenon in which electrons in a two-dimensional material experience a transverse magnetic field and become localized into Landau levels. The quantum Hall effect is closely related to topological insulators and provides a useful analogue for understanding their behavior.
Quantum Spin Hall Insulators: These are 2D topological insulators that conduct electricity on the edges, but are insulators within the bulk. Electrons traveling along the edge are protected by quantum spin numbers.
Three-dimensional Topological Insulators: These are bulk insulating materials, but the surface of these materials is metallic and has protected surface states that are topologically protected.
Weyl Semimetals: These are 3D materials that have a band crossing point called Weyl nodes. These nodes can have opposite chirality and are topologically protected, leading to unusual electronic properties.
Floquet Topological Insulators: These are periodically driven systems that can be topologically non-trivial even in the absence of a magnetic field. They have potential applications in photonics and optoelectronics.
Higher-order Topological Insulators: These are insulators that have topologically protected properties only at their corners or higher dimension boundaries. They can have potential applications in quantum computing and quantum information processing.
Majorana Fermions: These are topologically protected quasiparticle excitations that can exist in 1D topological superconductors. They are characterized by non-Abelian exchange statistics and have potential applications in building fault-tolerant quantum computers.
"The topological insulator cannot be continuously transformed into a trivial one without untwisting the bands, which closes the band gap and creates a conducting state."
"Since this results from a global property of the topological insulator's band structure, local (symmetry-preserving) perturbations cannot damage this surface state."
"While ordinary insulators can also support conductive surface states, only the surface states of topological insulators have this robustness property."
"An insulator which cannot be adiabatically transformed into an ordinary insulator without passing through an intermediate conducting state."
"Topological insulators and trivial insulators are separate regions in the phase diagram, connected only by conducting phases."
"The properties of topological insulators and their surface states are highly dependent on both the dimension of the material and its underlying symmetries."
"Topological insulators can be classified using the so-called periodic table of topological insulators."
"All topological insulators have at least U(1) symmetry from particle number conservation, and often have time-reversal symmetry from the absence of a magnetic field."
"So-called 'topological invariants', taking values in Z2 or Z, allow classification of insulators as trivial or topological, and can be computed by various methods."
"The surface states of topological insulators can have exotic properties."
"At a given energy, the only other available electronic states have different spin, so 'U'-turn scattering is strongly suppressed and conduction on the surface is highly metallic."
"Despite their origin in quantum mechanical systems, analogues of topological insulators can also be found in classical media."
"There exist photonic, magnetic, and acoustic topological insulators, among others."
"Electrons can only move along the surface of the material."
"There exists an energy gap between the valence and conduction bands of the material."
"Untwisting the bands closes the band gap and creates a conducting state."
"The local (symmetry-preserving) perturbations cannot damage this surface state. This is unique to topological insulators."
"Topological insulators provide an example of a state of matter not described by the Landau symmetry-breaking theory that defines ordinary states of matter."
"Topological insulators and trivial insulators are separate regions in the phase diagram, connected only by conducting phases."