"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."
Explores the properties of materials that have non-trivial topology, which means that their properties are not affected by small deformations or imperfections in their structure.
Topology and topological invariants: Topology is the branch of mathematics that deals with the properties of complex structures that are preserved under continuous transformations. In the context of condensed matter physics, topological invariants quantify the topological properties of materials that are insensitive to small perturbations.
Berry phase and geometric phase: Berry phase is a quantum mechanical phase that arises in the evolution of a quantum state undergoing adiabatic changes. The geometric phase is a more general concept that includes the Berry phase as a special case. These concepts are used to understand the topological properties of materials.
Dirac and Weyl semimetals: These are materials in which the electronic band structure shows degeneracy points that behave like massless fermions. Dirac and Weyl semimetals are examples of topological materials that have attracted a lot of attention in recent years.
Topological insulators: Topological insulators are materials that are insulating in the bulk but have conducting surface states that are protected by topology. These materials have potential applications in spintronics and quantum computing.
Chern insulators and quantum Hall effect: Chern insulators are two-dimensional materials that exhibit a topological quantum phase known as the quantum anomalous Hall effect. This is characterized by the appearance of quantized Hall conductance in the absence of an applied magnetic field.
Majorana fermions: Majorana fermions are exotic particles that are their own antiparticles. They have been predicted to exist in certain topological materials and have potential applications in quantum computing.
Topological phases of matter: Topological phases of matter refer to a broad class of quantum phases that are characterized by non-local properties and topological order. These phases have applications in quantum computing, condensed matter physics, and beyond.
Topological quantum field theory: Topological quantum field theory is a branch of theoretical physics that studies the topological properties of gauge theories and their applications to condensed matter physics, cosmology, and other areas.
Topological superconductors: These are materials that are superconducting in the bulk but have conducting surface states that are protected by topology. They have potential applications in quantum computing and other areas of technology.
Quantum spin Hall effect: The quantum spin Hall effect is a topological quantum phase that is characterized by the appearance of spin-polarized edge states in a two-dimensional electron gas. This has potential applications in spintronics and quantum computing.
Topological defects: Topological defects are topological entities that arise in certain physical systems, such as liquid crystals, cosmic strings, and superconductors. They are characterized by their non-trivial topology and have potential applications in materials science and technology.
Topological order: Topological order refers to the non-local properties of certain quantum phases that are characterized by topological invariants. These phases have potential applications in quantum computing and other areas of technology.
Topological insulators: These are materials that act as insulators in the bulk, but have conducting surface states that are protected by the topology of the material. The surface states can't be removed by perturbations that don't change the topology.
Topological superconductors: These are materials that exhibit superconductivity, but with topologically protected surface states. These states can't be removed by perturbations that don't change the topology, and they're expected to have unique properties useful for quantum computing.
Weyl semimetals: These are materials that have massless chiral fermions, called Weyl fermions, as quasiparticles. They're distinguished from conventional semimetals by their topological invariants.
Dirac semimetals: These are materials with linear touching points of conduction and valence bands, called Dirac points. They're protected by symmetry and topology, leading to unusual transport properties.
Higher-order topological insulators: These are materials where there's a higher order of topological protection: In addition to the topologically protected surface states, there are topologically protected corner or hinge states.
Floquet topological phases: These are materials undergoing periodic, driven evolution, meaning that they're not in equilibrium. They're defined by the topology of the Floquet operator that governs their dynamics.
Quantum anomalous Hall effect materials: These are materials that exhibit the quantum Hall effect without applying an external magnetic field. They require a combination of spin-orbit coupling and magnetization to achieve the topological phase.
Majorana fermion materials: These are materials that have Majorana fermions as quasiparticles. They're useful for developing topological quantum computers and have been found in topological superconductors.
Topological crystalline insulators: These are materials that have a crystalline symmetry that protects the surface states. This leads to the formation of new types of topological phases that are protected by crystalline symmetry rather than topology.
Topological photonics: These are photonic materials that mimic the behavior of electrons in topological insulators. They're useful for developing devices that can propagate light without losses or scattering.
"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."