Topological insulatorA 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. A topological insulator is an insulator for the same reason a "trivial" (ordinary) insulator is: there exists an energy gap between the valence and conduction bands of the material. But in a topological insulator, these bands are, in an informal sense, "twisted", relative to a trivial insulator.
Topological orderIn physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.
Topological quantum computerA topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions). These braids form the logic gates that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable.
Topological defectTopological defects or solitons are irregularities or disruptions that occur within continuous fields or ordered states of matter. These defects, which can take various forms such as points, lines, or surfaces, are characterized by their stability and the fact that they cannot be 'smoothed out' or removed through continuous transformations of the field or material. They play a significant role in various areas of physics, including condensed matter physics, cosmology, and quantum field theory, and can have profound effects on the properties and behavior of the systems in which they occur.
Topological spaceIn mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.
Topological propertyIn topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
Topological groupIn mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups.
Topological vector spaceIn mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness.
Tight bindingIn solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method (linear combination of atomic orbitals method) used in chemistry. Tight-binding models are applied to a wide variety of solids.
TopologyIn mathematics, topology (from the Greek words τόπος, and λόγος) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.
Phase diagramA phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous states) occur and coexist at equilibrium. Common components of a phase diagram are lines of equilibrium or phase boundaries, which refer to lines that mark conditions under which multiple phases can coexist at equilibrium. Phase transitions occur along lines of equilibrium.
Hubbard modelThe Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each electron experiences competing forces: one pushes it to tunnel to neighboring atoms, while the other pushes it away from its neighbors. Its Hamiltonian thus has two terms: a kinetic term allowing for tunneling ("hopping") of particles between lattice sites and a potential term reflecting on-site interaction.
Topological string theoryIn theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological quantum field theory. There are two main versions of topological string theory: the topological A-model and the topological B-model. The results of the calculations in topological string theory generically encode all holomorphic quantities within the full string theory whose values are protected by spacetime supersymmetry.
Topological ringIn mathematics, a topological ring is a ring that is also a topological space such that both the addition and the multiplication are continuous as maps: where carries the product topology. That means is an additive topological group and a multiplicative topological semigroup. Topological rings are fundamentally related to topological fields and arise naturally while studying them, since for example completion of a topological field may be a topological ring which is not a field.
Chern classIn mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, Gromov-Witten invariants. Chern classes were introduced by . Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold.
Topological quantum numberIn physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are topological invariants associated with topological defects or soliton-type solutions of some set of differential equations modeling a physical system, as the solitons themselves owe their stability to topological considerations.
Topological data analysisIn applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools.
Topological quantum field theoryIn gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for mathematical work related to topological field theory.
Phase (matter)In the physical sciences, a phase is a region of material that is chemically uniform, physically distinct, and (often) mechanically separable. In a system consisting of ice and water in a glass jar, the ice cubes are one phase, the water is a second phase, and the humid air is a third phase over the ice and water. The glass of the jar is another separate phase. (See .) More precisely, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform.
Topological manifoldIn topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable manifolds are topological manifolds equipped with a differential structure).
