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Periodic Table Of Topological Invariants
The periodic table of topological invariants is an application of topology to physics. It indicates the group of topological invariant for topological insulators and superconductors in each dimension and in each discrete symmetry class. Discrete symmetry classes There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices. They are defined by three symmetries of the Hamiltonian \hat = \sum_ H_ c_i^ c_j, (where c_i, and c_i^, are the annihilation and creation operators of mode i, in some arbitrary spatial basis) : time reversal symmetry, particle hole (or charge conjugation) symmetry, and chiral (or sublattice) symmetry. Chiral symmetry is a unitary operator S, that acts on c_i, as a unitary rotation (S c_i S^ = (U_S)_ c_j,) and satisfies S^2 = 1. A Hamiltonian H possesses chiral symmetry when S\hatS^=-\hat, for some choice of S (on the level of first-quantised Hamiltonians, this m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek words , and ) 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. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berry Connection And Curvature
In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. The concept was first introduced by S. Pancharatnam as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics. Berry phase and cyclic adiabatic evolution In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ... H(\mathbf R) depends on a (vector) parameter \mathbf R that v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superconductors
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike an ordinary metallic conductor, whose resistance decreases gradually as its temperature is lowered even down to near absolute zero, a superconductor has a characteristic critical temperature below which the resistance drops abruptly to zero. An electric current through a loop of superconducting wire can persist indefinitely with no power source. The superconductivity phenomenon was discovered in 1911 by Dutch physicist Heike Kamerlingh Onnes. Like ferromagnetism and atomic spectral lines, superconductivity is a phenomenon which can only be explained by quantum mechanics. It is characterized by the Meissner effect, the complete ejection of magnetic field lines from the interior of the superconductor during its transitions into the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Insulators
Insulator may refer to: * Insulator (electricity), a substance that resists electricity ** Pin insulator, a device that isolates a wire from a physical support such as a pin on a utility pole ** Strain insulator, a device that is designed to work in mechanical tension to withstand the pull of a suspended electrical wire or cable * Insulator (genetics), an element in the genetic code * Thermal insulation, a material used to resist the flow of heat * Building insulation Building insulation is any object in a building used as insulation for thermal management. While the majority of insulation in buildings is for thermal purposes, the term also applies to acoustic insulation, fire insulation, and impact ins ..., the material used in building construction to prevent heat loss * Mott insulator, a type of electrical insulator * Topological insulator, a material that behaves as an insulator in its interior while permitting the movement of charges on its boundary See also * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The N-Category Café
John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, applications of higher categories to physics, and applied category theory. Baez is also the author of ''This Week's Finds in Mathematical Physics'', an irregular column on the internet featuring mathematical exposition and criticism. He started ''This Week's Finds'' in 1993 for the Usenet community, and it now has a following in its new form, the blog "Azimuth". ''This Week's Finds'' anticipated the concept of a personal weblog. Additionally, Baez is known on the World Wide Web as the author of the crackpot index. Early life and education Baez was born in San Francisco, California. He graduated with an A.B. in mathematics from Princeton University in 1982 after completing a senior thesis, titled "Recursivity in quantum mechanics", under th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Review B
''Physical Review B: Condensed Matter and Materials Physics'' (also known as PRB) is a peer-reviewed, scientific journal, published by the American Physical Society (APS). The Editor of PRB is Laurens W. Molenkamp. It is part of the ''Physical Review'' family of journals. About the Physical Review Journals The current Editor in Chief is Michael Thoennessen. PRB currently publishes over 4500 papers a year, making it one of the largest physics journals in the world. PRB ranked by the Eigenfactor, University of Washington, 2012 Scop ...
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Symmetry-protected Topological Order
Symmetry-protected topological (SPT) order is a kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap. To derive the results in a most-invariant way, renormalization group methods are used (leading to equivalence classes corresponding to certain fixed points). The SPT order has the following defining properties: (a) ''distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry''. (b) ''however, they all can be smoothly deformed into the same trivial product state without a phase transition, if the symmetry is broken during the deformation''. The above definition works for both bosonic systems and fermionic systems, which leads to the notions of bosonic SPT order and fermionic SPT order. Using the notion of quantum entanglement, we can say that SPT states are short-range entangled states ''with a symmetry'' (by contrast: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.see for ex. Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by a vector space over a field , where is equipped with a quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bott Periodicity Theorem
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory. There are corresponding period-8 phenomena for the matching theories, ( real) KO-theory and ( quaternionic) KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable homotopy groups of spheres, which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of sph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannians
In mathematics, the Grassmannian is a space that parameterizes all - dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than . When is a real or complex vector space, Grassmannians are compact smooth manifolds. In general they have the structure of a smooth algebraic variety, of dimension k(n-k). The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general. Notations for the Grassmannian vary between authors; notations include , , , or to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space . Motivation By giving a collection of subspaces of some vector space a topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexei Kitaev
Alexei Yurievich Kitaev (russian: Алексей Юрьевич Китаев; born August 26, 1963) is a Russian–American professor of physics at the California Institute of Technology and permanent member of the Kavli Institute for Theoretical Physics. He is best known for introducing the quantum phase estimation algorithm and the concept of the topological quantum computer while working at the Landau Institute for Theoretical Physics. For this work, he was awarded a MacArthur Fellowship in 2008. He is also known for introducing the complexity class QMA and showing the 2-local Hamiltonian problem is QMA- complete, the most complete result for k-local Hamiltonians. Kitaev is also known for contributions to research on a model relevant to researchers of the AdS/CFT correspondence started by Subir Sachdev and Jinwu Ye; this model is known as the Sachdev–Ye–Kitaev (SYK) model. Kitaev was educated in Russia, receiving an M.Sc. from the Moscow Institute of Physics and Tec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
