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Spin Chain
A spin chain is a type of model in statistical physics. Spin chains were originally formulated to model magnetic systems, which typically consist of particles with magnetic spin located at fixed sites on a lattice. A prototypical example is the quantum Heisenberg model. Interactions between the sites are modelled by operators which act on two different sites, often neighboring sites. They can be seen as a quantum version of statistical lattice models, such as the Ising model, in the sense that the parameter describing the spin at each site is promoted from a variable taking values in a discrete set (typically \, representing 'spin up' and 'spin down') to a variable taking values in a vector space (typically the spin-1/2 or two-dimensional representation of \mathfrak(2)). History The prototypical example of a spin chain is the Heisenberg model, described by Werner Heisenberg in 1928. This models a one-dimensional lattice of fixed particles with spin 1/2. A simple version (the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Chain3
Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin (geometry), the rotation of an object around an internal axis * Spin (propaganda), an intentionally biased portrayal of something Spin, spinning or spinnin may also refer to: Physics and mathematics * Spin group, Spin(''n''), a particular double cover of the special orthogonal group SO(''n'') ** the corresponding spin algebra, \mathfrak(n) * Spin tensor, a tensor quantity for describing spinning motion in special relativity and general relativity * Spin (aerodynamics), autorotation of an aerodynamically stalled aeroplane * SPIN bibliographic database, an indexing and abstracting service focusing on physics research Textile arts * Spinning (polymers), a process for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bethe Ansatz
In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models. It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic isotropic (XXX) Heisenberg model. Since then the method has been extended to other spin chains and statistical lattice models. "Bethe ansatz problems" were one of the topics featuring in the "To learn" section of Richard Feynman's blackboard at the time of his death. Discussion In the framework of many-body quantum mechanics, models solvable by the Bethe ansatz can be contrasted with free fermion models. One can say that the dynamics of a free model is one-body reducible: the many-body wave function for fermions (bosons) is the anti-symmetrized (symmetrized) product of one-body wave functions. Models solvable by the Bethe ansatz are not free: the two-body sector has a non-trivial scat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. \begin \sigma_1 = \sigma_x &= \begin 0&1\\ 1&0 \end, \\ \sigma_2 = \sigma_y &= \begin 0& -i \\ i&0 \end, \\ \sigma_3 = \sigma_z &= \begin 1&0\\ 0&-1 \end. \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalues'', is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory. The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by \hat, where the hat indicates that it is an operator. It can also be written as H or \check. Introduction The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Configuration Space (physics)
In classical mechanics, the parameters that define the configuration of a system are called '' generalized coordinates,'' and the space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system. Notice that this is a notion of "unrestricted" configuration space, i.e. in which different point particles may occupy the same position. In mathematics, in particular in topology, a notion of "restricted" configuration space is mostly used, in which the diagonals, representing "colliding" particles, are removed. Examples A particle in 3D space The position of a single particle moving in ordinary Euclidean 3-space is defined by the vector q=(x,y,z), and therefore its ''configuration space'' is Q=\mathbb^3. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-simple Lie Algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals.) Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form \kappa(x, y) = \operatorname(\operatorname(x)\operatorname(y)) is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable ideals; * the radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michel Gaudin (physicist)
Michel Gaudin (2 December 1931 – 4 August 2023) was a French physicist, known for the Gaudin model, in which a central spin is coupled to many surrounding spins. Biography Michel Gaudin was born on 2 December 1931. After graduating with the degree of ''ingénieur des ponts et chaussées'' (civil engineer), Gaudin joined in 1956 the CEA in Saclay to work on neutron experiments. Two years later, he joined Claude Bloch's theorists' working group, to which he belonged for the rest of his career. In 1967 he received from the University of Paris-Sud in Orsay his doctoral degree in physics with thesis ''Étude d'un modèle à une dimension pour un système de fermions en interaction''. Gaudin's research deals with, among other topics, the quantum-mechanical description of many-body systems, in particular, spin systems. With M. L. Mehta in 1960 he published ''On the density of eigenvalues of a random matrix'', an important paper on random matrices. Gaudin received the Fondation Sai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaudin Model
In physics, the Gaudin model, sometimes known as the ''quantum'' Gaudin model, is a model, or a large class of models, in statistical mechanics first described in its simplest case by Michel Gaudin. They are exactly solvable models, and are also examples of quantum spin chains. History The simplest case was first described by Michel Gaudin in 1976, with the associated Lie algebra taken to be \mathfrak_2, the two-dimensional special linear group. Mathematical formulation Let \mathfrak be a semi-simple Lie algebra of finite dimension d. Let N be a positive integer. On the complex plane \mathbb, choose N different points, z_i. Denote by V_\lambda the finite-dimensional irreducible representation of \mathfrak corresponding to the dominant integral element \lambda. Let (\boldsymbol) := (\lambda_1, \cdots, \lambda_N) be a set of dominant integral weights of \mathfrak. Define the tensor product V_:=V_\otimes \cdots \otimes V_. The model is then specified by a set of operators H_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fa-Yueh Wu
Fa-Yueh Wu (January 5, 1932 – January 21, 2020) was a Chinese-born theoretical physicist, mathematical physicist, and mathematician who studied and contributed to solid-state physics and statistical mechanics. Life Early stage Born on January 5, 1932, in Shimen County, Hunan Province, Republic of China (1912–1949), Republic of China, with his father, a member of the Legislature, as his fourth child. The temporary capital of the Chiang Kai-shek administration of Nationalist government was placed in Chongqing in December 1938, but before that, in 1937, he evacuated to Chongqing with his father and stepmother and entered an elementary school there. However, due to repeated Bombing of Chongqing, he was unable to settle in one place. In 1943, he enrolled in Nankai University, Nankai Junior High School, which was evacuated to Chongqing at the time. He transferred to a high school in Nanjing, which became the capital of Chiang Kai-shek administration again in 1946, after the col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliott Lieb
Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist. He is a professor of mathematics and physics at Princeton University. Lieb's works pertain to quantum and classical many-body problem, atomic structure, the stability of matter, functional inequalities, the theory of magnetism, and the Hubbard model. Biography Lieb was born in Boston in 1932, the family moved to New York when he was five. His father came from Lithuania and was an accountant, his mother came from Bessarabia and worked as a secretary. Lieb received his B.S. in physics from the Massachusetts Institute of Technology in 1953 and his PhD in mathematical physics from the University of Birmingham in England in 1956. Lieb was a Fulbright Fellow at Kyoto University, Japan (1956–1957), and worked as the Staff Theoretical Physicist for IBM from 1960 to 1963. In 1961–1962, Lieb was on leave as professor of applied mathematics at Fourah Bay College, the University of Sierra Leone. In 1963, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
