|
Z N Model
The Z_N model (also known as the clock model) is a simplified statistical mechanical spin model. It is a generalization of the Ising model. Although it can be defined on an arbitrary graph, it is integrable only on one and two-dimensional lattices, in several special cases. Definition The Z_N model is defined by assigning a spin value at each node r on a graph, with the spins taking values s_r=\exp, where q\in \. The spins therefore take values in the form of complex roots of unity. Roughly speaking, we can think of the spins assigned to each node of the Z_N model as pointing in any one of N equidistant directions. The Boltzmann weights for a general edge rr' are: ::w\left(r,r'\right)=\sum_^x_^\left(s_s_^*\right)^k where * denotes complex conjugation and the x_^ are related to the interaction strength along the edge rr'. Note that x_^=x_^ and x_0 are often set to 1. The (real valued) Boltzmann weights are invariant under the transformations s_r \rightarrow \omega^k s_r and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions. This established the fields of statistical thermodynamics and statistical physics. The founding of the field of statistical mechanics is generally credited to three physicists: *Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates *James Clerk Maxwell, who developed models of probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
XY Model
The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for . Definition Given a -dimensional lattice , per each lattice site there is a two-dimensional, unit-length vector The ''spin configuration'', is an assignment of the angle for each . Given a ''translation-invariant'' interaction and a point dependent external field \mathbf_=(h_j,0), the ''configuration energy'' is : H(\mathbf) = - \sum_ J_\; \mathbf_i\cdot\mathbf_j -\sum_j \mathbf_j\cdot \mathbf_j =- \sum_ J_\; \cos(\theta_i-\theta_j) -\sum_j h_j\cos\theta_j The case in which except for nearest neighbor is called ''nearest neighbor'' case. The ''configuration probability'' is given by the Boltzmann distribution with inverse temperature : :P(\mathbf)=\frac \qquad Z=\int_ \prod_ d\theta_j\;e^. where is the normalization, or partition f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exactly Solvable Models
Exact may refer to: * Exaction, a concept in real property law * ''Ex'Act'', 2016 studio album by Exo * Schooner Exact, the ship which carried the founders of Seattle Companies * Exact (company), a Dutch software company * Exact Change, an American independent book publishing company * Exact Editions, a content management platform Mathematics * Exact differentials, in multivariate calculus * Exact algorithms, in computer science and operations research * Exact colorings, in graph theory * Exact couples, a general source of spectral sequences * Exact sequences, in homological algebra * Exact functor In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much ..., a function which preserves exact sequences See also * * Exactor (other) * XACT (other) * EXACTO, a sniper rif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Models
Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally biased portrayal of something Spin, spinning or spinnin may also refer to: Physics and mathematics * Spin, the rotation of an object around a central axis * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin group, a particular double cover of the special orthogonal group SO(''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 creating polymer fibres * Spinning (textiles), the creation of yarn or thread by twisting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
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 with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program, the economic policies advocated by Alexander Hamilton, the first United States Secretary of the Treasury See also * Alexander Hamilton (1755 or 1757–1804), American statesman and one of the Founding Fathers of the US * Hamilton (other) * List of things named after William Rowan Hamilton {{ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transverse-field Ising Model
The transverse field Ising model is a quantum version of the classical Ising model. It features a lattice with nearest neighbour interactions determined by the alignment or anti-alignment of spin projections along the z axis, as well as an external magnetic field perpendicular to the z axis (without loss of generality, along the x axis) which creates an energetic bias for one x-axis spin direction over the other. An important feature of this setup is that, in a quantum sense, the spin projection along the x axis and the spin projection along the z axis are not commuting observable quantities. That is, they cannot both be observed simultaneously. This means classical statistical mechanics cannot describe this model, and a quantum treatment is needed. Specifically, the model has the following quantum Hamiltonian: :H = -J\left(\sum_ Z_i Z_ + g \sum_j X_j \right) Here, the subscripts refer to lattice sites, and the sum \sum_ is done over pairs of nearest neighbour sites i and j. X_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Clock Model
The quantum clock model is a quantum lattice model. It is a generalisation of the transverse-field Ising model The transverse field Ising model is a quantum version of the classical Ising model. It features a lattice with nearest neighbour interactions determined by the alignment or anti-alignment of spin projections along the z axis, as well as an externa ... . It is defined on a lattice with N states on each site. The Hamiltonian of this model is :H = -J \left( \sum_ (Z^\dagger_i Z_j + Z_i Z^\dagger_j ) + g \sum_j (X_j + X^\dagger_j) \right) Here, the subscripts refer to lattice sites, and the sum \sum_ is done over pairs of nearest neighbour sites i and j. The clock matrices X_j and Z_j are N \times N generalisations of the Pauli matrices satisfying : Z_j X_k = e^ X_k Z_j and X_j^N = Z_j^N = 1 where \delta_ is 1 if j and k are the same site and zero otherwise. J is a prefactor with dimensions of energy, and g is another coupling coefficient that determines the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potts Model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid-state physics. The strength of the Potts model is not so much that it models these physical systems well; it is rather that the one-dimensional case is exactly solvable, and that it has a rich mathematical formulation that has been studied extensively. The model is named after Renfrey Potts, who described the model near the end of his 1951 Ph.D. thesis. The model was related to the "planar Potts" or " clock model", which was suggested to him by his advisor, Cyril Domb. The four-state Potts model is sometimes known as the Ashkin–Teller model, after Julius Ashkin and Edward Teller, who considered an equivalent model in 1943. The Potts model is related to, and generalized by, several other models, including the XY mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiral Potts Model
The chiral Potts model is a spin model on a planar lattice in statistical mechanics studied by Helen Au-Yang Perk and Jacques Perk, among others. It may be viewed as a generalization of the Potts model, and as with the Potts model, the model is defined by configurations which are assignments of ''spins'' to each vertex of a graph, where each spin can take one of N values. To each edge joining vertices with assigned spins n and n', a Boltzmann weight W(n,n') is assigned. For this model, chiral means that W(n,n') \neq W(n',n). When the weights satisfy the Yang–Baxter equation, it is integrable, in the sense that certain quantities can be exactly evaluated. For the integrable chiral Potts model, the weights are defined by a high genus curve, the chiral Potts curve. Unlike the other solvable models, whose weights are parametrized by curves of genus less or equal to one, so that they can be expressed in terms of trigonometric functions, rational functions for the genus zero case, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Model
A spin model is a mathematical model used in physics primarily to explain magnetism. Spin models may either be classical or quantum mechanical in nature. Spin models have been studied in quantum field theory as examples of integrable models. Spin models are also used in quantum information theory and computability theory in theoretical computer science. The theory of spin models is a far reaching and unifying topic that cuts across many fields. Introduction In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar. The study of the behavior of such "spin models" is a thriving area of research in co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
