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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] |
Ising Model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. The Ising model was invented by the physicist , who gave it as a pro ... [...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] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view 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, 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 term ''Hilbert space'' for the abstract concept ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferromagnetism
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials are the familiar metals noticeably attracted to a magnet, a consequence of their large magnetic permeability. Magnetic permeability describes the induced magnetization of a material due to the presence of an ''external'' magnetic field, and it is this temporarily induced magnetization inside a steel plate, for instance, which accounts for its attraction to the permanent magnet. Whether or not that steel plate acquires a permanent magnetization itself, depends not only on the strength of the applied field, but on the so-called coercivity of that material, which varies greatly among ferromagnetic materials. In physics, several different types of material magnetism are distinguished. Ferromagnetism (along with the similar effect ferrimagnetism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiferromagnetism
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism. The phenomenon of antiferromagnetism was first introduced by Lev Landau in 1933. Generally, antiferromagnetic order may exist at sufficiently low temperatures, but vanishes at and above the Néel temperature – named after Louis Néel, who had first identified this type of magnetic ordering. Above the Néel temperature, the material is typically paramagnetic. Measurement When no external field is applied, the antiferromagnetic structure corresponds to a vanishing total magnetization. In an external magnetic field, a kind of ferrimagnetic behavior may be displayed in the antiferromagnetic phase, with the absolute value of one of the sublattice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Model (physics)
In theoretical physics, a minimal model or Virasoro minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models have been classified and solved, and found to obey an ADE classification. The term minimal model can also refer to a rational CFT based on an algebra that is larger than the Virasoro algebra, such as a W-algebra. Relevant representations of the Virasoro algebra Representations In minimal models, the central charge of the Virasoro algebra takes values of the type : c_ = 1 - 6 \ . where p, q are coprime integers such that p,q \geq 2. Then the conformal dimensions of degenerate representations are : h_ = \frac\ , \quad \text\ r,s\in\mathbb^*\ , and they obey the identities : h_ = h_ = h_\ . The spectrums of minimal models are made of irreducible, degenerate lowest-weight representations of the Virasoro algebra, whose conformal dimensions are of the type h_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kramers–Wannier Duality
The Kramers–Wannier duality is a symmetry in statistical physics. It relates the free energy of a two-dimensional square-lattice Ising model at a low temperature to that of another Ising model at a high temperature. It was discovered by Hendrik Kramers and Gregory Wannier in 1941. With the aid of this duality Kramers and Wannier found the exact location of the critical point for the Ising model on the square lattice. Similar dualities establish relations between free energies of other statistical models. For instance, in 3 dimensions the Ising model is dual to an Ising gauge model. Intuitive idea The 2-dimensional Ising model exists on a lattice, which is a collection of squares in a chessboard pattern. With the finite lattice, the edges can be connected to form a torus. In theories of this kind, one constructs an involutive transform. For instance, Lars Onsager suggested that the Star-Triangle transformation could be used for the triangular lattice. Now the dual o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Potts Model
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantization". This means that the magnitude of the physical property can take on only discrete values consisting of integer multiples of one quantum. For example, a photon is a single quantum of light (or of any other form of electromagnetic radiation). Similarly, the energy of an electron bound within an atom is quantized and can exist only in certain discrete values. (Atoms and matter in general are stable because electrons can exist only at discrete energy levels within an atom.) Quantization is one of the foundations of the much broader physics of quantum mechanics. Quantization of energy and its influence on how energy and matter interact (quantum electrodynamics) is part of the fundamental framework for understanding and describing nature. E ... [...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] |
Lattice Models
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are quite popular in theoretical physics, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory. Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models (when they are solvable) includes the presence of solitons. Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang–Baxter equation and quantum groups. The solution of these models has g ... [...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] |
