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Quantum Three-state Potts Model
The three-state Potts CFT, also known as the \mathbb_3 parafermion CFT, is a conformal field theory in two dimensions. It is a minimal model with central charge c=4/5 . It is considered to be the simplest minimal model with a non-diagonal partition function in Virasoro characters, as well as the simplest non-trivial CFT with the W-algebra as a symmetry. Properties The critical three-state Potts model has a central charge of c = 4/5 , and thus belongs to the discrete family of unitary minimal models with central charge less than one. These conformal field theories are fully classified and for the most part well-understood. The modular partition function of the critical three-state Potts model is given by :: Z = , \chi_ + \chi_, ^2 + , \chi_ + \chi_, ^2 + 2, \chi_, ^2+2, \chi_, ^2 Here \chi_ (q) \equiv \textrm_ (q^) refers to the Virasoro character, found by taking the trace over the Verma module generated from the Virasoro primary operator labeled by integers r, s . Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Field Theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points. Scale invariance vs conformal invariance In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that ''local'' scale invariant theories have t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Model (physics)
In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ..., 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, giving rise to an ADE classification. Most minimal models have been solved, i.e. their 3-point structure constants have been computed analytically. 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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virasoro Algebra
In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel Virasoro. Structure The Virasoro algebra is spanned by generators for and the central charge . These generators satisfy ,L_n0 and The factor of \frac is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra or Schottenloher, Thm. 5.1, pp. 79. The Virasoro algebra has a presentation in terms of two generators (e.g. 3 and −2) and six relations. The generators L_ are called annihilation modes, while L_ are creation modes. A basis of creation generators of the Virasoro algebra's universal enveloping algebra is the set : \mathcal = \Big\_ For L\in \mathcal, let , L, = \sum_^k n_i, then _0,L= , L, L. Representation theory In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
W-algebra
In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples. Definition A W-algebra is an associative algebra that is generated by the modes of a finite number of meromorphic fields W^(z), including the energy-momentum tensor T(z)=W^(z). For h\neq 2, W^(z) is a primary field of conformal dimension h\in\frac12\mathbb^*. The generators (W^_n)_ of the algebra are related to the meromorphic fields by the mode expansions : W^(z) = \sum_ W^_n z^ The commutation relations of L_n=W^_n are given by the Virasoro algebra, which is parameterized by a central charge c\in \mathbb. This number is also called the central charge of the W-algebra. The commutation relations : _m, W^_n= ((h-1)m-n)W^_ are equivalent to the assumption that W^(z) i ... [...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 model, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Transition
In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic State of matter, states of matter: solid, liquid, and gas, and in rare cases, plasma (physics), plasma. A phase of a thermodynamic system and the states of matter have uniform physical property, physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. Types of phase transition States of matter Phase transitions commonly refer to when a substance tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
