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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 their ... [...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_ with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virasoro Algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. Definition The Virasoro algebra is spanned by generators for and the central charge . These generators satisfy ,L_n0 and The factor of 1/12 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. The Virasoro algebra has a presentation in terms of two generators (e.g. 3 and −2) and six relations. Representation theory Highest weight representations A highest weight representation of the Virasoro algebra is a representation generated by a primary state: a vector v such that : L_ v = 0, \quad L_0 v = hv, where the number is called the conformal dimension or conformal weight of v.P. Di Francesco, P. Mathieu, and D. S ... [...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, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |