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arXiv:hep-th/9705033 · Journal of Mathematical Physics

A non-Abelian square root of Abelian vertex operators

K. Frieler, Karl-Henning Rehren

Published Jun 1, 1998• 2 citations
arXivHTMLPDF

Abstract

The quantum field theoretical formulation of Kadanoff’s “correlations along a line” in the critical two-dimensional Ising model is given in terms of p-products of Wightman fields. It provides a quadratic factorization, in the sense of operator-valued distributions, of Abelian chiral vertex operators into non-Abelian exchange fields. This basic result implies p-polynomial relations between fields from various conformal models in two dimensions; notably the two-dimensional local fields with chiral SU(2) symmetry at level 2 are shown to share the nontrivial peculiarities of the Ising model order and disorder fields. These examples have interesting implications for the properties of p-products in general.

DOI
10.1063/1.532240

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•Jan 1, 2002•46 citations
The quantum field theoretical formulation of Kadanoff’s “correlations along a line” in the critical two-dimensional Ising model is given in terms of p-products of Wightman fields. It provides a quadratic factorization, in the sense of operator-valued distributions, of Abelian chiral vertex operators into non-Abelian exchange fields. This basic result implies p-polynomial relations between fields from various conformal models in two dimensions; notably the two-dimensional local fields with chiral SU(2) symmetry at level 2 are shown to share the nontrivial peculiarities of the Ising model order and disorder fields. These examples have interesting implications for the properties of p-products in general.
In this paper we review the critical Ising model by using the properties of conformal invariance. We use the known mapping of the Ising model to a theory of Majorana fermions and recognize that the self-duality property appears as a double degeneracy of the periodic ground state. The Ising model is doubled obtaining an Ashkin-Teller model (two Ising models coupled with a four-spin coupling) at the decoupling point. Using the bosonization technique this model is mapped onto the Gaussian model at a particular value of the temperature (K=1/\ensuremath{\pi}). This allows us to give the expressions for products of order, disorder, and energy operators of both Ising models in terms of operators in the Gaussian model. We compute several correlation functions of order, disorder, energy and spinor operators and show that they reproduce the operator product expansions predicted by conformal invariance. We explicitly discuss the physics by which the correlation functions obtained from the Gaussian model (a theory with conformal anomaly c=1) reproduce those of the Ising model (c=1/2). This provides a proof of Kadanoff and Brown's equivalences and conjectures. We provide a simple prescription to compute all n-point correlation functions at criticality, including mixed correlations of order, disorder, energy, and spinor operators. A bosonized form of the continuum limit of the transfer matrix for the critical Ising model is constructed. It is nonlocal and contains both spin-wave and vortex operators of the Gaussian model.
We consider systems of particles hopping stochastically on d-dimensional lattices with space-dependent probabilities. We map the master equation onto an evolution equation in a Fock space where the dynamics are given by a quantum Hamiltonian (continuous time) or a transfer matrix (discrete time). Using non-Abelian symmetries of these operators we derive duality relations, expressing the time evolution of a given initial configuration in terms of correlation functions of simpler dual processes. Particularly simple results are obtained for the time evolution of the density profile. As a special case we show that for any SU(2) symmetric system the two-point and three-point density correlation functions in the N-particle steady state can be computed from the probability distribution of a single particle moving in the same environment. We apply our results to various models, among them partial exclusion, a simple diffusion-reaction system, and the two-dimensional six-vertex model with space-dependent vertex weights. For a random distribution of the vertex weights one obtains a version of the random-barrier model describing diffusion of particles in disordered media. We derive exact expressions for the averaged two-point density correlation functions in the presence of weak, correlated disorder.
In this paper, we revisit the operator-state correspondence in the Majorana conformal field theory (CFT) with emphasis on its semion representation. Whereas the semion representation (or Z2{Z}_{2}Z2​ extension of the chiral Ising CFT) gives a concise ``Abelian'' (or invertible) representation at the level of fusion rules and quantum states, there exists a subtlety when considering the chiral multipoint correlation function. In this sense, the operator-state correspondence in the semion sector of the fermionic theory inevitably contains a difficulty coming from its anomalous conformal dimension 1/161/161/16 as a Z2{Z}_{2}Z2​ symmetry operator. By analyzing the asymptotic behaviors of the existing correlation functions, we propose a nontrivial correspondence between the chiral conformal blocks and bulk correlation functions containing both order and disorder fields. One can generalize this understanding to ZN{Z}_{N}ZN​ models or fractional supersymmetric models (for which there exist long-standing open problems). We expect that this may improve our understanding of the simple current extension of CFT, which commonly appears in the studies of topologically ordered systems.
By using conformal-field theory, we classify the possible irrelevant operators for the Ising model with nearest-neighbour interactions on the square and triangular lattices. We analyse the existing results for the free energy and its derivatives and for the correlation length, showing that they are in agreement with the conformal-field theory predictions. Moreover, these results imply that the nonlinear scaling field of the T operator, where T is the energy–momentum tensor, vanishes at the critical point. Several other peculiar cancellations are explained in terms of a number of general conjectures. We show that all existing results on the square and triangular lattices are consistent with the assumption that only nonzero-spin operators are present.
Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.
By using conformal-field theory, we classify the possible irrelevant operators for the Ising model on the square and triangular lattices. We analyze the existing results for the free energy and its derivatives and for the correlation length, showing that they are in agreement with the conformal-field theory predictions. Moreover, these results imply that the nonlinear scaling field of the energy-momentum tensor vanishes at the critical point. Several other peculiar cancellations are explained in terms of a number of general conjectures. We show that all existing results on the square and triangular lattice are consistent with the assumption that only nonzero spin operators are present.