Discrete holomorphicity and Ising model operator formalism

Hongler, Clément; Kytölä, Kalle; Zahabi, A.

doi:10.48550/arxiv.1211.7299

Cited by 4 publications

(11 citation statements)

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“…We have used these techniques to obtain the scaling limit of the correlation functions of Ising free fermions with specific boundary conditions. The continuum correlation functions are obtained from the scaling limit of the lattice correlation functions, [HKZ12]. By means of these methods, we find a proof of the Pfaffian formula for the correlation functions of free fermionic fields, in the scaling limit.…”

Section: Introductionmentioning

confidence: 91%

“…where P is any permutation of {1, 2, ..., 2n} and Sgn(P ) is the sign of the permutation. This result can be proved by using the Pfaffian formula for the lattice fermion correlation function (section (4.4) in [HKZ12]) and then taking the scaling limit.…”

Section: Fermionic Conformal Field Theory On Domain Dmentioning

confidence: 96%

“…The row-to-row propagation/time evolution of the fermions and their correlation functions in the transfer matrix analysis, and their relations to discrete holomorphicity are studied in [HKZ12]. It has been shown in [HKZ12], that the transfer matrix can be written in terms of Clifford algebra generators p k and q k and thus the time evolution of the free fermions can be explicitly calculated via the conjugation by the transfer matrix. This conjugation is called induced rotation and it is denoted by T (V M ).…”

Section: Ising Free Fermionsmentioning

confidence: 99%

“…They are solutions of the Riemann boundary value problem. For a general definition of Riemann boundary value problem see section (2) of [HKZ12]. Then we need to obtain the scaling limit of the above s-holomorphic functions.…”

Section: Ising Free Fermionsmentioning

confidence: 99%

“…The scaling limit of the correlation functions of Ising fermions on the upper-half plane can be determined by using first, the relation between lattice fermionic correlation functions and parafermionic observables and second, the scaling limit of the parafermionic observables. The relations between fermionic correlation functions and parafermionic observables are stated in Theorem (22) in [HKZ12]. In slightly different notation they are as follows,…”

Section: Ising Free Fermionsmentioning

confidence: 99%

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Conformal Field Theory, Vertex Operator Algebra and Stochastic Loewner Evolution in Ising Model

Zahabi¹

2015

Preprint

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We review the algebraic and analytic aspects of the conformal field theory (CFT) and its relation to the stochastic Loewner evolution (SLE) in an example of the Ising model. We obtain the scaling limit of the correlation functions of Ising free fermions on an arbitrary simply connected two-dimensional domain D. Then, we study the analytic and algebraic aspects of the fermionic CFT on D, using the Fock space formalism of fields, and the Clifford vertex operator algebra (VOA). These constructions lead to the conformal field theory of the Fock space fields and the fermionic Fock space of states and their relations in case of the Ising free fermions. Furthermore, we investigate the conformal structure of the fermionic Fock space fields and the Clifford VOA, namely the operator product expansions, correlation functions and differential equations. Finally, by using the Clifford VOA and the fermionic CFT, we investigate a rigorous realization of the CF T /SLE correspondence in the Ising model. First, by studying the relation between the operator formalism in the Clifford VOA and the SLE martingale generators, we find an explicit Fock space for the SLE martingale generators. Second, we obtain a subset of the SLE martingale observables in terms of the correlation functions of fermionic Fock space fields which are constructed from the Clifford VOA.

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Section: Introductionmentioning

confidence: 91%

Section: Fermionic Conformal Field Theory On Domain Dmentioning

confidence: 96%

Section: Ising Free Fermionsmentioning

confidence: 99%

Section: Ising Free Fermionsmentioning

confidence: 99%

Section: Ising Free Fermionsmentioning

confidence: 99%

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Conformal Field Theory, Vertex Operator Algebra and Stochastic Loewner Evolution in Ising Model

Zahabi¹

2015

Preprint

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The Fermionic Observable in the Ising Model and the Inverse Kac–Ward Operator

Lis¹

2013

Ann. Henri Poincaré

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Abstract. We show that the critical Kac-Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac-Ward operator on any planar graph is given by what we call a fermionic generating function. Furthermore, using bounds obtained in [20] for the spectral radius and operator norm of the Kac-Ward transition matrix, we provide a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac-Ward operators on isoradial graphs.

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Random loops and conformal field theory

Doyon¹

2014

J. Stat. Mech.

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Discrete holomorphicity and Ising model operator formalism

Cited by 4 publications

References 0 publications

Conformal Field Theory, Vertex Operator Algebra and Stochastic Loewner Evolution in Ising Model

Conformal Field Theory, Vertex Operator Algebra and Stochastic Loewner Evolution in Ising Model

The Fermionic Observable in the Ising Model and the Inverse Kac–Ward Operator

Random loops and conformal field theory

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