2015
DOI: 10.1090/conm/644/12795
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Discrete holomorphicity and Ising model operator formalism

Abstract: Abstract. We explore the connection between the transfer matrix formalism and discrete complex analysis approach to the two dimensional Ising model.We construct a discrete analytic continuation matrix, analyze its spectrum and establish a direct connection with the critical Ising transfer matrix. We show that the lattice fermion operators of the transfer matrix formalism satisfy, as operators, discrete holomorphicity, and we show that their correlation functions are Ising parafermionic observables. We extend t… Show more

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Cited by 12 publications
(16 citation statements)
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“…This allowed new developments concerning the fine understanding of the conformal invariance of the critical Ising model in general planar domains, both from the geometric (convergence of interfaces to SLE curves) and the analytic (confirming Conformal Field Theory predictions for the scaling limits of correlation functions) viewpoints; see [100,36,21,46,43,44,19,18,66,17,53,54]. A number of related results can be found in [35,6,30,13,14,45,38,16,7]. In parallel, further developments were made on various algorithmic and algebraic aspects of the model (e.g., see [78] and [70]).…”
mentioning
confidence: 87%
“…This allowed new developments concerning the fine understanding of the conformal invariance of the critical Ising model in general planar domains, both from the geometric (convergence of interfaces to SLE curves) and the analytic (confirming Conformal Field Theory predictions for the scaling limits of correlation functions) viewpoints; see [100,36,21,46,43,44,19,18,66,17,53,54]. A number of related results can be found in [35,6,30,13,14,45,38,16,7]. In parallel, further developments were made on various algorithmic and algebraic aspects of the model (e.g., see [78] and [70]).…”
mentioning
confidence: 87%
“…CFT provides a classification of local fields based on the Virasoro algebra. However, this is a priori difficult to link with local random variables in lattice models in the sense of (1), although much progress has been made in this direction recently with rigorous results in the Ising case [45,26,12,25]. This difficulty is particularly true for the stress-energy tensor T (w): it is expected to possess a wealth of important properties based on fundamental QFT principles, yet it is not easily identifiable as a local variable on the lattice.…”
Section: Conformal Field Theory and Conformal Loop Ensemblesmentioning
confidence: 99%
“…For self-avoiding walks, it provided a rigorous way to determine the bulk [14] and boundary [4] connectivity constants, and it has also proved very useful for numerical purposes [5]. In [17], discretely holomorphic observables for the Ising model have been related explicitly to the transfer-matrix formalism.…”
Section: Introductionmentioning
confidence: 99%