Direct construction of the quantum field operators of the D = 2 Ising model

Bert Schroer, Trong Tuong Truong

Daniela Cadamuro

The construction of pointlike fields in quantum integrable models is usually approached by constructing their n-point functions. However, convergence questions of the resulting infinite series remain unresolved even in the simplest case of the massive Ising model. On the other hand, the C*-algebraic approach to the construction considers semi-local bounded operators, but yields local operators only in a very abstract way. We use a novel approach to investigate the structure of these local observables, in the form of smeared pointlike quantum fields. Instead of considering their n-point functions and verifying the Wightman axioms, we establish them as closed operators affiliated with the local C*-algebras. We explicitly construct all pointlike fields of the Ising model, verify the Reeh-Schlieder property and show that they generate the local algebras by dualization.

Djordje Radicevic

Starting from the operator algebra of the (1+1)D Ising model on a spatial lattice, this paper explicitly constructs a subalgebra of smooth operators that are natural candidates for continuum fields in the scaling limit. At the critical value of the transverse field, these smooth operators are analytically shown to reproduce the operator product expansions found in the Ising conformal field theory.

David Evans, Alan L. Carey

Clément Hongler, Kalle Kytölä, Ali Zahabi

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 these correspondences also to outside the critical point. We show that critical Ising correlations can be computed with operators on discrete Cauchy data spaces, which encode the geometry and operator insertions in a manner analogous to the quantum states in the transfer matrix formalism.

Пронько А. Г., S. K. Syrygina