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.
In integrable models of quantum field theory, local fields are normally constructed by means of the bootstrap-formfactor program. However, the convergence of their $n$-point functions is unclear in this setting. An alternative approach uses fully convergent expressions for fields with weaker localization properties in spacelike wedges, and deduces existence of observables in bounded regions from there, but yields little information about their explicit form. We propose a new, hybrid construction: We aim to describe pointlike local quantum fields; but rather than exhibiting their $n$-point func…
We present a new viewpoint on the construction of pointlike local fields in integrable models of quantum field theory. As usual, we define these local observables by their form factors; but rather than exhibiting their $n$-point functions and verifying the Wightman axioms, we aim to establish them as closed operators affiliated with a net of local von Neumann algebras, which is defined indirectly via wedge-local quantities. We also investigate whether these fields have the Reeh-Schlieder property, and in which sense they generate the net of algebras. Our investigation focuses on scalar models …
We prove convergence of the 2- and 4-point fermionic observables of the FK-Ising model on simply connected domains discretised by a planar isoradial lattice in massive (near-critical) scaling limit. The former is alternatively known as a (fermionic) martingale observable (MO) for the massive interface, and in particular encapsulates boundary visit probabilties of the interface. The latter encodes connection probabilities in the 4-point alternating (generalised Dobrushin) boundary condition, whose exact convergence is then further analysed to yield crossing estimates for general boundary condit…
We consider many-point correlation functions of discrete fermions in the two-dimensional FK-Ising model and show that, despite not being commuting observable, they can be realized with a geometric-probabilistic approach in terms of loops of the model and their winding.
We compute rigorously the scaling limit of multi-point energy correlations in the critical Ising model on a torus. For the one-point function, averaged between horizontal and vertical edges of the square lattice, this result has been known since the 1969 work of Ferdinand and Fischer. We propose an alternative proof, in a slightly greater generality, via a new exact formula in terms of determinants of discrete Laplacians. We also compute the main term of the asymptotics of the difference $\mathbb{E}(ε_{V}-ε_{H})$ of the energy density on a vertical and a horizontal edge, which is of order of $…
We discuss the notion of s-embeddings $\mathcal{S}=\mathcal{S}_\mathcal{X}$ of planar graphs carrying a nearest-neighbor Ising model. The construction of $\mathcal{S}_\mathcal{X}$ is based upon a choice of a global complex-valued solution $\mathcal{X}$ of the propagation equation for Kadanoff-Ceva fermions. Each choice of $\mathcal{X}$ provides an interpretation of all other fermionic observables as s-holomorphic functions on $\mathcal{S}_\mathcal{X}$. We set up a general framework for the analysis of such functions on s-embeddings $\mathcal{S}^δ$ with $δ\to 0$. Throughout this analysis, a key…
We study the 2-dimensional Ising model at critical temperature on a simply connected subset $Ω_δ$ of the square grid $δ\mathbb{Z}^{2}$. The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to be a precise correspondence between local lattice fields of the Ising model and the local fields of Conformal Field Theory. Towards the proof of this correspondence, we analyze arbitrary spin pattern probabilities (probabilities of finite spin configurations occurring at the origin), explicitly obtain their infinite-volume…
