Non-integrable Ising models in cylindrical geometry: Grassmann representation and infinite volume limit

Abstract

In this paper, meant as a companion to arXiv:2006.04458, we consider a class of non-integrable $2D$ Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green's function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries.

69 pages. Minor revisions. Version accepted by Annales Henri Poincaré. arXiv admin note: substantial text overlap with arXiv:2006.04458

References in corpus (4)

• The scaling limit of the energy correlations in non integrable Ising models
• The Ising model on a cylinder: universal finite size corrections and diagonalized action
• Conformal invariance and Renormalization Group
• Interacting fermions on the half-line: boundary counterterms and boundary corrections

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