Non-integrable Ising Models in Cylindrical Geometry: Grassmann Representation and Infinite Volume Limit

Antinucci, Giovanni; Giuliani, Alessandro; Greenblatt, Rafael L.

doi:10.1007/s00023-021-01107-3

Cited by 6 publications

(18 citation statements)

References 25 publications

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“…Another common feature of the dimer and Ising models discussed in this paper is that their interacting, non-solvable, versions can be formulated exactly, at finite a and finite Ω, in terms of non-Gaussian Grassmann integrals [6,44,50]. For instance, the partition function of the interacting dimer model can be written as follows (again, the one of the non-planar Ising model admits an analogous representation):…”

Section: Methods and Ideas Behind The Proofsmentioning

confidence: 99%

“…The approach developed over the years to identify these cancellations in critical systems is based on the ideas of Wilsonian RG. More specifically, the constructive RG approach used in the proofs of Theorems 2.1 to 3.2 is the one developed by Benfatto, Gallavotti, Mastropietro and coworkers [16,17,19] and reviewed in, e.g., [43,75]; see also the more recent works [6], [48], and [47], which review and provide a pedagogical introduction to this method in the specific contexts of non-planar Ising models, interacting dimer models, and fermionic φ 4 d theories with long range interactions, respectively. At a very coarse level, the idea is to compute (4.2) recursively, first integrating out the degrees of freedom at length scales ℓ 0 2 −N ≃ a (here ℓ 0 is the length unit and −N = ⌊log 2 (a/ℓ 0 )⌋), then those at length scales ℓ 0 2 −N +1 , ℓ 0 2 −N +2 , .…”

Section: Methods and Ideas Behind The Proofsmentioning

confidence: 99%

“…This is a severe limitation for the rigorous construction of scaling limits in finite domains and for the study of their conformal covariance with respect to deformations of the domain. Recently, we managed to overcome several of these technical issues and to provide the first construction of non-planar Ising models in a domain with boundary, in cylindrical geometry: 5,6]). Fix V as in Theorem 2.1 and let λ 0 , β c = β c (λ) and Z 2 = Z 2 (λ) be the same introduced there.…”

Section: The Non-planar Casementioning

confidence: 99%

“…Moreover, it is not yet flexible enough for dealing with non-translationally invariant situations, including geometric perturbations of the domain or of the underlying lattice. However, recent progress in simple domains with boundaries [5,6] opens new perspectives for applications to general geometries and for an effective combinations of constructive RG ideas with probabilistic ones.…”

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Scaling limits and universality of Ising and dimer models

Giuliani¹

2021

Preprint

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After having introduced the notion of universality in statistical mechanics and its importance for our comprehension of the macroscopic behavior of interacting systems, I review recent progress in the understanding of the scaling limit of lattice critical models, including a quantitative characterization of the limiting distribution and the robustness of the limit under perturbations of the microscopic Hamiltonian. Specifically, I focus on two classes of non-exactly-solvable two-dimensional systems: non-planar Ising models and interacting dimers. In both settings, I describe the conjectures on the expected structure of the scaling limit, review the progress towards their proof, and state some of the recent results on the universality of the limit, which I contributed to. Finally, I outline the ideas and methods involved in the proofs, describe some of the perspectives opened by these results, and propose several open problems.

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Section: Methods and Ideas Behind The Proofsmentioning

confidence: 99%

Section: Methods and Ideas Behind The Proofsmentioning

confidence: 99%

Section: The Non-planar Casementioning

confidence: 99%

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confidence: 99%

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Scaling limits and universality of Ising and dimer models

Giuliani¹

2021

Preprint

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“…From a technical point of view, going in this direction requires a non-trivial extension of the RG multiscale methods to the non-translationally-invariant setting, and a sharp control of the RG flow of the marginal boundary running coupling constants. Promising results in this direction have been recently obtained in the context of non-planar critical Ising models in the half-plane [5].…”

Section: Theorem 2 [23]mentioning

confidence: 97%

Non-integrable dimer models: Universality and scaling relations

Giuliani

Toninelli

2019

Journal of Mathematical Physics

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In the last few years, the methods of constructive Fermionic Renormalization Group have been successfully applied to the study of the scaling limit of several two-dimensional statistical mechanics models at the critical point, including: weakly non-integrable 2D Ising models, Ashkin-Teller, 8-Vertex, and close-packed interacting dimer models. In this note, we focus on the illustrative example of the interacting dimer model and review some of the universality results derived in this context. In particular, we discuss the massless Gaussian free field (GFF) behavior of the height fluctuations. It turns out that GFF behavior is connected with a remarkable identity ('Haldane' or 'Kadanoff relation') between an amplitude and an anomalous critical exponent, characterizing the large distance behavior of the dimer-dimer correlations. 0

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Energy Correlations of Non-Integrable Ising Models: The Scaling Limit in the Cylinder

Antinucci

Giuliani

Greenblatt

2022

Commun. Math. Phys.

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We consider a class of non-integrable 2D Ising models whose Hamiltonian, in addition to the standard nearest neighbor couplings, includes additional weak multi-spin interactions which are even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and compute the multipoint energy correlations at the critical temperature via a multiscale expansion, uniformly convergent in the domain size and in the lattice spacing. We prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the nearest neighbor Ising model in a finite cylinder with renormalized horizontal and vertical couplings, up to an overall multiplicative constant independent of the shape of the domain. The proof is based on a representation of the generating function of correlations in terms of a non-Gaussian Grassmann integral, and a constructive Renormalization Group (RG) analysis thereof. A key technical novelty compared with previous works is a systematic analysis of the effect of the boundary corrections to the RG flow, in particular a proof that the scaling dimension of boundary operators is better by one dimension than their bulk counterparts. In addition, a cancellation mechanism based on an approximate image rule for the fermionic Green’s function is of crucial importance for controlling the flow of the (superficially) marginal boundary terms under RG iterations.

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Non-integrable Ising Models in Cylindrical Geometry: Grassmann Representation and Infinite Volume Limit

Cited by 6 publications

References 25 publications

Scaling limits and universality of Ising and dimer models

Scaling limits and universality of Ising and dimer models

Non-integrable dimer models: Universality and scaling relations

Energy Correlations of Non-Integrable Ising Models: The Scaling Limit in the Cylinder

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