Non-integrable dimer models: Universality and scaling relations

Giuliani, Alessandro; Toninelli, Fabio Lucio

doi:10.1063/1.5109618

Cited by 4 publications

(4 citation statements)

References 53 publications

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“…where, letting x = (x 1 , x 2 ), Even if not indicated explicitly, the functions ν, α ω , β ω , K ω,j, , H ω,j, , p ω all depend non-trivially on the edge weights {t e }. In particular, generically, ν = 1 + c 1 λ + O(λ 2 ), with c 1 a non zero coefficient, which depends upon the edge weights (this was already observed in [26] for interacting dimers on planar graphs); therefore, generically, ν is larger or smaller than 1, depending on the sign of λ.…”

Section: 1mentioning

confidence: 74%

“…In a previous series of works [24,25,26,27], in collaboration with V. Mastropietro, we started developing methods for the treatment of interacting, non-solvable, dimer models via constructive, fermionic, Renormalization Group (RG) techniques. We exhibited an explicit class of models, which include the 6-vertex model close to its free Fermi point as well as several non-integrable versions thereof, for which we proved the GFF nature of the scaling limit of the height fluctuations, as well as the validity of a 'Kadanoff' or 'Haldane' scaling relation connecting the critical exponent of the so-called electric correlator with the one of the dimer-dimer correlation.…”

Section: Introductionmentioning

confidence: 99%

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Weakly non-planar dimers

Giuliani¹,

Bruno²,

Toninelli³

2022

Preprint

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We study a model of fully-packed dimer configurations (or perfect matchings) on a bipartite periodic graph that is two-dimensional but not planar. The graph is obtained from Z 2 via the addition of an extensive number of extra edges that break planarity (but not bipartiteness). We prove that, if the weight λ of the non-planar edges is small enough, the height function scales on large distances to the Gaussian Free Field with a λ-dependent amplitude, that coincides with the anomalous exponent of dimer-dimer correlations. Because of non-planarity, Kasteleyn's theory does not apply: the model is non-determinantal. Rather, we map the model to a system of interacting lattice fermions in the Luttinger universality class, that we then analyze via fermionic Renormalization Group methods.

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Section: 1mentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Weakly non-planar dimers

Giuliani¹,

Bruno²,

Toninelli³

2022

Preprint

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“…This is not true any more for the interacting model. Indeed, an explicit calculation of ν at first order in λ for the model with plaquette interaction shows a non-trivial dependence both on λ and on the weights [27].…”

Section: Remarkmentioning

confidence: 99%

Non-integrable Dimers: Universal Fluctuations of Tilted Height Profiles

Giuliani

Mastropietro

Toninelli

2020

Commun. Math. Phys.

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We study a class of close-packed dimer models on the square lattice, in the presence of small but extensive perturbations that make them non-determinantal. Examples include the 6-vertex model close to the free-fermion point, and the dimer model with plaquette interaction previously analyzed in previous works. By tuning the edge weights, we can impose a non-zero average tilt for the height function, so that the considered models are in general not symmetric under discrete rotations and reflections. In the determinantal case, height fluctuations in the massless (or 'liquid') phase scale to a Gaussian log-correlated field and their amplitude is a universal constant, independent of the tilt. When the perturbation strength λ is sufficiently small we prove, by fermionic constructive Renormalization Group methods, that log-correlations survive, with amplitude A that, generically, depends non-trivially and non-universally on λ and on the tilt. On the other hand, A satisfies a universal scaling relation ('Haldane' or 'Kadanoff' relation), saying that it equals the anomalous exponent of the dimer-dimer correlation.

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“…the non-oscillatory term. Let us remark that an explicit computation [51] shows that, generically, not only ν ′ (0) is different from zero, but it also depends explicitly upon the average slope ρ j = ρ j (t 1 , t 2 , t 3 , λ) = e∈C η→η+aê j σ e (lim L→∞ E λ a,Ω (1 e ) − 1/4). Once that the sharp asymptotics for the two-point dimer correlation is know, we can compute the variance of height fluctuations, in analogy with (3.14) then we plug the asymptotics (3.17) in the right side of this equation, and by choosing the paths C ηx 1 →ηx 2 , C ηx 3 →ηx 4 well separated, we find that the contributions to the variance from the second and third terms in the right side of (3.17) vanish as a → 0.…”

Section: Interacting Dimer Modelsmentioning

confidence: 99%

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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Non-integrable dimer models: Universality and scaling relations

Cited by 4 publications

References 53 publications

Weakly non-planar dimers

Weakly non-planar dimers

Non-integrable Dimers: Universal Fluctuations of Tilted Height Profiles

Scaling limits and universality of Ising and dimer models

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