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Lecture 1 - The interacting dimer model

Presented by: 
Fabio Toninelli Université Claude Bernard Lyon 1
Date: 
Monday 8th October 2018 - 11:00 to 12:30
Venue: 
INI Seminar Room 2
Abstract: 
The aim of this minicourse is to present recent results, obtained together with Vieri Mastropietro (arXiv:1406.7710 and arXiv:1612.01274), on non-integrable perturbations of the classical dimer model on the square lattice. In the integrable situation, the model is free-fermionic and the large-scale fluctuations of its height function tend to a two-dimensional massless Gaussian field (GFF). We prove that convergence to GFF holds also for sufficiently small non-integrable perturbations. At the same time, we show that the dimer-dimer correlations exhibit non-trivial critical exponents, continuously depending upon the strength of the interaction: the model belongs, in a suitable sense, to the `Luttinger liquid' universality class. The proofs are based on constructive Renormalization Group for interacting fermions in two dimensions.   Contents:   1. Basics: the model, height function, interacting dimer model. The main results for the interacting model: GFF fluctuations and    Haldane relation.   2. The non-interacting dimer model: Kasteleyn theory, thermodynamiclimit, long-distance asymptotics of correlations, GFF fluctuations. Fermionic representation of the non-interacting and of the interacting dimer model.   3. Multi-scale analysis of the free propagator, Feynman diagrams and dimensional estimates. Determinant expansion.   Non-renormalized multiscale expansion.   4. Renormalized multiscale expansion. Running coupling constants. Beta function.   5. The reference continuum model (the `infrared fixed point'): the Luttinger model. Exact solvability of the Luttinger model. Bosonization.   6. Ward identities and anomalies. Schwinger-Dyson equation. Closed equation for the correlation functions. Comparison of the lattice model with the reference one.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons