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Lecture 6 - The interacting dimer model

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SRQ - Scaling limits, rough paths, quantum field theory

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.

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