Lecture 6 - The interacting dimer model
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| Description: |
Giuliani, A
Friday 19th October 2018 - 11:00 to 12:30 |
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| Created: | 2018-10-23 16:45 |
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| Collection: | Scaling limits, rough paths, quantum field theory |
| Publisher: | Isaac Newton Institute |
| Copyright: | Giuliani, A |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
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| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| 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.
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