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Lecture 4  The interacting dimer model
http://sms.cam.ac.uk/media/2845158
Giuliani, A
Monday 15th October 2018  11:00 to 12:30
40
Lecture 4  The interacting dimer model
http://sms.cam.ac.uk/media/2845158
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Lecture 4  The interacting dimer model
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http://sms.cam.ac.uk/media/2845158
Lecture 4  The interacting dimer model
Giuliani, A
Monday 15th October 2018  11:00 to 12:30
Tue, 16 Oct 2018 13:09:24 +0100
Isaac Newton Institute
Giuliani, A
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Giuliani, A
Monday 15th October 2018  11:00 to 12:30
Giuliani, A
Monday 15th October 2018  11:00 to 12:30
Cambridge University
6000
http://sms.cam.ac.uk/media/2845158
Lecture 4  The interacting dimer model
Giuliani, A
Monday 15th October 2018  11:00 to 12:30
The aim of this minicourse is to present recent results, obtained together with Vieri Mastropietro (arXiv:1406.7710 and arXiv:1612.01274), on nonintegrable perturbations of the classical dimer model on the square lattice. In the integrable situation, the model is freefermionic and the largescale fluctuations of its height function tend to a twodimensional massless Gaussian field (GFF). We prove that convergence to GFF holds also for sufficiently small nonintegrable perturbations. At the same time, we show that the dimerdimer correlations exhibit nontrivial 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 noninteracting dimer model: Kasteleyn theory, thermodynamiclimit, longdistance asymptotics of correlations, GFF fluctuations. Fermionic representation of the noninteracting and of the interacting dimer model. 3. Multiscale analysis of the free propagator, Feynman diagrams and dimensional estimates. Determinant expansion. Nonrenormalized 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. SchwingerDyson equation. Closed equation for the correlation functions. Comparison of the lattice model with the reference one.
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