Presented by:
Fabio Toninelli
Date:
Wednesday 10th October 2018 - 15:30 to 17:00
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.
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