Non-integrable Dimers: Universal Fluctuations of Tilted Height Profiles

• Published in: Communications in mathematical physics Vol. 377; no. 3; pp. 1883 - 1959
• Main Authors: Giuliani, Alessandro, Mastropietro, Vieri, Toninelli, Fabio Lucio
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2020; Springer Nature B.V; Springer Verlag
• Subjects: Amplitudes; Classical and Quantum Gravitation; Complex Systems; Correlation; Dimers; Fermions; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Perturbation; Physics; Physics and Astronomy; Probability; Quantum Physics; Relativity Theory; Renormalization group methods; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-020-03760-x

We study a class of close-packed dimer models on the square lattice, in the presence of small but extensive perturbations that make them non-determinantal. Examples include the 6-vertex model close to the free-fermion point, and the dimer model with plaquette interaction previously analyzed in previous works. By tuning the edge weights, we can impose a non-zero average tilt for the height function, so that the considered models are in general not symmetric under discrete rotations and reflections. In the determinantal case, height fluctuations in the massless (or ‘liquid’) phase scale to a Gaussian log-correlated field and their amplitude is a universal constant, independent of the tilt. When the perturbation strength λ is sufficiently small we prove, by fermionic constructive Renormalization Group methods, that log-correlations survive, with amplitude A that, generically, depends non-trivially and non-universally on λ and on the tilt. On the other hand, A satisfies a universal scaling relation (‘Haldane’ or ‘Kadanoff’ relation), saying that it equals the anomalous exponent of the dimer–dimer correlation.