# Haldane relation for interacting dimers

In this talk I will review some recent results on the existence and nature
of the scaling limit of interacting, close-packed, dimers on the
two-dimensional square lattice. By constructive Renormalization Group
techniques, we compute: the multipoint dimer correlations, which display
non-trivial critical exponents, continuously varying with the interaction
strength; and the height fluctuations, which, after proper coarse graining
and rescaling, converge to the massless Gaussian field with a suitable
interaction-dependent pre-factor (`amplitude’). We also prove the identity
between the critical exponent of the two-point dimer correlation and the
amplitude of this massless Gaussian field. This identity is the
restatement, in the context of interacting dimers, of one of the Haldane
universality relations, part of his Luttinger liquid conjecture,
originally formulated in the context of one-dimensional interacting Fermi
systems. Its proof requires the combined use of an exact lattice Ward
Identity for the lattice theory, with the chiral Ward Identites of a
continuum reference model, which describes the infrared fixed point of the
interacting theory.
Joint work with V. Mastropietro and F. Toninelli.