Haldane relation for interacting dimers

Abstract

We consider a model of weakly interacting, close-packed, dimers on the two-dimensional square lattice. In a previous paper, we computed both 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'). In this paper, we 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 validity is a strong confirmation of the effective massless Gaussian field description of the interacting dimer model, which was guessed on the basis of formal bosonization arguments. We also conjecture that a certain discrete curve defined at the lattice level via the Temperley bijection converges in the scaling limit to an SLE${}_{\kappa }$ process, with $\kappa$ depending non-trivially on the interaction and related in a simple way to the amplitude of the limiting Gaussian field.

43 pages, 3 figures

References in corpus (6)

• Classical dimers with aligning interactions on the square lattice
• Quantum criticality, lines of fixed points, and phase separation in doped two-dimensional quantum dimer models
• Non perturbative Adler-Bardeen Theorem
• The six-vertex model and Schramm-Loewner evolution
• Asymptotics of height change on toroidal Temperleyan dimer models
• Height fluctuations in non-integrable classical dimers

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• Extreme boundary conditions and random tilings
• Universal edge transport in interacting Hall systems
• Nonexistence and Uniqueness for Pure States of Ferroelectric Six-Vertex Models
• Non-integrable dimer models: universality and scaling relations
• Derivation of field theory for the classical dimer model using bosonization
• The scaling limit of boundary spin correlations in non-integrable Ising models