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Abstract:

We study the classical hard-core dimer model on the triangular lattice. Following Kasteleyn's fundamental theorem on planar graphs, this problem is soluble using Pfaffians. This model is particularly interesting for, unlike the dimer problems on the bipartite square and hexagonal lattices, its correlations are short ranged with a correlation length of less than one lattice constant. We compute the dimer-dimer and monomer-monomer correlators, and find that the model is deconfining: the monomer-monomer correlator falls off exponentially to a constant value 0.1494..., only slightly below the nearest-neighbor value of 1/6. We also consider the anisotropic triangular lattice model in which the square lattice is perturbed by diagonal bonds of one orientation and small fugacity. We show that the model becomes noncritical immediately and that this perturbation is equivalent to adding a mass term to each of two Majorana fermions that are present in the long wavelength limit of the square lattice problem.

Abstract:

We study the behavior of Heisenberg, antiferromagnetic, integer-spin chains in the presence of a magnetic field exceeding the attendant spin gap. For temperatures much smaller than the gap, the spin chains exhibit Luttinger liquid behavior. We compute exactly both the corresponding Luttinger parameter and the Fermi velocity as a function of magnetic field. This enables the computation of a number of correlators from which we derive the spin conductance, the expected form of the dynamic structure factor relevant to inelastic neutron-scattering experiments, and NMR relaxation rates. We also comment upon the robustness of the magnetically induced gapless phase both to finite temperature and finite couplings between neighboring chains.