A numerical adaptation of SAW identities from the honeycomb to other 2D lattices

Recently, Duminil-Copin and Smirnov proved a long-standing conjecture by Nienhuis that the connective constant of self-avoiding walks on the honeycomb lattice is $\sqrt{2+\sqrt{2}}.$ A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed on the honeycomb lattice persist on other lattices. This permits the accurate estimation, though not an exact evaluation, of certain critical amplitudes, as well as critical points, for these lattices. For the honeycomb lattice an exact amplitude for loops is proved.Comment: 21 pages, 7 figures. Changes in v2: Improved numerical analysis, giving greater precision. Explanation of why we observe what we do. Extra reference

Classical correlations of defects in lattices with geometrical frustration in the motion of a particle

We map certain highly correlated electron systems on lattices with geometrical frustration in the motion of added particles or holes to the spatial defect-defect correlations of dimer models in different geometries. These models are studied analytically and numerically. We consider different coverings for four different lattices: square, honeycomb, triangular, and diamond. In the case of hard-core dimer covering, we verify the existed results for the square and triangular lattice and obtain new ones for the honeycomb and the diamond lattices while in the case of loop covering we obtain new numerical results for all the lattices and use the existing analytical Liouville field theory for the case of square lattice.The results show power-law correlations for the square and honeycomb lattice, while exponential decay with distance is found for the triangular and exponential decay with the inverse distance on the diamond lattice. We relate this fact with the lack of bipartiteness of the triangular lattice and in the latter case with the three-dimensionality of the diamond. The connection of our findings to the problem of fractionalized charge in such lattices is pointed out.Comment: 6 pages, 6 figures, 1 tabl

Prefect Klein tunneling in anisotropic graphene-like photonic lattices

We study the scattering of waves off a potential step in deformed honeycomb lattices. For small deformations below a critical value, perfect Klein tunneling is obtained. This means that a potential step in any direction transmits waves at normal incidence with unit transmission probability, irrespective of the details of the potential. Beyond the critical deformation, a gap in the spectrum is formed, and a potential step in the deformation direction reflects all normal-incidence waves, exhibiting a dramatic transition form unit transmission to total reflection. These phenomena are generic to honeycomb lattice systems, and apply to electromagnetic waves in photonic lattices, quasi-particles in graphene, cold atoms in optical lattices