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

TL;DR

This paper investigates defect correlations in geometrically frustrated lattices by mapping electron systems to dimer models, revealing different correlation decay behaviors across various lattice geometries.

Contribution

It provides new analytical and numerical results on defect correlations in honeycomb and diamond lattices, extending previous findings for square and triangular lattices.

Findings

Power-law correlations in square and honeycomb lattices

Exponential decay in triangular lattice

Inverse distance decay in diamond lattice

Abstract

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.