Properties of Laughlin states on fractal lattices

Abstract

Laughlin states have recently been constructed on fractal lattices and have been shown to be topological in such systems. Some of their properties are, however, quite different from the two-dimensional case. On the Sierpinski triangle, for instance, the entanglement entropy shows oscillations as a function of particle number and does not obey the area law despite being topologically ordered, and the particle density is non-uniform in the bulk. Here, we investigate these deviant properties in greater detail on the Sierpinski triangle, and we also study the properties on the Sierpinski carpet and the T-fractal. We find that the density variations across the fractal are present for all the considered fractal lattices and for most choices of the number of particles. The size of anyons inserted into the lattice Laughlin state also varies with position on the fractal lattice. We observe that quasiholes and quasiparticles have similar sizes and that the size of the anyons typically increases with decreasing Hausdorff dimension. As opposed to periodic lattices in two dimensions, the Sierpinski triangle and carpet have inner edges. We construct trial states for both inner and outer edge states. We find that oscillations of the entropy as a function of particle number are present for the T-fractal, but not for the Sierpinski carpet. Finally, we observe deviations from the area law for several different bipartitions on the Sierpinski triangle.