Abstract
The fractional quantum Hall effect is a paradigm of topological order and has been studied thoroughly in two dimensions. Here, we construct a new type of fractional quantum Hall system, which has the special property that it lives in fractal dimensions. We provide analytical wave functions and exact few-body parent Hamiltonians, and we show numerically for several different Hausdorff dimensions between 1 and 2 that the systems host anyons. We also find examples of fractional quantum Hall physics in fractals with Hausdorff dimension 1 and ln(4)/ln(5). Our results suggest that the local structure of the investigated fractals is more important than the Hausdorff dimension to determine whether the systems are in the desired topological phase. The study paves the way for further investigations of strongly-correlated topological systems in fractal dimensions.
Highlights
Elementary particles that exist in three dimensions are all either bosons or fermions
These developments are important, since each new type of system has its own properties: graphene gave a relativistic version of the fractional quantum Hall effect, the introduction of lattices eliminated the need for a physical magnetic field, and generalized anyons in three or four dimensions are quite different from anyons in two dimensions
Our results suggest that anyons and fractional quantum Hall physics can exist in all dimensions between 1 and 2
Summary
Sourav Manna,* Biplab Pal ,* Wei Wang ( ) ,* and Anne E. Among the most important developments in the field are the discoveries of new types of fractional quantum Hall effects and new types of systems where anyons can be realized. The much harder question of whether interacting topological phases, fractional quantum Hall physics, and anyons can exist in fractal space is, still unanswered. We answer this important question in the affirmative by constructing a different type of fractional quantum Hall models, which have the special property that they live in fractal space. We find fractional quantum Hall physics on certain fractals with Hausdorff dimension 1 and below This demonstration of topology in interacting systems with fractal dimensions opens up several possibilities for further investigations. A Sierpinski gasket of generation zero is a single triangle, and the Sierpinski gasket of generation n + 1 is obtained from the Sierpinski gasket of generation n by applying the operation shown in Fig. 1 to all
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