Anyons and Fractional Quantum Hall Effect in Fractal Dimensions

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 Laughlin type 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 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 Laughlin type 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.
Elementary particles that exist in three dimensions are all either bosons or fermions. Nevertheless, Leinaas and Myrheim argued that in two dimensions it is allowed mathematically to have particles that are neither bosons, nor fermions, but anyons [1]. Anyons have unusual exchange properties and can have fractional charge. It was later discovered that anyons are realized physically as quasiparticles in the fractional quantum Hall effect [2,3]. The investigations of anyons have led to more important new insights, including a large development within the description of quantum phases and phase transitions [4], and ideas to use anyons to store and process quantum information in a topologically protected way [5].
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. This includes the observation of fractional quantum Hall physics in graphene [6,7], fractional quantum Hall physics in lattice systems [8], and generalizations of anyons and the fractional quantum Hall effect to three-or four-dimensional systems [9][10][11]. 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.
The possibility of changing the dimension is particularly interesting, since the properties of a system in general depend strongly on the dimension of the system. Introducing fractal structures, it is even possible to consider non-integer dimensions. In the past, classical and singleparticle quantum models have been studied on fractal lattices [12][13][14], and renewed interest in the topic, in the form of investigations of noninteracting quantum models, has appeared recently [15][16][17][18][19][20]. This interest is, in part, motivated by experimental developments to prepare fractal structures in molecules and nanomaterials [21,22]. Ultracold atoms in optical lattices also provide an interesting direction for realizing fractal models, in particular given the current efforts on creating innovative lattice potentials [23]. The question about topological quantum models on fractals has been taken up recently [17,18,20], where noninteracting Chern insulator models have been investigated. The much harder question of whether interacting topological phases, fractional quantum Hall physics, and anyons can exist in fractal space is, however, still unanswered.
In this Letter, we answer this important question in the affirmative by constructing a new type of fractional quantum Hall models, which have the special property that they live in fractal space. We provide analytical wavefunctions and corresponding few-body parent Hamiltonians. We also show how to construct anyons in the models and that the anyons are screened and have the correct charge and braiding properties. Our results suggest that anyons and fractional quantum Hall physics can exist in all dimensions between 1 and 2. We also find fractional quantum Hall physics on certain fractals with Hausdorff dimension 1 and below.
The numerical resources needed to study stronglycorrelated quantum many-body systems generally grow exponentially with the system size. Two-dimensional systems are therefore difficult to handle. One-dimensional systems are easier, but the physics is often quite different, e.g. because there are strong restrictions on the possible directions a particle can move. Systems with dimensions between 1 and 2 constitute an intriguing intermediate regime, where interesting physics is likely to happen. The present work is a first example in this direction.
Fractal lattices.-Spaces having fractal dimension are realized in fractals. A famous example is the Sierpinski gasket ( Fig. 1) with dimension D = ln(3)/ ln(2) ≈ 1.5850. A Sierpinski gasket of generation 0 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 on the left in Fig. 1 to all triangles in the gasket. The full fractal is obtained in the limit of infinite generation. In any physical system, there is a limit to how small the smallest scales of a fractal can be and a limit to how large the total fractal can be, so the generation of a physical fractal is always finite. As long as we are investigating the fractal at a length scale, which is large compared to the smallest structures of the fractal and small compared to the total size of the fractal, it does, however, not make a difference whether the generation of the fractal is finite or infinite, and the system will effectively be in a space with the dimension of the fractal.
Here, we consider a lattice model on the fractal, where there is one lattice site at the center of each of the smallest triangles. In the limit of large enough generation, it does not make a significant difference, whether we treat each triangle as a triangle or a single point, since the triangles are much smaller than the length scales of interest.
Quantum states.-We start our search for anyons in fractal space by constructing fractional quantum Hall states on fractal lattices. The standard fractional quantum Hall effect is realized in a two-dimensional electron gas, and an important ingredient is a strong magnetic field perpendicular to the plane. Just taking a fractional quantum Hall state and restricting the possible particle positions to be on the fractal lattice is not enough to obtain the desired states. The main trick is that we also need to restrict the magnetic flux to only go through the lattice sites as shown in Fig. 1. This means that the Gaussian factor, present, e.g., in the Laughlin state, is modified. We can find the appropriate modification by using the conformal field theory approach [24,25] to construct the states.
We here consider the Laughlin state with q fluxes per particle, where q is an integer. We associate a vertex operator V nj (z j ) = : e i(qnj −η)φ(zj )/ √ q : to each of the lattice sites. Here, n j ∈ {0, 1} is the number of particles on the site, z j is the position of the site in the plane written as a complex number, −η is the magnetic flux through the site, φ(z j ) is the chiral part of a free massless boson, and : . . . : means normal ordering. We The lattice sites are drawn as circles, and the centers of the two anyons are on the sites marked by a green ring. The color of the jth circle gives ρj = ψ2,2,29|nj|ψ2,2,29 − ψ2,0,30|nj|ψ2,0,30 , i.e., how much the presence of the anyons affect the number of particles on each site. The anyons are seen to be screened. Note also that when we keep the total number of particles fixed and increase the generation of the fractal, the anyons are spread over more sites, but the total size of the anyons remain the same. take −η < 0, since the magnetic field points inwards (as in Fig. 1). In the two-dimensional case, we can insert anyons with charge p k /q, where p k is an integer, into the states at the positions w k by including a vertex operator H(w k ) = : e ip k φ(w k )/ √ q : for each of the anyons. We do the same here to investigate whether these operators also produce anyons on the fractal lattice. The Laughlin state on a fractal lattice with N sites, M particles, and K "anyons" is then defined as |ψ q,K,M ∝ n1,...,n N 0| where |0 is the vacuum state. This evaluates to Here, φ j are undetermined phases, which do not influence the results below, and δ n is one if M = j n j = (ηN − k p k )/q and zero otherwise. Note that the wavefunction has a well-defined limit for w k → z j , so the anyon coordinates are allowed to be on the lattice sites. In all our numerical computations below, we take p k = 1. When we go from one generation to the next, we keep the number of particles M and the total flux −ηN fixed.
Topological properties.-Although the analytical expression for the state (1) is similar to the Laughlin state, there is no guarantee that the state has the correct topological properties to qualify for being a Laughlin type state. We now show numerically that it has. We do this by showing that the anyons are screened and have the same charge and braiding properties as the normal Laughlin anyons.
The particle density difference on site j, defined as ρ j = ψ q,K,M − k p k /q |n j |ψ q,K,M − k p k /q − ψ q,0,M |n j |ψ q,0,M , gives the expectation value of the number of particles on the site, when there are anyons in the system, relative to the same quantity, when there are no anyons in the system. Note that the number of particles in the two states is chosen such that the magnetic flux −η is the same for the two states. Therefore, we have a system with screened anyons, if ρ j is only different from zero in a small region around each anyon. The charge of the kth anyon is defined as Q k = − j∈R k ρ j , where R k is a small region around the anyon, which is large enough to enclose the anyon, but small enough to not enclose other anyons. We compute ρ j numerically using the Metropolis Monte Carlo algorithm [26]. The results for q = 2, M = 30, and 2 anyons in Fig. 2 show that the anyons are, indeed, screened and have the expected charge 0.5. The property that determines the relevant length scale is the density of particles in the system, and we have chosen the number of particles such that the typical distance between two particles is large compared to the smallest lattice spacing and small compared to the complete fractal. We can also see this at the level of the anyons. The two generations shown in the figure have the same typical distance between the particles. The size of the anyons do not change significantly, when going from generation 4 to generation 5. At the same time, the anyons are small compared to the full lattice. This means that the anyons would not be affected significantly, if we replaced the finite generation fractal with an infinite generation fractal.
Computing ρ j for different positions of the anyons show that the size of the anyons depends on the local distribution of lattice sites around the anyons. It is reasonable that the screening is affected by, at which points close to the anyons, we allow particles to be present. In all cases, however, the anyons are screened and have the correct charge. We have done similar computations for q = 3, and the conclusions are the same.
To braid the anyons, we need to move the w k along a continuous path. We can do this by allowing the w k to be at any point in the two-dimensional plane. We note that even when w k is between the lattice sites, the anyon itself is still present only on the fractal lattice. This is so because all the particles forming the quantum state are only allowed to be on the fractal lattice. For Laughlin states in two dimensions, it is known analytically that the anyons have the correct braiding properties as long as the anyons are screened and have the correct charge [3,27]. The derivation of the braiding properties in [27] also holds for the fractal lattice. We hence conclude that the constructed states, indeed, have the desired topological properties.
Other dimensions.-We have now demonstrated that anyons and fractional quantum Hall physics can be realized for a particular fractal dimension. This raises the question whether the effects can be seen in any dimension between 1 and 2, or only for this special dimension. We therefore investigate a family of fractals that allows us to vary the dimension. We start from a square, and we go from one generation to the next by dividing each of the squares present into L × L squares of equal size and only keeping U of the squares in a particular pattern. The dimension of this fractal is ln(U )/ ln(L). We study  If we put the model on a one-dimensional line ( Fig.  5(a)), we find that the anyons are not screened. This is expected, given that this model is critical [25]. Interestingly, however, it is possible to have screened anyons and fractional quantum Hall physics also in one dimension ( Fig. 5(b)), if we choose the fractal as in Fig. 4. A similar computation for L = 5 and U = 4, with the 4 squares sitting next to each other and forming a square, also shows screening, so that fractional quantum Hall physics is also possible in dimension ln(4)/ ln (5). This again shows that the distribution of lattice points in the vicinity of the anyons is more important for the screening than the Hausdorff dimension of the space.
Exact Hamiltonian.-So far, we have shown that anyons can exist in fractal dimensions, and we have constructed fractional quantum Hall states on fractal lattices hosting anyons. As long as η < 1 + q/N + k p k /N , it is possible to use the conformal field theory properties of the states to construct a Hamiltonian, which has the state |ψ q,K,M , defined on a general lattice, as ground state [27]. This results in the Hamiltonian Here, b j is the operator that annihilates a hardcore boson (fermion) on site j, when q is even (odd), n j = b † j b j is the number operator, and We find numerically that the ground state of H is unique, when the number of particles in the system is fixed to M .
Discussion.-We have constructed a new type of fractional quantum Hall models that are defined on fractals, and we have shown that anyons can exist in dimensions between 1 and 2. We have also shown that fractional quantum Hall physics can appear in systems with Hausdorff dimension less than 1. These results are an important first step that opens up more interesting directions for further studies: Comparisons between fractional quantum Hall physics in continuous systems and in lattices have revealed important differences and new possibilities. The fractal lattices also provide new possibilities, and this strongly motivates a detailed investigation of the interplay between fractional quantum Hall physics and the structure of fractals. It seems particularly promising to study properties that have a strong dependence on the dimension of space, such as transport and entanglement.
The idea presented in this work of restricting the magnetic field to a fractal lattice may give some hints on how to construct fractional Chern insulator type Hamiltonians on fractal lattices. A natural choice would be a model with interactions and complex hopping terms that realize the magnetic field. Such models could pave the way for implementations in ultracold atoms in optical lattices in the future.
Tools to detect topological order in quantum systems often rely on defining the investigated models on closed surfaces. This is, however, not possible for fractals, and the work hence motivates the development of additional methods to test for topology.
Finally, as discussed above, the study of stronglycorrelated quantum systems on fractals constitute an interesting intermediate regime between one and two dimensions, where little is currently known.