In recent years, there has been much interest in finding fractional quantum Hall models in lattice systems, both because it may lead to new ways to realize fractional quantum Hall physics, and because the lattice gives rise to new effects and possibilities. The lattice model could, e.g., be a spin model as shown on the right or particles on a lattice as shown below.
It has been demonstrated numerically that it is possible to obtain fractional quantum Hall models in lattice systems by constructing a lattice Hamiltonian having many of the same features as the Hamiltonian in the corresponding continuum systems. We have proposed a different route to obtain fractional quantum Hall models in lattice systems, in which the approach instead is to consider analytical wavefunctions having practically the same form as in the continuum and then derive Hamiltonians for which these lattice states are ground states. This includes, e.g., models with Laughlin and Moore-Read ground states. The analytical form of the wavefunctions allows us to compute interesting properties, including topological properties, of the states for system sizes much larger than what can be reached with exact diagonalization. The exact Hamiltonians of the models contain interactions between spins that are far apart, but in a number of cases we have demonstrated that the Hamiltonian can be transformed into a short-range Hamiltonian without significantly altering the ground state. We have used this observation to propose a scheme to implement a related model in ultracold atoms in optical lattices.
Anyons are quasiparticles with peculiar properties that can appear in topologically ordered quantum many-body systems. They can, for instance, have a charge that is only a fraction of the charge of an electron, and when two anyons are exchanged, the many-body wavefunction is transformed in a more complicated way than just multiplication by a plus or a minus sign. Anyons are hence neither bosons nor fermions.
We construct models with analytical ground states that can host anyons. By changing the interaction strengths in the Hamiltonian, it is possible to create different types of anyons at specified positions, move the anyons around in the system in a controlled way to do braiding operations, and finally fuse the anyons by bringing them together. The anyons can be studied in great detail using Monte Carlo simulations. We have, for instance, investigated the size and shape of different anyons, computed braiding properties, and studied the effective magnetic field that anyons experience in the systems. We have also found the interesting result that quasielectrons can be constructed in a much simpler way in lattice models than in the continuum.
A typical approach to investigate a quantum system is to first write down the Hamiltonian and then compute the ground state and the properties of the ground state. Interacting quantum systems are, however, difficult to analyze because the resources needed to solve the Schrödinger equation numerically generally grow exponentially with system size, and models that can be solved analytically are therefore particularly helpful to gain insight.
We may then ask how we can find Hamiltonians for which at least the ground state can be found analytically. A more effective approach to find exactly solvable models, however, is to start by writing down an interesting wavefunction and then find a Hamiltonian for which the wavefunction is the ground state.
We have demonstrated that conformal field theory can be used to obtain an interesting family of quantum many-body models (including those discussed above and different types of critical models in 1D) with analytical ground states and few-body Hamiltonians, and we have used Monte Carlo simulations to compute interesting properties of these models. We are also investigating how the physical states inherit properties from the underlying conformal field theory.
Fractional quantum Hall models on lattices typically require the number of particles per lattice site to have some fixed value. Since, however, it is possible to have fractional quantum Hall states both in the continuum and in lattices, it is natural to ask, whether we can interpolate between the two cases. To change a lattice model towards the continuum, we should add lattice sites while keeping the number of particles fixed as shown in the figure. For models constructed from conformal field theory, we have found out how to modify the definition of the wavefunctions to add more lattice sites, and we have also proposed a construction that allows the Hamiltonian to be found at lower lattice filling factors. The construction allows us in a direct way to study the effects of the lattice, and it makes comparison to other models easier.
Ultracold atoms in optical lattices provide a versatile setup to investigate the physics of quantum many-body systems. We are interested in describing dynamics of trapped ultracold atoms and the possibility of implementing topological models in such systems.