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 instead derived various lattice fractional quantum Hall models, where the ground states are known analytically and have practically the same form as in the continuum. This includes, e.g., models with Laughlin and Moore-Read ground states. The analytical form of the wavefunction allows us to compute interesting properties, including topological properties, of the states for quite large system sizes. 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.
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 grow exponentially with system size, and models that can be solved analytically are therefore very 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) with analytical ground states and few-body Hamiltonians, and we have used Monte Carlo simulations to compute interesting properties of these models.
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.
Can we also express the excited states of the models analytically in terms of conformal field theory?
In conformal field theory one can construct a tower of descendant states from a primary state by applying modes of the current operator (shown in red in the figure) to the state. The ground state of the quantum model can be obtained from the primary state of the conformal field theory by applying a certain transformation (blue), and we can hence get a family of quantum states by applying the blue transformation to the descendants.
We have investigated this family of states for a model constructed from the SU(2)1 WZW conformal field theory, and we have found that the family span the complete Hilbert space. For the particular case of a uniform lattice in 1D, this model coincides with the Haldane-Shastry model, and all the excited states can be shown to be linear combinations of states with a fixed number k of current operators.