Abstract. We propose to use null vectors in conformal field theories to derive model Hamiltonians of quantum spin chains and corresponding ground state wave function(s). The approach is quite general, and we illustrate it by constructing a family of Hamiltonians whose ground states are the chiral correlators of the SU (2) k WZW model for integer values of the level k. The simplest example corresponds to k = 1 and is essentially a nonuniform generalization of the Haldane-Shastry model with long-range exchange couplings. At level k = 2, we analyze the model for N spin 1 fields. We find that the Renyi entropy and the two-point spin correlator show, respectively, logarithmic growth and algebraic decay. Furthermore, we use the null vectors to derive a set of algebraic, linear equations relating spin correlators within each model. At level k = 1, these equations allow us to compute the two-point spin correlators analytically for the finite chain uniform Haldane-Shastry model and to obtain numerical results for the nonuniform case and for higher-point spin correlators in a very simple way and without resorting to Monte Carlo techniques.

We propose a set of spin system wave functions that is very similar to lattice versions of the Laughlin states. The wave functions are conformal blocks of conformal field theories and for a filling factor of ν = 1/2 we provide a parent Hamiltonian, which is valid for any even number of spins and is at the same time a 2D generalization of the Haldane-Shastry model. We also demonstrate that the Kalmeyer-Laughlin state is reproduced as a particular case of this model. Finally, we discuss various properties of the spin states and point out several analogies to known results for the Laughlin states.

The fractional quantum Hall effect is one of the most striking phenomena in condensed matter physics. It is described by a simple Laughlin wavefunction and has been thoroughly studied both theoretically and experimentally. In lattice systems, however, much less is currently known, and only few models and mechanisms leading to it have been identified. Here we propose a new way of constructing lattice Hamiltonians with local interactions and fractional quantum Hall like ground states. In particular, we obtain a spin 1/2 model with a bosonic Laughlin-like ground state, displaying a variety of topological features. We also demonstrate how such a model naturally emerges out of a Fermi-Hubbard-like model at half filling, in which the kinetic energy part possesses bands with non-zero Chern number, and we show how this model can be implemented in an optical lattice setup with present or planned technologies.

We introduce a two-parameter family of strongly-correlated wave functions for bosons and fermions in lattices. One parameter, q, is connected to the filling fraction. The other one, η, allows us to interpolate between the lattice limit (η = 1) and the continuum limit (η → 0 + ) of families of states appearing in the context of the fractional quantum Hall effect or the Calogero-Sutherland model. We give evidence that the main physical properties along the interpolation remain the same. Finally, in the lattice limit, we derive parent Hamiltonians for those wave functions and in 1D, we determine part of the low energy spectrum.

We introduce a family of strongly-correlated spin wave functions on arbitrary spin-1 2 and spin-1 lattices in one and two dimensions. These states are lattice analogues of Moore-Read states of particles at filling fraction q 1 , which are non-Abelian fractional quantum Hall states in 2D. One parameter enables us to perform an interpolation between the continuum limit, where the states become continuum Moore-Read states of bosons (odd q) and fermions (even q), and the lattice limit. We show numerical evidence that the topological entanglement entropy stays the same along the interpolation for some of the states we introduce in 2D, which suggests that the topological properties of the lattice states are the same as in the continuum, while the 1D states are critical states. We then derive exact parent Hamiltonians for these states on lattices of arbitrary size. By deforming these parent Hamiltonians, we construct local Hamiltonians that stabilize some of the states we introduce in 1D and in 2D.