Conformal field theory has turned out to be a powerful tool to derive interesting lattice models with analytical ground states. Here, we investigate a class of critical, one-dimensional lattice models of fermions and hardcore bosons related to the Laughlin states. The Hamiltonians of the exact models involve interactions over long distances that are difficult to realize experimentally. This motivates us to study the properties of models with the same type of interactions, but now only between nearest and possibly next-nearest neighbor sites. Based on computations of wavefunction overlaps, entanglement entropies, and two-site correlation functions for systems of up to 32 sites, we find that the ground state is close to the ground state of the exact model. There is also a high overlap per site between the lowest excited states for the local and the exact models, although the energies of the low-lying excited states are modified to some extent for the system sizes considered. We also briefly discuss possibilities for realizing the local models in ultracold atoms in optical lattices.
Conformal field theory has turned out to be a powerful tool to derive interesting lattice models with analytical ground states. Here, we investigate a class of critical, one-dimensional lattice models of fermions and hardcore bosons related to the Laughlin states. The Hamiltonians of the exact models involve interactions over long distances that are difficult to realize experimentally. This motivates us to study the properties of models with the same type of interactions, but now only between nearest and possibly next-nearest neighbor sites. Based on computations of wavefunction overlaps, entanglement entropies, and two-site correlation functions for systems of up to 32 sites, we find that the ground state is close to the ground state of the exact model. There is also a high overlap per site between the lowest excited states for the local and the exact models, although the energies of the low-lying excited states are modified to some extent for the system sizes considered. We also briefly discuss possibilities for realizing the local models in ultracold atoms in optical lattices.
Abstract Conformal field theory has turned out to be a powerful tool to derive interesting lattice models with analytical ground states. Here, we investigate a class of critical, one-dimensional lattice models of fermions and hardcore bosons related to the Laughlin states. The Hamiltonians of the exact models involve interactions over long distances that are difficult to realize experimentally. This motivates us to study the properties of models with the same type of interactions, but now only between nearest and possibly next-nearest neighbor sites. Based on computations of wavefunction overlaps, entanglement entropies, and two-site correlation functions for systems of up to 32 sites, we find that the ground state is close to the ground state of the exact model. There is also a high overlap per site between the lowest excited states for the local and the exact models, although the energies of the low-lying excited states are modified to some extent for the system sizes considered. We also briefly discuss possibilities for realizing the local models in ultracold atoms in optical lattices.
Conformal field theory has recently been applied to derive few-body Hamiltonians whose ground states are lattice versions of fractional quantum Hall states. The exact lattice models involve interactions over long distances, which is difficult to realize in experiments. It seems, however, that such long-range interactions should not be necessary, as the correlations decay exponentially in the bulk. This poses the question, whether the Hamiltonians can be truncated to contain only local interactions without changing the physics of the ground state. Previous studies have, in a couple of cases with particularly much symmetry, obtained such local Hamiltonians by keeping only a few local terms and numerically optimizing the coefficients. Here, we investigate a different strategy to construct truncated Hamiltonians, which does not rely on optimization, and which can be applied independently of the choice of lattice. We test the approach on two models with bosonic Laughlin-like ground states with filling factor 1/2 and 1/4, respectively. We first investigate how the coupling strengths in the exact Hamiltonians depend on distance, and then we study the truncated models. For the case of 1/2 filling, we find that the truncated model with truncation radius 2 lattice constants on the square lattice and 1 lattice constant on the triangular lattice has an approximate twofold ground-state degeneracy on the torus, and the overlap per site between these states and the states constructed from conformal field theory is higher than 0.99 for the lattices considered. For the model at 1/4 filling, our results give some hints that a truncation radius of 5 on the square lattice and 7 on the triangular lattice might be enough, but the finite-size effects are too large to judge whether the topology is, indeed, present in the thermodynamic limit. The states with high overlap also have the expected topological entanglement entropies.
We introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors $\ensuremath{\nu}=1/q$) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the Z2 properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflect gapless edge states as was discussed by Li and Haldane.
Recently, it was shown that fractional quantum Hall states can be defined on fractal lattices. Proposed exact parent Hamiltonians for these states are nonlocal and contain three-site terms. In this work, we look for simpler, approximate parent Hamiltonians for bosonic Laughlin states at half filling, which contain only onsite potentials and two-site hopping with the interaction generated implicitly by hardcore constraints (as in the Hofstadter and Kapit-Mueller models on periodic lattices). We use an ``inverse method'' to determine such Hamiltonians on finite-generation Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet and triangle lattices. The ground states of some of the resulting models display relatively high overlap with the model states if up to third-neighbor hopping terms are considered, and by increasing the maximum hopping distance one can achieve nearly perfect overlaps. When the number of particles is reduced and additional potentials are introduced to trap quasiholes, the overlap with a model quasihole wave function is also high in some cases, especially for the nonlocal Hamiltonians. We also study how the small system size affects the braiding properties for the model quasihole wave functions and perform analogous computations for Hamiltonian models.
We study lattice wave functions obtained from the SU(2)1 Wess-Zumino-Witten conformal field theory. Following Moore and Read's construction, the Kalmeyer-Laughlin fractional quantum Hall state is defined as a correlation function of primary fields. By an additional insertion of Kac-Moody currents, we associate a wave function with each state of the conformal field theory. These wave functions span the complete Hilbert space of the lattice system. On the cylinder, we study global properties of the lattice states analytically and correlation functions numerically using a Metropolis Monte Carlo method. By comparing short-range bulk correlations, numerical evidence is provided that the states with one current operator represent edge states in the thermodynamic limit. We show that the edge states with one Kac-Moody current of lowest order have a good overlap with low-energy excited states of a local Hamiltonian, for which the Kalmeyer-Laughlin state approximates the ground state. For some states, exact parent Hamiltonians are derived on the cylinder. These Hamiltonians are SU(2) invariant and nonlocal with up to four-body interactions.
We study lattice models of charged particles in uniform magnetic fields. We show how longer range hopping can be engineered to produce a massively degenerate manifold of single-particle ground states with wave functions identical to those making up the lowest Landau level of continuum electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, Laughlin's fractional quantum Hall wave function is an exact many-body ground state of our lattice model. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next-nearest neighbor hoppings. We suggest how to realize this model using atoms in optical lattices, and describe observable consequences of the resulting fractional quantum Hall physics.