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arXiv:1002.4099 · Physical Review A

Fractional quantum Hall states in the vicinity of Mott plateaus

R. O. Umucalılar, Erich J. Mueller

Published May 24, 2010• 28 citations
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Abstract

We perform variational Monte Carlo calculations to show that bosons in a rotating optical lattice will form analogs of fractional quantum Hall states when the tunneling is sufficiently weak compared to the interactions, and the deviation of density from an integer is commensurate with the effective magnetic field. We compare the energies of superfluid and correlated states to one another and to the energies found in full configuration-interaction calculations on small systems. We look at overlaps between our variational states and the exact ground state, characterizing the ways in which fractional quantum Hall effect correlations manifest themselves near the Mott insulating state. We explore the experimental signatures of these states.

DOI
10.1103/physreva.81.053628

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We perform variational Monte Carlo calculations to show that bosons in a rotating optical lattice will form analogs of fractional quantum Hall states when the tunneling is sufficiently weak compared to the interactions, and the deviation of density from an integer is commensurate with the effective magnetic field. We compare the energies of superfluid and correlated states to one another and to the energies found in full configuration-interaction calculations on small systems. We look at overlaps between our variational states and the exact ground state, characterizing the ways in which fractional quantum Hall effect correlations manifest themselves near the Mott insulating state. We explore the experimental signatures of these states.
In this paper, the Bose-Hubbard model (BHM) with the nearest-neighbor (NN) repulsions is studied from the viewpoint of possible bosonic analogs of the fractional quantum Hall (FQH) state in the vicinity of the Mott insulator (MI). First, by means of the Gutzwiller approximation, we obtain the phase diagram of the BHM in a magnetic field. Then, we introduce an effective Hamiltonian describing excess particles on a MI and calculate the vortex density, momentum distribution, and the energy gap. These calculations indicate that the vortex solid forms for small NN repulsions, but a homogeneous featureless ``Bose metal'' takes the place of it as the NN repulsion increases. We consider particular filling factors at which the bosonic FQH state is expected to form. Chern-Simons (CS) gauge theory to the excess particle is introduced, and a modified Gutzwiller wave function, which describes bosons with attached flux quanta, is introduced. The energy of the excess particles in the bosonic FQH state is calculated using that wave function, and it is compared with the energy of the vortex solid and Bose metal. We found that the energy of the bosonic FQH state is lower than that of the Bose metal and comparable with the vortex solid. Finally, we clarify the condition that the composite fermion appears by using CS theory on the lattice that we previously proposed for studying the electron FQH effect.
In this research, we study the bosonic fractional quantum Hall (FQH) states in a system of ultracold bosons in a two-dimensional optical lattice in the presence of a synthetic magnetic field, described by the bosonic Harper–Hofstadter Hamiltonian. We use the cluster Gutzwiller mean-field and exact diagonalization techniques in our work. We obtain incompressible states as ground states at various filling factors similar to those of the FQH states. We focus in particular on the ν = 1/2 FQH state, and it is characterized by the two-point correlation function and the many-body Chern number. We further investigate the effect of dipolar interaction on the ν = 1/2 FQH state. We find that the dipolar interaction stabilizes the FQH state against the competing superfluid state.
We analyze a recently proposed method to create fractional quantum Hall (FQH) states of atoms confined in optical lattices [A. S\o{}rensen et al., Phys. Rev. Lett. 94, 086803 (2005)]. Extending the previous work, we investigate conditions under which the FQH effect can be achieved for bosons on a lattice with an effective magnetic field and finite on-site interaction. Furthermore, we characterize the ground state in such systems by calculating Chern numbers which can provide direct signatures of topological order and explore regimes where the characterization in terms of wave-function overlap fails. We also discuss various issues which are relevant for the practical realization of such FQH states with ultracold atoms in an optical lattice, including the presence of a long-range dipole interaction which can improve the energy gap and stabilize the ground state. We also investigate a detection technique based on Bragg spectroscopy to probe these systems in an experimental realization.
We describe a method to create fractional quantum Hall states of atoms confined in optical lattices. We show that the dynamics of the atoms in the lattice is analogous to the motion of a charged particle in a magnetic field if an oscillating quadrupole potential is applied together with a periodic modulation of the tunneling between lattice sites. In a suitable parameter regime the ground state in the lattice is of the fractional quantum Hall type, and we show how these states can be reached by melting a Mott-insulator state in a superlattice potential. Finally, we discuss techniques to observe these strongly correlated states.
The Bose-Hubbard model is extended to include nearest- and far-neighbor interactions and is related to the fractional quantum Hall effect (FQHE). Both models may be studied in optical lattices with quantum gases. The ground state is calculated for the extended Bose-Hubbard model with strong repulsive interactions (weak hopping). Incompressible Mott insulator states are found at rational filling fractions compatible with the principal and secondary FQHE filling fractions of the lowest Landau levels observed experimentally. It is discussed to which extent these states at fractional filling survive or undergoes a Mott insulator transition to a superfluid as hopping terms are included.
We consider the Bose-Hubbard model in a two-dimensional rotating optical lattice and investigate the consequences of the effective magnetic field created by rotation. Using a Gutzwiller-type variational wave function, we find an analytical expression for the Mott insulator (MI)--superfluid (SF) transition boundary in terms of the maximum eigenvalue of the Hofstadter butterfly. The dependence of phase boundary on the effective magnetic field is complex, reflecting the self-similar properties of the single particle energy spectrum. Finally, we argue that fractional quantum Hall phases exist close to the MI-SF transition boundaries, including MI states with particle densities greater than one.