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.
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.
Extensive research interest has been attracted by the fact that recent experiments in cold atoms simulate the chiral edge states of (1 + 1)-dimensional quantum Hall systems with laser-assisted transitions between different atomic nuclear spin states in a one-dimensional optical lattice. Within such an optical lattice, we study the phase diagrams of ν = 1 / 3 filling hard-core fermions and ν = 1 / 2 hard-core bosons with internal spin states in the presence of dipole-dipolar interactions. Our results show that the interaction drives the fermionic (bosonic) systems from the normal (superfluid) phase into the fractional topological phase with possible crystalline orderings. The topological nature of the latter phase is identified by the numerical evaluation of the many-body Chern number. Remarkably, we find that a considerable spectrum gap stabilizes the fractional topological phase with a moderate interaction strength, while, for the normal/superfluid phase, the Thouless charge pumping fluctuates strongly for small spin systems but approaches its universal value 1/3 for fermionic and 1/2 for bosonic systems with large spins.
Strongly interacting topological matter exhibits fundamentally new phenomena with potential applications in quantum information technology. Emblematic instances are fractional quantum Hall states, where the interplay of magnetic fields and strong interactions gives rise to fractionally charged quasi-particles, long-ranged entanglement, and anyonic exchange statistics. Progress in engineering synthetic magnetic fields has raised the hope to create these exotic states in controlled quantum systems. However, except for a recent Laughlin state of light, preparing fractional quantum Hall states in engineered systems remains elusive. Here, we realize a fractional quantum Hall (FQH) state with ultracold atoms in an optical lattice. The state is a lattice version of a bosonic ν=1/2 Laughlin state with two particles on sixteen sites. This minimal system already captures many hallmark features of Laughlin-type FQH states: we observe a suppression of two-body interactions, we find a distinctive vortex structure in the density correlations, and we measure a fractional Hall conductivity of σH/σ0=0.6(2) via the bulk response to a magnetic perturbation. Furthermore, by tuning the magnetic field we map out the transition point between the normal and the FQH regime through a spectroscopic probe of the many-body gap. Our work provides a starting point for exploring highly entangled topological matter with ultracold atoms.
Quantum Hall (QH) states of two-dimensional (2D) single-layer optical lattices are examined using the Bose-Hubbard model (BHM) in the presence of an artificial gauge field. We study the QH states of both the homogeneous and inhomogeneous systems. For the homogeneous case, we use cluster Gutzwiller mean-field (CGMF) theory with cluster sizes ranging from $2\ifmmode\times\else\texttimes\fi{}2$ to $5\ifmmode\times\else\texttimes\fi{}5$. We then consider the inhomogeneous case, which is relevant to experimental realization. In this case, we use CGMF and exact diagonalization (ED). The ED studies are using lattice sizes ranging from $3\ifmmode\times\else\texttimes\fi{}3$ to $4\ifmmode\times\else\texttimes\fi{}12$. Our results show that the geometries of the QH states are sensitive to the magnetic flux $\ensuremath{\alpha}$ and cluster sizes. For homogeneous systems, among various combinations of $1/5\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1/2$ and filling factor $\ensuremath{\nu}$, only the QH state of $\ensuremath{\alpha}=1/4$ with $\ensuremath{\nu}=1/2,\phantom{\rule{0.16em}{0ex}}1,\phantom{\rule{0.16em}{0ex}}3/2$, and 2 occur as ground states. For other combinations, the competing superfluid (SF) state is the ground state and QH state is metastable. For BHM with envelope potential, all the QH states observed in homogeneous systems exist for box potentials, but none exist for the harmonic potential. The QH states also persist for very shallow Gaussian envelope potential. As a possible experimental signature, we study the two-point correlations of the QH and SF states.
We study fermions and hard-core bosons with long-range dipolar interactions at fractional fillings in a topological checkerboard lattice with short-range hoppings up to next-next-nearest neighbors [T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys. Rev. Lett. 106, 236804 (2011)]. We consider the case that the dipoles are aligned in the perpendicular direction by an external field without the complication of anisotropic interaction. Using exact diagonalization, we find clear signatures of fractional quantum Hall (FQH) states at filling factors 1/3 and 1/5 for fermions (1/2 and 1/4 for bosons) in the lowest Chern band with a robust spectrum gap at moderate dipolar interaction strength. The robustness of these FQH states against a long-range interaction tail and band flatness is investigated. When the dipolar interaction decreases, the fermionic FQH states turn into normal states, and the bosonic 1/4 FQH state turns into a superfluid state. The bosonic 1/2 FQH state survives even in the absence of the dipolar interaction, but vanishes when the hard core becomes a soft core with a critical on-site repulsion. In the thin torus limit, the static density structure factors indicate that the FQH state turns into a commensurate charge density wave state.
We show via tensor network methods that the Harper-Hofstadter Hamiltonian for hard-core bosons on a square geometry supports a topological phase realizing the $\ensuremath{\nu}=\frac{1}{2}$ fractional quantum Hall (FQH) effect on the lattice. We address the robustness of the ground-state degeneracy and of the energy gap, measure the many-body Chern number, and characterize the system using Green functions, showing that they decay algebraically at the edges of open geometries, indicating the presence of gapless edge modes. Moreover, we estimate the topological entanglement entropy by taking a combination of lattice bipartitions that reproduces the topological structure of the original proposals by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and Levin and Wen [Phys. Rev. Lett. 96, 110405 (2006)]. The numerical results show that the topological contribution is compatible with the expected value $\ensuremath{\gamma}=\frac{1}{2}$. Our results provide extensive evidence that FQH states are within reach of state-of-the-art cold-atom experiments.