High-field fractional quantum Hall effect in optical lattices

Measuring Topological Entanglement Entropy Using Maxwell Relations

Topological entanglement entropy (TEE) is a key diagnostic of topological order, allowing one to detect the presence of Abelian or non-Abelian anyons. However, there are currently no experimentally feasible protocols to measure TEE in condensed matter systems. Here, we propose a scheme to measure the TEE of chiral topological phases, carrying protected edge states, based on a nontrivial connection with the thermodynamic entropy change occurring in a quantum point contact (QPC) as it pinches off the topological liquid into two. We show how this entropy change can be extracted using Maxwell relations from charge detection of a nearby quantum dot. We demonstrate this explicitly for the Abelian Laughlin states, using an exact solution of the sine-Gordon model describing the universal crossover in the QPC. Our approach might open a new thermodynamic detection scheme of topological states also with non-Abelian statistics.

Topological bands for ultracold atoms

There have been significant recent advances in realizing band structures with geometrical and topological features in experiments on cold atomic gases. This review summarizes these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. Descriptions are given of the different methods that have been used to generate these novel band structures for cold atoms and of the physical observables that have allowed their characterization. The focus is on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. Also described is how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, descriptions are given of the forms of interparticle interactions that emerge when atoms are subjected to these energy bands and of some of the many-body phases that may be sought in future experiments.

Observation of fractional Chern insulators in a van der Waals heterostructure

Topologically ordered phases are characterized by long-range quantum entanglement and fractional statistics rather than by symmetry breaking. First observed in a fractionally filled continuum Landau level, topological order has since been proposed to arise more generally at fractional fillings of topologically nontrivial Chern bands. Here we report the observation of gapped states at fractional fillings of Harper-Hofstadter bands arising from the interplay of a magnetic field and a superlattice potential in a bilayer graphene-hexagonal boron nitride heterostructure. We observed phases at fractional filling of bands with Chern indices [Formula: see text] Some of these phases, in [Formula: see text] and [Formula: see text] bands, are characterized by fractional Hall conductance-that is, they are known as fractional Chern insulators and constitute an example of topological order beyond Landau levels.

Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number

The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number C. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with |C|>1. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor ν, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors ν=r/(r|C|+1) for bosons, or ν=r/(2r|C|+1) for fermions. This series includes a bosonic integer quantum Hall state in |C|=2 bands. We construct specific cases where a single band of the Harper-Hofstadter model is occupied. For these cases, we provide numerical evidence that several states in this series are realized as incompressible quantum liquids for bosons with contact interactions.

Bosonic integer quantum Hall effect in optical flux lattices

In two dimensions strongly interacting bosons in a magnetic field can realize a bosonic integer quantum Hall state, the simplest two-dimensional example of a symmetry-protected topological phase. We propose a realistic implementation of this phase using an optical flux lattice. Through exact diagonalization calculations, we show that the system exhibits a clear bulk gap and the topological signature of the bosonic integer quantum Hall state. In particular, the calculation of the many-body Chern number leads to a quantized Hall conductance in agreement with the analytical predictions. We also study the stability of the phase with respect to some of the experimentally relevant parameters.

Topological growing of Laughlin states in synthetic gauge fields

We suggest a scheme for the preparation of highly correlated Laughlin states in the presence of synthetic gauge fields, realizing an analogue of the fractional quantum Hall effect in photonic or atomic systems of interacting bosons. It is based on the idea of growing such states by adding weakly interacting composite fermions along with magnetic flux quanta one by one. The topologically protected Thouless pump ("Laughlin's argument") is used to create two localized flux quanta and the resulting hole excitation is subsequently filled by a single boson, which, together with one of the flux quanta, forms a composite fermion. Using our protocol, filling 1/2 Laughlin states can be grown with particle number N increasing linearly in time and strongly suppressed number fluctuations. To demonstrate the feasibility of our scheme, we consider two-dimensional lattices subject to effective magnetic fields and strong on-site interactions. We present numerical simulations of small lattice systems and also discuss the influence of losses.

Topological edge states and fractional quantum Hall effect from umklapp scattering

We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. In electron gases with stripes, these gaps result in plateaus for the Hall conductivity exactly at the known fillings n/m (both positive integers and m odd) for the integer and fractional quantum Hall effect. For double strips, we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological effects predicted here could be tested directly in optical lattices.

Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices

We demonstrate the experimental implementation of an optical lattice that allows for the generation of large homogeneous and tunable artificial magnetic fields with ultracold atoms. Using laser-assisted tunneling in a tilted optical potential, we engineer spatially dependent complex tunneling amplitudes. Thereby, atoms hopping in the lattice accumulate a phase shift equivalent to the Aharonov-Bohm phase of charged particles in a magnetic field. We determine the local distribution of fluxes through the observation of cyclotron orbits of the atoms on lattice plaquettes, showing that the system is described by the Hofstadter model. Furthermore, we show that for two atomic spin states with opposite magnetic moments, our system naturally realizes the time-reversal-symmetric Hamiltonian underlying the quantum spin Hall effect; i.e., two different spin components experience opposite directions of the magnetic field.

Tunable band topology reflected by fractional quantum Hall States in two-dimensional lattices

Two-dimensional lattice models subjected to an external effective magnetic field can form nontrivial band topologies characterized by nonzero integer band Chern numbers. In this Letter, we investigate such a lattice model originating from the Hofstadter model and demonstrate that the band topology transitions can be realized by simply introducing tunable longer-range hopping. The rich phase diagram of band Chern numbers is obtained for the simple rational flux density and a classification of phases is presented. In the presence of interactions, the existence of fractional quantum Hall states in both |C| = 1 and |C| > 1 bands is confirmed, which can reflect the band topologies in different phases. In contrast, when our model reduces to a one-dimensional lattice, the ground states are crucially different from fractional quantum Hall states. Our results may provide insights into the study of new fractional quantum Hall states and experimental realizations of various topological phases in optical lattices.

Precise shaping of laser light by an acousto-optic deflector

We present a laser beam shaping method using acousto-optic deflection of light and discuss its application to dipole trapping of ultracold atoms. By driving the acousto-optic deflector with multiple frequencies, we generate an array of overlapping diffraction-limited beams that combine to form an arbitrary-shaped smooth and continuous trapping potential. Confinement of atoms in a flat-bottomed potential formed by a laser beam with uniform intensity over its central region confers numerous advantages over the harmonic confinement intrinsic to Gaussian beam dipole traps and many other trapping schemes. We demonstrate the versatility of this beam shaping method by generating potentials with large flat-topped regions as well as intensity patterns that compensate for residual external potentials to create a uniform background to which the trapping potential of experimental interest can be added.

Adiabatic continuation of fractional Chern insulators to fractional quantum Hall States

We show how the phases of interacting particles in topological flat bands, known as fractional Chern insulators, can be adiabatically connected to incompressible fractional quantum Hall liquids in the lowest Landau level of an externally applied magnetic field. Unlike previous evidence suggesting the similarity of these systems, our approach enables a formal proof of the equality of their topological orders, and furthermore this proof robustly extends to the thermodynamic limit. We achieve this result using the hybrid Wannier orbital basis proposed by Qi [Phys. Rev. Lett. 107, 126803 (2011)] in order to construct interpolation Hamiltonians that provide continuous deformations between the two models. We illustrate the validity of our approach for the ground state of bosons in the half filled Chern band of the Haldane model, showing that it is adiabatically connected to the ν=1/2 Laughlin state of bosons in the continuum fractional quantum Hall problem.

Fractional quantum Hall effect of lattice bosons near commensurate flux

We study interacting bosons on a lattice in a magnetic field. When the number of flux quanta per plaquette is close to a rational fraction, the low-energy physics is mapped to a multispecies continuum model: bosons in the lowest Landau level where each boson is given an internal degree of freedom, or pseudospin. We find that the interaction potential between the bosons involves terms that do not conserve pseudospin, corresponding to umklapp processes, which in some cases can also be seen as BCS-type pairing terms. We argue that in experimentally realistic regimes for bosonic atoms in optical lattices with synthetic magnetic fields, these terms are crucial for determining the nature of allowed ground states. In particular, we show numerically that certain paired wave functions related to the Moore-Read Pfaffian state are stabilized by these terms, whereas certain other wave functions can be destabilized when umklapp processes become strong.

Fractional quantum Hall states of photons in an array of dissipative coupled cavities

We report a theoretical study of the collective optical response of a two-dimensional array of nonlinear cavities in the impenetrable photon regime under a strong artificial magnetic field. Taking advantage of the nonequilibrium nature of the photon gas, we propose an experimentally viable all-optical scheme to generate and detect strongly correlated photon states which are optical analogs of the Laughlin states of fractional quantum Hall physics.

Correlated phases of bosons in the flat lowest band of the dice lattice

We study correlated phases occurring in the flat lowest band of the dice-lattice model at flux density one-half. We discuss how to realize this model, also referred to as the T(3) lattice, in cold atomic gases. We construct the projection of the model to the lowest dice band, which yields a Hubbard Hamiltonian with interaction-assisted hopping processes. We solve this model for bosons in two limits. In the limit of large density, we use Gross-Pitaevskii mean-field theory to reveal time-reversal symmetry breaking vortex lattice phases. At low density, we use exact diagonalization to identify three stable phases at fractional filling factors ν of the lowest band, including a classical crystal at ν = 1/3, a supersolid state at ν = 1/2, and a Mott insulator at ν = 1.

Two-dimensional imaging of gauge fields in optical lattices

We propose a scheme to generate an arbitrary Abelian vector potential for atoms trapped in a two-dimensional optical lattice. By making the optical lattice potential dependent on the atomic state, we transform the problem into that of a two-dimensional imaging. It is shown that an arbitrarily fine pattern of the gauge field in the lattice can be realized without need of diffraction-limited imaging.

Experimental realization of strong effective magnetic fields in an optical lattice

We use Raman-assisted tunneling in an optical superlattice to generate large tunable effective magnetic fields for ultracold atoms. When hopping in the lattice, the accumulated phase shift by an atom is equivalent to the Aharonov-Bohm phase of a charged particle exposed to a staggered magnetic field of large magnitude, on the order of 1 flux quantum per plaquette. We study the ground state of this system and observe that the frustration induced by the magnetic field can lead to a degenerate ground state for noninteracting particles. We provide a measurement of the local phase acquired from Raman-induced tunneling, demonstrating time-reversal symmetry breaking of the underlying Hamiltonian. Furthermore, the quantum cyclotron orbit of single atoms in the lattice exposed to the magnetic field is directly revealed.

Fractional quantum Hall effect of hard-core bosons in topological flat bands

Recent proposals of topological flat band models have provided a new route to realize the fractional quantum Hall effect without Landau levels. We study hard-core bosons with short-range interactions in two representative topological flat band models, one of which is the well-known Haldane model (but with different parameters). We demonstrate that fractional quantum Hall states emerge with signatures of an even number of quasidegenerate ground states on a torus and a robust spectrum gap separating these states from the higher energy spectrum. We also establish quantum phase diagrams for the filling factor 1/2 and illustrate quantum phase transitions to other competing symmetry-breaking phases.

Exact parent Hamiltonian for the quantum Hall states in a lattice

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.

Observation of vortex nucleation in a rotating two-dimensional lattice of Bose-Einstein condensates

We report the observation of vortex nucleation in a rotating optical lattice. A 87Rb Bose-Einstein condensate was loaded into a static two-dimensional lattice and the rotation frequency of the lattice was then increased from zero. We studied how vortex nucleation depended on optical lattice depth and rotation frequency. For deep lattices above the chemical potential of the condensate we observed a linear dependence of the number of vortices created with the rotation frequency, even below the thermodynamic critical frequency required for vortex nucleation. At these lattice depths the system formed an array of Josephson-coupled condensates. The effective magnetic field produced by rotation introduced characteristic relative phases between neighboring condensates, such that vortices were observed upon ramping down the lattice depth and recombining the condensates.

Composite fermion theory for bosonic quantum Hall states on lattices

We study the ground states of the Bose-Hubbard model in a uniform magnetic field, motivated by the physics of cold atomic gases on lattices at high vortex density. Mapping the bosons to composite fermions (CF) leads to the prediction of quantum Hall fluids that have no counterpart in the continuum. We construct trial states for these phases and test numerically the predictions of the CF model. We establish the existence of strongly correlated phases beyond those in the continuum limit and provide evidence for a wider scope of the composite fermion approach beyond its application to the lowest Landau level.

Realizing and detecting the quantum Hall effect without landau levels by using ultracold atoms

We design an ingenious scheme to realize Haldane's quantum Hall model without Landau levels by using ultracold atoms trapped in an optical lattice. Three standing-wave laser beams are used to construct a wanted honeycomb lattice, where different on site energies in two sublattices required in the model can be implemented through tuning the phase of one laser beam. The staggered magnetic field is generated from the light-induced Berry phase. Moreover, we establish a relation between the Hall conductivity and the atomic density, enabling us to detect the Chern number with the typical density-profile-measurement technique.

Dynamic optical lattices: two-dimensional rotating and accordion lattices for ultracold atoms

We demonstrate a novel experimental arrangement which can rotate a 2D optical lattice at frequencies up to several kilohertz. Ultracold atoms in such a rotating lattice can be used for the direct quantum simulation of strongly correlated systems under large effective magnetic fields, allowing investigation of phenomena such as the fractional quantum Hall effect. Our arrangement also allows the periodicity of a 2D optical lattice to be varied dynamically, producing a 2D accordion lattice.

Electron polarizability of crystalline solids in quantizing magnetic fields and topological gap numbers

A theory of the static electron polarizability of crystals whose energy spectrum is modified by quantizing magnetic fields is presented. The polarizability is strongly affected by nondissipative Hall currents induced by the presence of crossed electric and magnetic fields: these can even change its sign. Results are illustrated in detail for a two-dimensional square lattice. The polarizability and the Hall conductivity are, respectively, linked to the two topological quantum numbers entering the so-called Diophantine equation. These numbers could in principle be detected in actual experiments.

Staggered-vortex superfluid of ultracold bosons in an optical lattice

We show that the dynamics of cold bosonic atoms in a two-dimensional square optical lattice produced by a bichromatic light-shift potential is described by a Bose-Hubbard model with an additional effective staggered magnetic field. In addition to the known uniform superfluid and Mott insulating phases, the zero-temperature phase diagram exhibits a novel kind of finite-momentum superfluid phase, characterized by a quantized staggered rotational flux. An extension for fermionic atoms leads to an anisotropic Dirac spectrum, which is relevant to graphene and high-T(c) superconductors.

Trapped Fermi gases in rotating optical lattices: realization and detection of the topological hofstadter insulator

We consider a gas of noninteracting spinless fermions in a rotating optical lattice and calculate the density profile of the gas in an external confinement potential. The density profile exhibits distinct plateaus, which correspond to gaps in the single particle spectrum known as the Hofstadter butterfly. The plateaus result from insulating behavior whenever the Fermi energy lies within a gap. We discuss the necessary conditions to realize the Hofstadter insulator in a cold atom setup and show how the quantized Hall conductance can be measured from density profiles using the Streda formula.

Excitation of a d-Density wave in an optical lattice with driven tunneling

Quantum phases with unusual symmetries may play a key role in the understanding of solid state systems at low temperatures. We propose a realistic scenario, well in reach of present experimental techniques, which should permit us to produce a stationary quantum state with d x2-y2 symmetry in a two-dimensional bosonic optical square lattice. This state, characterized by alternating rotational flux in each plaquette, arises from driven tunneling implemented by a stimulated Raman scattering process. We discuss bosons in a square lattice; however, more complex systems involving other lattice geometries appear possible.