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

Phase- and temperature-driven chiral topological superfluids on a honeycomb lattice

The correlated spinful Haldane model exhibits rich topological phases consisting of chiral topological superfluids (TSFs) and topological spin density waves. However, most of previous studies mainly focus on the case with the fixed hopping phase or at zero temperature. In this paper, we study the attractive spinful Haldane model with arbitrary phase at finite temperature. The chiral TSFs with Chern numberC = 2 and 4 emerge driven by the phase and temperature. In particular, the temperature can drive aC = 2 topological superfluid from a trivial normal insulator phase at an appropriate interaction. The bulk topology of all TSFs is uncovered by the Wilson loop method, and confirmed by the responses of edge dislocations.

Measurement of Spin Chern Numbers in Quantum Simulated Topological Insulators

The topology of quantum systems has become a topic of great interest since the discovery of topological insulators. However, as a hallmark of the topological insulators, the spin Chern number has not yet been experimentally detected. The challenge to directly measure this topological invariant lies in the fact that this spin Chern number is defined based on artificially constructed wave functions. Here we experimentally mimic the celebrated Bernevig-Hughes-Zhang model with cold atoms, and then measure the spin Chern number with the linear response theory. We observe that, although the Chern number for each spin component is ill defined, the spin Chern number measured by their difference is still well defined when both energy and spin gaps are nonvanished.

Phase Diagram of the Attractive Kane-Mele-Hubbard Model at Half Filling

Motivated by recent developments in the experimental study of ultracold atoms in graphene-like honeycomb optical lattices, we investigate superconductivity of the attractive Kane-Mele-Habbard (KMH) model with the next-nearest-neighbor (NNN) hoping at half filling. The mean-field approximation is used to study the phase diagram which interpolates the trivial and the non-trivial topological states. It is shown that: (a) when the NNN hoping is taken into account, one has to introduce two mean-field gap equations for the two sublattices, instead of a single gap when the NNN hopping is neglected, and (b) in the non-trivial topological region the phase diagram with the NNN hopping is significantly different compared to the phase diagram calculated previously, but without the NNN term. We also discuss the superconducting instability of the attractive KMH model that is driven by condensation of Cooperons.

Topological phases and pumps in the Su-Schrieffer-Heeger model periodically modulated in time

By the Floquet theory, we transform the Su-Schrieffer-Heeger model with the periodically modulated nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions into an effective 2D model, which holds the total Chern number of  ±1 modulated by the parameter θ. Under a staggered electric potential, the topological phase diagrams of the effective 2D model are reshaped and similar to the well-known Haldane model. While under a staggered Zeeman field, the topological phase diagram has the same shape as the former case, but with different Chern numbers, such as the spin and valley Chern numbers. With the combination of the staggered Zeeman field and the electric field, the effective 2D model holds even richer topological phases. At last, we find some types of topological pump, which can generate the time-averaged current without any bias voltage. The current depends on their different Chern numbers. In other words, we can modulate the parameters to obtain various Chern numbers to control the topological pump.

Topological quantum phase transitions of Chern insulators in disk geometry

We investigate topological quantum phase transitions (TQPTs) of Chern insulators in two-dimensional honeycomb-lattice disk with six-fold rotational symmetry. By considering the nearest-neighbor, next-nearest-neighbor hopping parameters and the staggered-flux parameter of the Haldane model, we can obtain rich topological quantum phases. The trivial and non-trivial phases of the Haldane model in disk geometry can be distinguished based on chiral edge states, real-space particle densities and local density of states. We also explore the TQPTs of Chern insulators with an external potential which varies with the radius of the disk geometry. Some interesting topological phases with large Chern numbers can be observed when we consider long-distance hoppings. Furthermore, we use a machine learning algorithm as an effective way to automatically identify various topological phases and phase diagrams for the Haldane model in disk geometry.

Generalized Haldane models on laser-coupling optical lattices

We propose two generalized Haldane models on laser-coupling optical lattices. Laser-assisted nearest neighbour tunnelings generate artificial staggered magnetic flux, facilitating the realization of topological nontrivial band structures. As generalizations of Haldane model, these models support topological insulator and semimetal phases featuring high Chern numbers. We show simple rules for computing Chern numbers of our models and display the phase diagrams. Moreover, numerical calculations of energy spectra are in perfect agreement with our theoretical expectations. Our models may serve as two new family members for generalizing Haldane model on optical lattices.

Quantum anomalous Hall phase in a one-dimensional optical lattice

We propose to simulate and detect quantum anomalous Hall phase with ultracold atoms in a one-dimensional optical lattice, with the other synthetic dimension being realized by modulating spin-orbit coupling. We show that the system manifests a topologically nontrivial phase with two chiral edge states which can be readily detected in this synthetic two-dimensional system. Moreover, it is interesting that at the phase transition point there is a flat energy band and this system can also be in a topologically nontrivial phase with two Fermi zero modes existing at the boundaries by considering the synthetic dimension as a modulated parameter. We also show how to measure these topological phases experimentally in ultracold atoms. Another model with a random Rashba and Dresselhaus spin-orbit coupling strength is also found to exhibit topological nontrivial phase, and the impact of the disorder to the system is revealed.

Topological phases of the kicked Harper-Kitaev model with ultracold atoms

We propose using ultracold atoms trapped in a one-dimensional periodically driven optical lattice to realize the Harper-Kitaev model, where the on-site energies are periodically kicked. Such a system provides a natural platform to study both Chern insulators and Majorana fermions. Based on calculating the quasienergy spectra, we find that both Floquet Majorana modes and Hall chiral edge modes could appear at the sample boundary in the gaps between the quasienergy bands. We also study the competition of topological superconductor and Chern insulator states in the model. We calculate the [Formula: see text] index and Floquet Chern number to characterize the above two different topological states, including the topological phase transitions in the kicked Harper-Kitaev model with the increase in the strength of the kick.

Topological phases in two-dimensional materials: a review

Topological phases with insulating bulk and gapless surface or edge modes have attracted intensive attention because of their fundamental physics implications and potential applications in dissipationless electronics and spintronics. In this review, we mainly focus on recent progress in the engineering of topologically nontrivial phases (such as [Formula: see text] topological insulators, quantum anomalous Hall effects, quantum valley Hall effects etc) in two-dimensional systems, including quantum wells, atomic crystal layers of elements from group III to group VII, and the transition metal compounds.

Dynamical Generation of Topological Magnetic Lattices for Ultracold Atoms

We propose a scheme to dynamically synthesize a space-periodic effective magnetic field for neutral atoms by time-periodic magnetic field pulses. When atomic spin adiabatically follows the direction of the effective magnetic field, an adiabatic scalar potential together with a geometric vector potential emerges for the atomic center-of-mass motion, due to the Berry phase effect. While atoms hop between honeycomb lattice sites formed by the minima of the adiabatic potential, complex Peierls phase factors in the hopping coefficients are induced by the vector potential, and these phase factors facilitate a topological Chern insulator. With further tuning of external parameters, both a topological phase transition and topological flat bands can be achieved, highlighting realistic prospects for studying strongly correlated phenomena in this system. Our Letter presents an alternative pathway towards creating and manipulating topological states of ultracold atoms by magnetic fields.

Light-induced gauge fields for ultracold atoms

Gauge fields are central in our modern understanding of physics at all scales. At the highest energy scales known, the microscopic universe is governed by particles interacting with each other through the exchange of gauge bosons. At the largest length scales, our Universe is ruled by gravity, whose gauge structure suggests the existence of a particle-the graviton-that mediates the gravitational force. At the mesoscopic scale, solid-state systems are subjected to gauge fields of different nature: materials can be immersed in external electromagnetic fields, but they can also feature emerging gauge fields in their low-energy description. In this review, we focus on another kind of gauge field: those engineered in systems of ultracold neutral atoms. In these setups, atoms are suitably coupled to laser fields that generate effective gauge potentials in their description. Neutral atoms 'feeling' laser-induced gauge potentials can potentially mimic the behavior of an electron gas subjected to a magnetic field, but also, the interaction of elementary particles with non-Abelian gauge fields. Here, we review different realized and proposed techniques for creating gauge potentials-both Abelian and non-Abelian-in atomic systems and discuss their implication in the context of quantum simulation. While most of these setups concern the realization of background and classical gauge potentials, we conclude with more exotic proposals where these synthetic fields might be made dynamical, in view of simulating interacting gauge theories with cold atoms.

Mass and chirality inversion of a Dirac cone pair in Stückelberg interferometry

We show that a Stückelberg interferometer made of two massive Dirac cones can reveal information on band eigenstates such as the chirality and mass sign of the cones. For a given spectrum with two gapped cones, we propose several low-energy Hamiltonians differing by their eigenstates properties. The corresponding interband transition probability is affected by such differences in its interference fringes being shifted by a new phase of geometrical origin. This phase can be a useful bulk probe for topological band structures realized with artificial crystals.

Extracting the Chern number from the dynamics of a Fermi gas: implementing a quantum Hall bar for cold atoms

We propose a scheme to measure the quantized Hall conductivity of an ultracold Fermi gas initially prepared in a topological Chern insulating phase and driven by a constant force. We show that the time evolution of the center of mass, after releasing the cloud, provides a direct and clear signature of the topologically invariant Chern number. We discuss the validity of this scheme, highlighting the importance of driving the system with a sufficiently strong force to displace the cloud over measurable distances while avoiding band-mixing effects. The unusual shapes of the driven atomic cloud are qualitatively discussed in terms of a semiclassical approach.

Detecting topological phases in cold atoms

Chern insulators are band insulators which exhibit a gap in the bulk and gapless excitations in the edge. Detection of Chern insulators is a serious challenge in cold atoms since the Hall transport measurements are technically unrealistic for neutral atoms. By establishing a natural correspondence between the time-reversal invariant topological insulator and the quantum anomalous Hall system, we show for a class of Chern insulators that the topology can be determined by only measuring Bloch eigenstates at highly symmetric points of the Brillouin zone. Furthermore, we introduce two experimental schemes, including the spin-resolved Bloch oscillation, to carry out the measurement. These schemes are highly feasible under realistic experimental conditions. Our results may provide a powerful tool to detect topological phases in cold atoms.

Topological quantum phase transitions on the kagomé and square-octagon lattices

We study the topological quantum phase transitions of tight-binding electrons on two representative multi-band lattice models with superimposed staggered fluxes: the kagomé lattice and the square-octagon lattice. By considering the nearest-neighbor and next-nearest-neighbor hopping parameters (t1 and t2) and the staggered flux parameter φ, we obtain rich topological quantum phase transitions for both the lattices. The whole phase diagram for the kagomé lattice in the t2-φ parameter space is obtained. We also illustrate a series of topological quantum phase transitions as well as topological bands of high Chern numbers for the square-octagon lattice. Furthermore, interesting topological flat bands with high flatness ratios (e.g. of about 43) are found in the square-octagon lattice, especially when the next-next-nearest-neighbor hopping (t3) is included.

Fractional topological states of dipolar fermions in one-dimensional optical superlattices

We study the properties of dipolar fermions trapped in one-dimensional bichromatic optical lattices and show the existence of fractional topological states in the presence of strong dipole-dipole interactions. We find some interesting connections between fractional topological states in one-dimensional superlattices and the fractional quantum Hall states: (i) the one-dimensional fractional topological states for systems at filling factor ν=1/p have p-fold degeneracy, (ii) the quasihole excitations fulfill the same counting rule as that of fractional quantum Hall states, and (iii) the total Chern number of p-fold degenerate states is a nonzero integer. The existence of crystalline order in our system is also consistent with the thin-torus limit of the fractional quantum Hall state on a torus. The possible experimental realization in cold atomic systems offers a new platform for the study of fractional topological phases in one-dimensional superlattice systems.

Topological Bose-Mott insulators in a one-dimensional optical superlattice

We study topological properties of the Bose-Hubbard model with repulsive interactions in a one-dimensional optical superlattice. We find that the Mott insulator states of the single-component (two-component) Bose-Hubbard model under fractional fillings are topological insulators characterized by a nonzero charge (or spin) Chern number with nontrivial edge states. For ultracold atomic experiments, we show that the topological Chern number can be detected through measuring the density profiles of the bosonic atoms in a harmonic trap.

Fractional Chern insulators in topological flat bands with higher Chern number

Lattice models forming bands with higher Chern number offer an intriguing possibility for new phases of matter with no analogue in continuum Landau levels. Here, we establish the existence of a number of new bulk insulating states at fractional filling in flat bands with a Chern number C = N > 1, forming in a recently proposed pyrochlore model with strong spin-orbit coupling. In particular, we find compelling evidence for a series of stable states at ν = 1/(2N + 1) for fermions as well as bosonic states at ν = 1/(N + 1). By examining the topological ground state degeneracies and the excitation structure as well as the entanglement spectrum, we conclude that these states are Abelian. We also explicitly demonstrate that these states are nevertheless qualitatively different from conventional quantum Hall (multilayer) states due to the novel properties of the underlying band structure.

Topological quantum phase transitions and topological flat bands on the kagomé lattice

We investigate the topological properties of the tight-binding electrons on the two-dimensional kagomé lattice with two kinds of short-range hopping integral and two kinds of staggered magnetic flux. Considering the nearest-neighbor hopping (t(1)) with the staggered flux parameter φ(1) and the next nearest-neighbor hopping (t(2)) with the staggered flux parameter φ(2), we demonstrate a series of topological quantum phase transitions and find some topological bands with high Chern numbers, when tuning one parameter (t(2) or φ(2)) while the others are fixed. We have also found that, in some parameter regions, the system exhibits interesting topological flat bands with Chern number C =± 1 and a large gap above them, and the flatness ratio can reach a high value of about 170.

Edge states and topological phases in one-dimensional optical superlattices

We show that one-dimensional quasiperiodic optical lattice systems can exhibit edge states and topological phases which are generally believed to appear in two-dimensional systems. When the Fermi energy lies in gaps, the Fermi system on the optical superlattice is a topological insulator characterized by a nonzero topological invariant. The topological nature can be revealed by observing the density profile of a trapped fermion system, which displays plateaus with their positions uniquely determined by the ration of wavelengths of the bichromatic optical lattice. The butterflylike spectrum of the superlattice system can be also determined from the finite-temperature density profiles of the trapped fermion system. This finding opens an alternative avenue to study the topological phases and Hofstadter-like spectrum in one-dimensional optical lattices.

Tunable gauge potential for neutral and spinless particles in driven optical lattices

We present a universal method to create a tunable, artificial vector gauge potential for neutral particles trapped in an optical lattice. The necessary Peierls phase of the hopping parameters between neighboring lattice sites is generated by applying a suitable periodic inertial force such that the method does not rely on any internal structure of the particles. We experimentally demonstrate the realization of such artificial potentials, which generate ground-state superfluids at arbitrary nonzero quasimomentum. We furthermore investigate possible implementations of this scheme to create tunable magnetic fluxes, going towards model systems for strong-field physics.

Quantum Hall effect in kagomé lattices under staggered magnetic field

The interplay of staggered magnetic field (SMF) and uniform magnetic field (UMF) on the quantum Hall effect (QHE) in kagomé lattices is investigated in the weak UMF limit. The topological band gaps coming from SMF are robust against UMF although the extended bands split into a series of Landau levels. With SMF applied, in the unconventional QHE region, one plateau of Hall conductance becomes wider and the others are compressed. Meanwhile, one of the two series of integer Hall plateaus splits and the resulting two series of Hall plateaus still exhibit the integer behavior. The Hall conductance varies with SMF step by step with the step height being e(2)/h or 2e(2)/h according to the QHE being conventional or unconventional. In the transitional regions, redistribution of Chern numbers happens even in the weak UMF limit.

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.

Topological Fermi liquids from Coulomb interactions in the doped honeycomb lattice

We propose a simple method for obtaining time reversal symmetry (T) broken phases in simple lattice models based on enlarging the unit cell. As an example we study the honeycomb lattice with nearest neighbor hopping and a local nearest neighbor Coulomb interaction V. We show that when the unit cell is enlarged to host six atoms that permits Kekulé distortions, self-consistent currents spontaneously form creating nontrivial magnetic configurations with total zero flux at high electron densities. A very rich phase diagram is obtained within a variational mean field approach that includes metallic phases with broken time reversal symmetry (T). The predominant (T) breaking configuration is an anomalous Hall phase, a realization of a topological Fermi liquid.

Tight-binding electrons on triangular and kagomé lattices under staggered modulated magnetic fields: quantum Hall effects and Hofstadter butterflies

We consider the tight-binding models of electrons on a two-dimensional triangular lattice and kagomé lattice under staggered modulated magnetic fields. Such fields have two components: a uniform-flux part with strength φ, and a staggered-flux part with strength Δφ. Various properties of the Hall conductances and Hofstadter butterflies are studied. When φ is fixed, variation of Δφ leads to the quantum Hall transitions and Chern numbers of Landau subbands being redistributed between neighboring pairs. The energy spectra with nonzero Δφs have similar fractal structures but quite different energy gaps compared with the original Hofstadter butterflies of Δφ = 0. Moreover, the fan-like structure of Landau levels in the low magnetic field region is also modified appreciably by Δφ.

Simulating the Wess-Zumino supersymmetry model in optical lattices

We study a cold atom-molecule mixture in two-dimensional optical lattices. We show that, by fine-tuning the atomic and molecular interactions, the Wess-Zumino supersymmetry (SUSY) model in 2+1 dimensions emerges in the low-energy limit and can be simulated in such mixtures. At zero temperature, SUSY is not spontaneously broken, which implies identical relativistic dispersions of the atom and its superpartner, a bosonic diatom molecule. This defining signature of SUSY can be probed by single-particle spectroscopies. Thermal breaking of SUSY at a finite temperature is accompanied by a thermal Goldstone fermion, i.e., phonino excitation. This and other signatures of broken SUSY can also be probed experimentally.

Making massless Dirac fermions from a patterned two-dimensional electron gas

Analysis of the electronic structure of an ordinary two-dimensional electron gas (2DEG) under an appropriate external periodic potential of hexagonal symmetry reveals that massless Dirac fermions are generated near the corners of the supercell Brillouin zone. The required potential parameters are found to be achievable under or close to laboratory conditions. Moreover, the group velocity is tunable by changing either the effective mass of the 2DEG or the lattice parameter of the external potential, and it is insensitive to the potential amplitude. The finding should provide a new class of systems other than graphene for investigating and exploiting massless Dirac fermions using 2DEGs in semiconductors.