We propose a simple scheme for the realization of a topological quasienergy band structure with ultracold atoms in a periodically driven optical square lattice. It is based on a circular lattice shaking in the presence of a superlattice that lowers the energy on every other site. The topological band gap, which separates the two bands with Chern numbers $\ifmmode\pm\else\textpm\fi{}1$, is opened in a way characteristic of Floquet topological insulators, namely, by terms of the effective Hamiltonian that appear in subleading order of a high-frequency expansion. These terms correspond to processes where a particle tunnels several times during one driving period. The interplay of such processes with particle interactions also gives rise to new interaction terms of several distinct types. For bosonic atoms with on-site interactions, they include nearest-neighbor density-density interactions introduced at the cost of weakened on-site repulsion as well as density-assisted tunneling. Using exact diagonalization, we investigate the impact of the individual induced interaction terms on the stability of a bosonic fractional Chern insulator state at half filling of the lowest band.
We propose a simple scheme for the realization of a topological quasienergy band structure with ultracold atoms in a periodically driven optical square lattice. It is based on a circular lattice shaking in the presence of a superlattice that lowers the energy on every other site. The topological band gap, which separates the two bands with Chern numbers \ifmmode±\else\textpm\fi1, is opened in a way characteristic of Floquet topological insulators, namely, by terms of the effective Hamiltonian that appear in subleading order of a high-frequency expansion. These terms correspond to processes where a particle tunnels several times during one driving period. The interplay of such processes with particle interactions also gives rise to new interaction terms of several distinct types. For bosonic atoms with on-site interactions, they include nearest-neighbor density-density interactions introduced at the cost of weakened on-site repulsion as well as density-assisted tunneling. Using exact diagonalization, we investigate the impact of the individual induced interaction terms on the stability of a bosonic fractional Chern insulator state at half filling of the lowest band.
Fractional Chern insulators are the proposed phases of matter mimicking the physics of fractional quantum Hall states on a lattice without an overall magnetic field. The notion of Floquet fractional Chern insulators refers to the potential possibilities to generate the underlying topological band structure by means of Floquet engineering. In these schemes, a highly controllable and strongly interacting system is periodically driven by an external force at a frequency such that double tunneling events during one forcing period become important and contribute to shaping the required effective energy bands. We show that in the described circumstances it is necessary to take into account also third order processes combining two tunneling events with interactions. Referring to the obtained contributions as micromotion-induced interactions, we find that those interactions tend to have a negative impact on the stability of fractional Chern insulating phases and discuss implications for future experiments.
We show theoretically that periodically driven systems with short range Hubbard interactions offer a feasible platform to experimentally realize fractional Chern insulator states. We exemplify the procedure for both the driven honeycomb and the square lattice, where we derive the effective steady state band structure of the driven system by using the Floquet theory and subsequently study the interacting system with exact numerical diagonalization. The fractional Chern insulator state equivalent to the 1/3 Laughlin state appears at 7/12 total filling (1/6 filling of the upper band). The state also features spontaneous ferromagnetism and is thus an example of the spontaneous breaking of a continuous symmetry along with a topological phase transition. We discuss light-driven graphene and shaken optical lattices as possible experimental systems that can realize such a state.
This thesis reports on the realization of Floquet topological systems with ultracold atoms in an optical honeycomb lattice. Using periodically driven quantum systems, topological phases of matter can be simulated by an effective, static Hamiltonian related to the time-evolution at integer multiples of the driving period, which is known as Floquet engineering. However, periodic driving can also give rise to genuinely time-dependent settings without static counterparts. One example is the anomalous Floquet phase in two dimensions, in which all bulk bands have a Chern number of zero but nevertheless robust chiral edge modes appear, which would be precluded by the bulk-edge correspondence in a static system. In an optical honeycomb lattice, anomalous Floquet systems can be created by continuous, periodic modulation of the laser intensities. This driving scheme results in different topological regimes, three of which are investigated, including the anomalous Floquet phase and a Haldane-like phase. Periodically driven systems feature periodic quasienergies and can be characterized in terms of winding numbers, which count the number of chiral edge modes in each quasienergy gap. By interferometric measurements of the quasienergy gaps between the first two Floquet bands, the topological phase transitions, emerging as gap closings, are located. For periodically driven systems, a modified bulk-edge correspondence can be formulated. In particular, the change of the winding number at a phase transition is related to the sign change of the local Berry curvature. The Berry curvature at the gap closing points is probed by Hall deflection measurements to obtain the winding numbers in each of the topological regimes, revealing the existence of chiral edge modes also in a setting with smooth boundaries as used in the experiments. The measured quasienergy gaps and transverse deflections are quantitatively well described by a numerical calculation of the Floquet bandstructure that includes coupling to higher bands during the driving period. To derive the spectrum of the modulated lattice in a geometry with edges, a tight-binding description of the system is discussed. Circular phase modulation of the honeycomb lattice can also give rise to an anomalous Floquet system when inversion symmetry is broken. The topological phase diagrams and the experimental feasibility are compared for both modulation schemes. Due to the vanishing Chern numbers, the bulk states in the anomalous Floquet regime can be fully localized by a disorder potential. This could prevent heating in interacting, periodically driven systems, resulting in a many-body localized bulk coexisting with thermalizing edge states. The experimental realization of disorder and strong interactions, as well as independent probes of the edge states are investigated. Bloch band geometry can be extended to multiband systems, using Wilson lines. The possible symmetry protection of their eigenvalues is discussed, along with measurements in the optical honeycomb lattice. Moreover, the onset of heating in weakly-interacting, periodically driven systems, triggered by parametric instabilities, is studied experimentally in a one-dimensional lattice.
We study the many-body physics in twisted bilayer graphene coupled to periodic driving of a circularly polarized light when electron-electron interactions are taken into account. In the limit of high driving frequency \Omega Ω , we use Floquet theory to formulate the system by an effective static Hamiltonian truncated to the order of \Omega^{-2} Ω − 2 , which consists of a single-electron part and the screened Coulomb interaction. We numerically simulate this effective Hamiltonian by extensive exact diagonalization in the parameter space spanned by the twist angle and the driving strength. Remarkably, in a wide region of the parameter space, we identify Floquet fractional Chern insulator states in the partially filled Floquet valence bands. We characterize these topologically ordered states by ground-state degeneracy, spectral flow, and entanglement spectrum. In regions of the parameter space where fractional Chern insulator states are absent, we find topologically trivial charge density waves and band-dispersion-induced Fermi liquids which strongly compete with fractional Chern insulator states.
We investigate an interaction-driven transition between crystalline and liquid states of strongly correlated spinless fermions within topological flat bands at low density (with filling factors $\ensuremath{\nu}=\frac{1}{5}, \frac{1}{7}, \frac{1}{9}$). Using exact diagonalization for finite-size systems with periodic boundary conditions, we distinguish different phases, whose stability depends on the interaction range, controlled by the screening parameter of the Coulomb interaction. The crystalline phases are identified by a crystallization strength, calculated from the Fourier transforms of pair correlation density, while the fractional Chern insulator (FCI) phases are characterized using momentum counting rules, entanglement spectrum, and overlaps with corresponding fractional quantum Hall states. The type of the phase depends on a particular single-particle model and its topological properties. We show that for $\ensuremath{\nu}=\frac{1}{7}$ and 51 it is possible to tune between the Wigner crystal and fractional Chern insulator phase in the kagome lattice model with the band carrying the Chern number C=1. In contrast, in the C=2 models, the Wigner crystallization was absent at $\ensuremath{\nu}=\frac{1}{5}$, and appeared at $\ensuremath{\nu}=\frac{1}{9}$, suggesting that C=2 FCIs are more stable against the formation of crystalline order.
Complex quantum many-body systems are ubiquitous in nature, yet their behaviour often remains very challenging to predict with analytical or numerical calculations.In this thesis a different approach is presented, using ultracold fermionic 40 K atoms in a lattice with a flexible geometry, formed by interfering laser beams.This system constitutes a precisely tunable, yet very accessible complex system, which can be used to probe -or "quantum simulate" -phenomena predicted for condensed-matter systems.We demonstrate, theoretically and experimentally, how a periodically modulated system can be described by an effective Hamiltonian on longer time-scales.This approach based on Floquet theory applies even when driving the system far from equilibrium.Using oscillating magnetic field gradients, we create an effectively spin-dependent lattice, where the tunnelling rate for different internal atomic states can be freely tuned.The flexible geometry of the lattice is used to create a honeycomb lattice and study the linear band crossings -known as Dirac cones -characteristic of its band structure.We distort the lattice and use momentum-resolved interband transitions probe the deformation and movement of the Dirac cones and their eventual disappearance when moving towards a square or dimerised geometry.We show that circular modulation of a honeycomb lattice induces imaginary next-nearest neighbour tunnelling, which breaks time-reversal symmetry.In addition, a tunable site-offset is used to break inversion symmetry.We thereby realize a model proposed by F.D.M. Haldane in 1988 and study the predicted topological transition from a trivial to a Chern-insulating regime, characterised by a closing band gap.Furthermore, effects of the Berry curvature are observed in both regimes, using transverse drifts of a driven atomic cloud.Using a mixture of two internal states with tunable scattering between them, we study the Hubbard model on a variety of geometries and with a range of observables.The crossover between metal, Mott-insulator and band insulator in a two-and threedimensional Honeycomb lattice is probed by observing the double occupancy, compressibility and excitation spectrum.The addition of a site-offset leads to the appearance of a competing type of density order -an explicit charge-density wave -which is directly probed and leads to new types of excitations.At lower temperatures, magnetic correlations appear.By freezing out the dynamics of the system and adiabatically merging neighbouring sites,the probability to find singlet or triplet states on different patterns of bonds is directly probed.Using this method we study the dependence of magnetic correlations on geometry in seven different lattice structures, and crossovers between them.We find that lower connectivity quantum effects enhance short-range correlations and compare our results to sophisticated numerical approximations.Ramping the lattice from one geometry to another allows for probing the quantum dynamics of re-distributing magnetic correlations, from the sudden to the adiabatic regime.v