High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective

We derive a systematic high-frequency expansion for the effective Hamiltonian and the micromotion operator of periodically driven quantum systems. Our approach is based on the block diagonalization of the quasienergy operator in the extended Floquet Hilbert space by means of degenerate perturbation theory. The final results are equivalent to those obtained within a different approach [Phys.\ Rev.\ A {\bf 68}, 013820 (2003), Phys.\ Rev.\ X {\bf 4}, 031027 (2014)] and can also be related to the Floquet-Magnus expansion [J.\ Phys.\ A {\bf 34}, 3379 (2000)]. We discuss that the dependence on the driving phase, which plagues the latter, can lead to artifactual symmetry breaking. The high-frequency approach is illustrated using the example of a periodically driven Hubbard model. Moreover, we discuss the nature of the approximation and its limitations for systems of many interacting particles.Comment: 48 pages, 7 figure

Dimensional transitions in small Yukawa clusters

We provide the detailed analysis of structural transitions leading to the rapid changes in dimensionality of small Yukawa clusters. These transformations are induced by the variations in the shape of confinement as well as the screening strength. We show, that even in the most primitive systems composed of only a few strongly interacting particles, the order parameter exhibits a power-law behavior in the vicinity of the critical point of the continuous transition. The critical exponent \gamma=1/2 is found to be universal in all studied cases, which is consistent with the general theory of continuous phase transitions.Comment: 9 pages, 11 figures. Submitted to Phys. Rev.

Modified interactions in a Floquet topological system on a square lattice and their impact on a bosonic fractional Chern insulator state

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 $\pm 1$, is opened in a way characteristic to 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.Comment: 10 pages, 4 figures, submitted to Physical Review

The role of real-space micromotion for bosonic and fermionic Floquet fractional Chern insulators

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 bandstructure 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 of fractional Chern insulating phases and discuss implications for future experiments.Comment: 13 pages, 7 figure

Semi-synthetic zigzag optical lattice for ultracold bosons

We consider a one-dimensional "zigzag" lattice, pictured as a two-site wide single strip taken from a triangular lattice, affected by a tunable homogeneous magnetic flux piercing its triangular plaquettes. We focus on a semi-synthetic lattice produced by combining a one-dimensional spin-dependent lattice in the long direction with laser-induced transitions between atomic internal states that define the short synthetic dimension. In contrast to previous studies on semi-synthetic lattices, the atom-atom interactions are nonlocal in both lattice directions. We investigate the ground-state properties of the system for the case of strongly interacting bosons, and find that the interplay between the frustration induced by the magnetic field and the interactions gives rise to an exotic gapped phase at fractional filling factors corresponding to one particle per magnetic unit cell.Comment: 9 pages, 6 figures; v3: final version to appear in PR

Six-dimensional time-space crystalline structures

Time crystalline structures are characterized by regularity that single-particle or many-body systems manifest in the time domain, closely resembling the spatial regularity of ordinary space crystals. Here we show that time and space crystalline structures can be combined together and even six-dimensional time-space lattices can be realized. As an example, we demonstrate that such time-space crystalline structures can reveal the six-dimensional quantum Hall effect quantified by the third Chern number.Comment: 7 pages + extended version of supplemental materia

Butterfly-like spectra and collective modes of antidot superlattices in magnetic fields

We calculate the energy band structure for electrons in an external periodic potential combined with a perpendicular magnetic field. Electron-electron interactions are included within a Hartree approximation. The calculated energy spectra display a considerable degree of self-similarity, just as the ``Hofstadter butterfly.'' However, screening affects the butterfly, most importantly the bandwidths oscillate with magnetic field in a characteristic way. We also investigate the dynamic response of the electron system in the far-infrared (FIR) regime. Some of the peaks in the FIR absorption spectra can be interpreted mainly in semiclassical terms, while others originate from inter(sub)band transitions.Comment: 4 pages with 2 embeded eps figures. Uses revtex, multicol and graphicx styles. Accepted for publication in PRB Brief Report

Lateral Superlattices in Commensurate Magnetic Fields: Electronic Structure, Transport and Optical Properties

The present thesis is devoted to the theoretical study of two-dimensional electron systems moving in a modulated lateral periodic potential and a competing perpendicular magnetic field. The three introductory chapters offer a brief exposition of the basic issues in this area -- the description of the electron motion in a prependicular magnetic field, the group of magnetic translations and the commensurability phenomena -- at an elementary level. The second part of the thesis is a collection of original papers. A bandstructure calculation scheme based on the ray group of magnetic translation operators is introduced and used to investigate the electronic structure of modulated two-dimensional electron systems. In addition to the expected intricate internal structure of the energy subbands, the results reveal characteristic oscillations of the band-widths as a function of the magnetic field strength. The quantized Hall conductance in periodic systems is extracted from the bandstructure data. Its study predicts a possible rearrangement of the energy bands in superlattices composed of narrow and steep antidots. This rearrangement facilitates an easier access to the energy subgaps characterized by nontrivial, that is, different from 0 or 1, quantum Hall indices. The study of tunneling between two parallel layers of modulated two-dimensional electron systems has lead to the formulation of the methodological approach to the spectroscopic measurement of the electronic structure properties. The optical spectra of the considered systems are addressed both using a microscopic quantum mechanical approach based on the random-phase approximation and simplified semiclassical methods. A number of collective modes are identified and analyzed