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arXiv:2012.01822 · Physical Review A

Intertwined space-time symmetry, orbital magnetism, and dynamical Berry connection in a circularly shaken optical lattice

Hua Chen, W. Vincent Liu

Published Jul 9, 2021• 5 citations
arXivHTMLPDF

Abstract

We study the circular shaking of a two-dimensional optical lattice, which is essentially a $(2+1)$-dimensional space-time lattice exhibiting periodicities in both spatial and temporal dimensions. The near-resonant optical shaking considered here dynamically couples the low-lying $s$ band and the first excited $p$ bands by transferring a photon of shaking frequency. The intertwined space-time symmetries are further uncovered to elucidate the degeneracy in the spectrum solved with the generalized Bloch-Floquet theorem. Setting the chirality of circular shaking explicitly breaks time-reversal symmetry and lifts the degeneracy of ${p}_{\ifmmode\pm\else\textpm\fi{}}={p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ orbitals, leading to the local circulation of orbital magnetism, i.e., the imbalanced occupation in ${p}_{\ifmmode\pm\else\textpm\fi{}}$ orbitals. Moreover, the dynamics of Berry connection is revealed by the time evolution of the Berry curvature and the polarization, which have physical observable effects in experiments. Interestingly, the dynamics is found characterized by a universal phase shift, governed by the time screw rotational symmetry involving a fractional translation of time. These findings suggest that the present lattice-shaking scheme provides a versatile platform for the investigation of the orbital physics and the symmetry-protected dynamics.

DOI
10.1103/physreva.104.013308

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We study the circular shaking of a two-dimensional optical lattice, which is essentially a (2+1)(2+1)(2+1)-dimensional space-time lattice exhibiting periodicities in both spatial and temporal dimensions. The near-resonant optical shaking considered here dynamically couples the low-lying sss band and the first excited ppp bands by transferring a photon of shaking frequency. The intertwined space-time symmetries are further uncovered to elucidate the degeneracy in the spectrum solved with the generalized Bloch-Floquet theorem. Setting the chirality of circular shaking explicitly breaks time-reversal symmetry and lifts the degeneracy of p\ifmmode±\else\textpm\fi=px\ifmmode±\else\textpm\fiipy{p}_{\ifmmode\pm\else\textpm\fi{}}={p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}p\ifmmode±\else\textpm\fi​=px​\ifmmode±\else\textpm\fiipy​ orbitals, leading to the local circulation of orbital magnetism, i.e., the imbalanced occupation in p\ifmmode±\else\textpm\fi{p}_{\ifmmode\pm\else\textpm\fi{}}p\ifmmode±\else\textpm\fi​ orbitals. Moreover, the dynamics of Berry connection is revealed by the time evolution of the Berry curvature and the polarization, which have physical observable effects in experiments. Interestingly, the dynamics is found characterized by a universal phase shift, governed by the time screw rotational symmetry involving a fractional translation of time. These findings suggest that the present lattice-shaking scheme provides a versatile platform for the investigation of the orbital physics and the symmetry-protected dynamics.
We investigate the topological properties of a resonantly shaken one-dimensional optical lattice system, where the lattice position is periodically driven with two harmonic frequencies to generate one- and two-photon couplings between the two lowest orbitals. In a two-band approximation, we numerically show that degenerate edge states appear under a certain driving condition and that the corresponding topological phase is protected by the chiral symmetry of the periodically driven system. The system's micromotion is characterized with oscillating Zak phases and we find that the Zak phases are quantized only at the time when the chiral symmetry condition is explicitly satisfied. Finally, we describe the topological charge pumping effect which arises when the driving parameters are slowly modulated around a critical point, and investigate its adiabaticity with increasing modulation frequency.
We develop a theory of weakly interacting fermionic atoms in shaken optical lattices based on the orbital mixing in the presence of time-periodic modulations. Specifically, we focus on fermionic atoms in a circularly shaken square lattice with near-resonance frequencies, i.e., tuned close to the energy separation between the sss band and the ppp bands. First, we derive a time-independent four-band effective Hamiltonian in the noninteracting limit. Diagonalization of the effective Hamiltonian yields a quasienergy spectrum consistent with the full numerical Floquet solution that includes all higher bands. In particular, we find that the hybridized sss band develops multiple minima and therefore nontrivial Fermi surfaces at different fillings. We then obtain the effective interactions for atoms in the hybridized sss band analytically and show that they acquire momentum dependence on the Fermi surface even though the bare interaction is contactlike. We apply the theory to find the phase diagram of fermions with weak attractive interactions and demonstrate that the pairing symmetry is s+ds+ds+d wave. Our theory is valid for a range of shaking frequencies near resonance, and it can be generalized to other phases of interacting fermions in shaken lattices.
The geometry of Bloch bands affects many physical properties of crystalline solids and other spatially periodic systems. Direct experimental determination of such geometry is an active area of research. In this work, we focus on the fundamental connection between optical excitations and the relative geometry of pairs of Bloch bands, as characterized by the interband Berry connection. We simulate the response of electrons in solids to optical excitation by the response of ultracold fermionic atoms in optical lattices to periodic modulation of the lattice position. The strength of resonant excitation between bands, measured at each quasimomentum and for various lattice-shaking polarizations, directly maps out the interband Berry connection. We apply this method to the optical honeycomb lattice, driving excitations between the ground n=1n=1n=1 band and the excited n′={2,3,4}n'=\{2,3,4\}n′={2,3,4} bands. We observe transparency lines of quasimomenta at which the response to excitation of specific polarization is zero. Further, the interband Berry connection between bands 1 and 3 shows irreducible Dirac strings connecting the KKK and K′K'K′ points in the Brillouin zone, lines along which the interband Berry connection abruptly changes orientation. Our work establishes optical response as a powerful tool for characterizing geometrical and topological properties of band structure.
As the research of topological semimetals develops, semimetals with nodal-line rings have come into people's vision as a new platform for studying electronic topology. We propose a method that uses ultracold atoms in a two-dimensional amplitude-shaken bipartite hexagonal optical lattice to simulate the nodal-line semimetal, which can be achieved in the experiment by attaching one triangular optical lattice to a hexagonal optical lattice and periodically modulating the intensity and position of the triangular lattice. By adjusting the shaking frequency and well depth difference of the hexagonal optical lattice, a transformation from Dirac semimetal to the nodal-line semimetal is observed in our system. This transformation can be demonstrated by the Berry phase and Berry curvature, which guides the measurement. Furthermore, the amplitude shaking and C3{C}_{3}C3​ symmetry generate a time-reversal-symmetry-unstable mode, and the proportion of the mode and the trivial mode of the hexagonal lattice controls the transformation. This proposal paves a way to study nodal-line semimetals in two-dimensional systems, and it contributes to the application of nodal-line semimetals.
Chiral magnetism is a fascinating quantum phenomena that has been found in low-dimensional magnetic materials. It is not only interesting for understanding the concept of chirality, but also important for potential applications in spintronics. Past studies show that chiral magnets require both a lack of inversion symmetry and spin-orbit coupling to induce the Dzyaloshinskii-Moriya interaction. Here we report that the combination of inversion symmetry breaking and quantum degeneracy of orbital degrees of freedom will provide a new paradigm to achieve chiral orbital magnetism. By means of density matrix renormalization group calculation, we demonstrate that chiral orbital magnetism can be found when considering bosonic atoms loaded in the p band of an optical lattice in the Mott regime. The high tunability of our scheme is also illustrated through simply manipulating the inversion symmetry of the system for cold atom experimental conditions.
The realization of band structures with nontrivial topological properties in an optical lattice is an exciting topic in current studies of ultracold atoms. Here we point out that this lofty goal can be achieved by using a simple scheme of shaking an optical lattice, which is directly applicable in current experiments. The photon-assisted band hybridization leads to the production of an effective spin-orbit coupling, in which the band index represents the pseudospin. When this spin-orbit coupling has finite strengths in multiple directions, nontrivial topological structures emerge in the Brillouin zone, such as topological defects with a winding number of 1 or 2 in a shaken square lattice. The shaken lattice also allows one to study the transition between two band structures with distinct topological properties.