Non-Abelian Gauge Fields and Topological Insulators in Shaken Optical Lattices

Time-periodic driving like lattice shaking offers a low-demanding method to generate artificial gauge fields in optical lattices. We identify the relevant symmetries that have to be broken by the driving function for that purpose and demonstrate the power of this method by making concrete proposals for its application to two-dimensional lattice systems: We show how to tune frustration and how to create and control band touching points like Dirac cones in the shaken kagome lattice. We propose the realization of a topological and a quantum spin Hall insulator in a shaken spin-dependent hexagonal lattice. We describe how strong artificial magnetic fields can be achieved for example in a square lattice by employing superlattice modulation. Finally, exemplified on a shaken spin-dependent square lattice, we develop a method to create strong non-Abelian gauge fields.

Figure 1

(a)–(c) Lattice geometries involving triangular plaquettes pierced by an artificial magnetic flux ${\varphi }_{\nabla ,\Delta }=\pm \varphi$ (indicated by $+$ and $-$): (a) triangular lattice, (b) kagome lattice with tunneling $J_1$ ($J_2$) along the darker horizontal (lighter diagonal) bonds, and (c) hexagonal lattice with nearest-neighbor ACT (solid lines) between shallow $A$ and deep $B$ sites and next-nearest-neighbor tunneling between $A$ sites (dashed lines). (d) Driving function $F(t)$ breaking symmetries ($r$) and ($s$).

Figure 2

(a) The topology of band touching for the kagome lattice can be controlled by anisotropy $J_1/J_2$ and plaquette flux $\varphi$. The way and how often the three bands touch is depicted by the iconographic symbols. (b) Phase diagram of the hexagonal lattice as in Fig. 1c with bare/undriven (next) nearest neighbor tunneling matrix elements $J$ ($J^{\prime }$), subjected to a symmetry-breaking force $F(t)$ of amplitude $\alpha$ and direction ${\mathit{e}}_F=\cos ({\phi }_F){\mathit{e}}_x+\sin ({\phi }_F){\mathit{e}}_y$. White, no Dirac points are present; gray, a small nearest-neighbor tunneling $<0.02J$ renders the physics effectively 1D. The color bar encodes the masses at two Dirac points, labeled as $|m_{<}|\le |m_{>}|$. In the diagonally (horizontally) hatched region both masses are positive (negative). When the masses have opposite sign (unhatched), the system is a topological insulator (or a quantum spin Hall insulator for two spin states). Inset: Position of Dirac points in $k$ space for ${\phi }_F=\pi /30$, indicating how they move and merge with $\alpha$.

Figure 3

Artificial magnetic fluxes $\varphi$ through the plaquettes of a square lattice (lattice constant $d$, indicated by the grid) resulting from superlattice modulation. Stripes or larger patches with strong, rectified magnetic fluxes can be achieved. (a),(b) Single-component superlattices with (a) ${\mathit{q}}_1=(\pi /d)({\mathit{e}}_x+{\mathit{e}}_y/2)$, $V_1=4\hslash \omega$; (b) ${\mathit{q}}_1=\frac{1}{10}(\pi /d)({\mathit{e}}_x+{\mathit{e}}_y/2)$ and $V_1=20\hslash \omega$. (c) Two components with ${\mathit{q}}_{1/2}=\frac{1}{10}(\pi /d)({\mathit{e}}_x\pm {\mathit{e}}_y)$, $V_{1/2}=12\hslash \omega$. (d) Like (c), but with wavelengths and amplitudes doubled. Always ${\phi }_s=0$.

Figure 4

Non-Abelian SU(2) gauge fields. (a) Two standing laser waves (with a phase shift of $\pi /2$ and in-plane polarization as denoted in the figure) create a bipartite square lattice with alternating ${\sigma }^{+}$ and ${\sigma }^{-}$ polarized sites ($A$ and $B$) [43]. $m_F=\pm 1$ particles feel an energy difference of $\pm \Delta E$ between $A$ and $B$ sites. (b) The resulting level scheme. A constant $B$ field realizes an additional on-site energy splitting $\Delta E^{\prime }$ (green arrow) such that $|\Delta E_{A,B}|=|\pm \Delta E+\Delta E^{\prime }|$ becomes sublattice dependent. The coupling $\Omega$ of both spin states can be realized by magnetic or microwave fields. (c) Trace of the Wilson loop $L$ in parameter space. Deviations from 2 imply non-Abelian physics [$K_{\widehat{y}}=1.814$; outside the white (black) regions, $|\mathrm{tr}L|<1.9$ ($<1.99$)].