te of the art techniques.

![Zig-zag chains formed by an incoherent superposition between a triangular lattice [@Becker2010] $V_1({\vec r}\equiv(x,y))=V_{10}\left [ \sin^2\left ({\vec b}_1\cdot{\vec r}/2\right )+\sin^2\left ({\vec b}_2\cdot {\vec r}/2\right )
+\sin^2\left (({\vec b}_1-{\vec b}_2)\cdot{\vec r}/2\right )\right ]$, with $k$ the laser wavenumber, ${\vec b}_1=\sqrt{3}k {\vec e}_y$ and ${\vec b}_2=\sqrt{3}k(\sqrt{3}{\vec e}_x/2-{\vec e}_y/2)$, and an additional lattice $V_2({\vec r})=V_{20}\sin^2(\sqrt{3}ky/4 - \pi/4)$. In the figure, in which $V_{20}/V_{10}=2$, darker regions mean lower potential. The hopping rate between nearest (next-nearest) neighbors is $t>$ ($t'$).](fig1.eps){width="5.0cm"}

\[fig:1\]

#### Zig-zag lattices.

In the following we consider bosons in zig-zag optical lattices. As shown in Fig. \[fig:1\], this particular geometry may result from the incoherent superposition of a triangular lattice with elementary cell vectors $\vec{a}_1=a {\vec e}_x$ and $\vec{a}_2=a(\frac{1}{2}{\vec e_x}+\frac{\sqrt{3}}{2}{\vec e}_y)$ (formed by three laser beams of wavenumber $k=4\pi/3a$ oriented at $120$ degrees from each other, as discussed in Ref. [@Becker2010]) and a superlattice with lattice spacing $\sqrt{3}a$ oriented along $y$. For a sufficiently strong superlattice, zig-zag ladders are formed, and the hopping between ladders may be neglected. We will hence concentrate in the following on the physics of bosons in a single zig-zag ladder, which is to a large extent given by the rates $t$ and $t'$ characterizing the hopping along the two directions ${\vec a}_{1,2}$ (Fig. \[fig:1\]). As shown in Ref. [@Struck2011], a periodic lattice shaking may be employed to control the value of $t$ and $t'$ independently. Interestingly, their sign may be controlled as well. In the following we consider an inverted sign for both hoppings, which result in an anti-ferromagnetic coupling between sites [@Struck2011].

#### Model.

Ordering the sites as indicated in Fig. \[fig:1\], the physics of the sys