Strong Interaction Effects and Criticality of Bosons in Shaken Optical Lattices

We study the quantum phase transitions and identify a tricritical point between a normal Bose superfluid, a superfluid that breaks additional $Z_2$ Ising symmetry, and a Mott insulator in a recent shaken optical lattice experiment. We show that near the transition between normal and $Z_2$ symmetry breaking superfluids, bosons can condense into a momentum state with high or even locally maximum kinetic energies due to the interaction effect. We present a general low-energy effective field theory that treats both the superfluid transition and the Ising transition in a uniform framework. Using the perturbative renormalization group method, we find that the critical behavior of the quantum phase transition belongs to a universality class different from that of a dilute Bose gas.

Figure 1

Band structure of a shaken lattice. (a) Band structure before shaking. The red solid line on the top is the dressed $s$ band with energy shifted by an energy $\hslash \omega$. (b) The solid line is the dispersion for small shaking amplitude $f<f_{\mathrm{c}}^0$ and the dashed line is the dispersion for large shaking amplitude $f>f_{\mathrm{c}}^0$. Energy is plotted in units of lattice recoil energy $E_{\mathrm{r}}={\hslash }^2{k}_0^2/(2m)$.

Figure 2

Interaction shifts of SF-$Z_2$ SF quantum critical point. (a),(c) Deep lattice with $V/E_{\mathrm{r}}=16$ and $\hslash \omega /E_{\mathrm{r}}=7.1$; (b),(d) Shallow lattice with $V/E_{\mathrm{r}}=4$ and $\hslash \omega /E_{\mathrm{r}}=4.4$. (a) and (b) Condensate momentum $k_{\mathrm{c}}$ as a function of $f$. Blue dashed line is for noninteracting and the red solid line is for the interacting case with $gn/E_{\mathrm{r}}=1$. (c) and (d) Interaction ($gn$)-shaking amplitude($f$) phase diagram for a fixed frequency $\omega$. Gray shaded areas show regions where atoms condense to states with finite kinetic energy.

Figure 3

Schematic of the tricritical point of interacting bosons in a shaken lattice. The three phases are the Mott insulator phase (MI), normal superfluid (SF) phase, and the superfluid phase that breaks $Z_2$ symmetry ($Z_2$ SF). $g_{\mathrm{c}}$ defines the critical interaction strength for the normal superfluid to Mott insulator transition.

Figure 4

Renormalization group flow diagram of (a) case $A$ (no particle-hole symmetry) and (b) case $B$ (particle-hole symmetry).