Transition from a Two-Dimensional Superfluid to a One-Dimensional Mott Insulator

A two-dimensional system of atoms in an anisotropic optical lattice is studied theoretically. If the system is finite in one direction, it is shown to exhibit a transition between a two-dimensional superfluid and a one-dimensional Mott insulating chain of superfluid tubes. Monte Carlo simulations are consistent with the expectation that the phase transition is of Kosterlitz-Thouless type. The effect of the transition on experimental time-of-flight images is discussed.

Figure 1

Phase diagram for the finite anisotropic Hubbard model at number density $n=1$. The 2D superfluid and Mott insulating phases are denoted by “2D SF” and “2D MI,” respectively. The size of the 1D Mott insulating phase “1D MI” has been exaggerated in the figure. The vertical double arrow indicates the interval along which the MC calculations in this study have been performed.

Figure 2

Velocity distribution along the weakly coupled lattice direction. The curves corresponds to a ratio of the tunneling matrix element to the interaction strength, $t_y/U=0.012$ (dashed line), 0.0025 (solid line), and $1.5\times {10}^{-3}$ (dotted line). The inset shows the full 2D profile of the velocity distribution for the case $t_y/U=1.5\times {10}^{-3}$. The system size is $12\times 12$ sites.

Figure 3

From top to bottom, the density fluctuation, the correlation, and the superfluid density are shown as a function of the size-independent parameter $t_y{L}_x^{1.75}/U$, for different system sizes $L_x\times L_y$ measured in lattice sites, as indicated in the legend. The scale is logarithmic on both axes. The correlation function and the compressibility are multiplied by arbitrarily chosen numerical factors to separate the three groups of lines. The superfluid density is divided by the tunneling matrix element for better visual clarity.

Figure 4

(a) Superfluid density in the dual $XY$ model, Eq. (4), scaled according to Weber-Minnhagen scaling. (b) Exponent of the algebraic decay of the correlation ${\Gamma }_y$, as a function of the dimensionless coupling parameter $t_y{L}_x^{1.75}/U$. The horizontal line shows the critical value expected for a 1D Mott transition at fixed density, $1/(2K)=0.25$. Different curves correspond to different system sizes; the same symbols are used in both parts of the figure.