Tunneling-Induced Restoration of the Degeneracy and the Time-Reversal Symmetry Breaking in Optical Lattices

We study the ground-state properties of bosons loaded into the $p$ band of a one-dimensional optical lattice. We show that the phase diagram of the system is substantially affected by the anharmonicity of the lattice potential. In particular, for a certain range of tunneling strength, the full many-body ground state of the system becomes degenerate. In this region, an additional symmetry of the system, namely, the parity of the occupation number of the chosen orbital, is spontaneously broken. The state with a nonvanishing staggered angular momentum, which breaks the time-reversal symmetry, becomes the true ground state of the system.

Figure 1

Influence of the anharmonicity on the parameters of the Hamiltonian. (a) The ratio $a_x/a_y$ in the HA (dotted blue line) and in the Wannier basis (solid red line) that equalizes single-particle energies $E_x=E_y$. (b) The ratio $U_{xx}/U_{yy}$ (thick red line) and $3U_{xy}/U_{yy}$ (thin black line) calculated for exact Wannier functions. In both plots, we take $V_y/E_{R,y}=24$. The dotted vertical line indicates the case $V_x/E_{R,x}=6$ studied in the text. In the HA, both quantities are equal to 1.

Figure 2

(a) Filling and hopping of the $p_x$ (red line) and $p_y$ (blue line) orbitals for $\nu =3/2$ obtained with the ED method on a lattice with $L=6$ sites. Results agree with corresponding results obtained for $L=4$ (thick black lines) and DMRG calculations (not shown since practically indistinguishable from ED data). (b) The energy difference between the two ground states in even and odd subspaces of the eigenstates of the symmetry operator $\mathcal{S}$. The energies are obtained with the ED method on the lattice with $L=4$, 6, and 8 sites (thin black, dashed blue, and thick red lines, respectively). Note that corresponding lines cross the zero energy $L$ times. (c) Expectation value of the staggered angular momentum ${\widehat{\mathcal{L}}}_z/L$ as a function of tunneling obtained with DMRG on the lattice with $L=64$ sites. Nonvanishing value of ${\widehat{\mathcal{L}}}_z$ is present only in the region where the ground state is degenerate. In all figures, the shaded region denotes the range of tunnelings where the ground state of the system is degenerate in the thermodynamic limit.