Spontaneous rotation in one-dimensional systems of cold atoms

We theoretically study harmonically trapped one-dimensional Bose gases (e.g., ${}^7\text{L}\text{i}$, ${}^{23}\text{N}\text{a}$, ${}^{39}\text{K}$, ${}^{87}\text{R}\text{b}$, etc.) with multibands occupied, focusing on effects of higher-energy bands. Combining the Ginzburg-Landau theory with bosonization techniques, we predict that the repulsive interaction between higher-band bosons and the quantum fluctuation can induce the ground state with a finite angular momentum around the trapped axis. In this state, the $Z_2$ reflection symmetry (clockwise or anticlockwise rotations) is spontaneously broken.

Figure 1

(Color online) One-dimensional trapped Bose gas confined along the $x$ axis. In the text, we predict that the Bose gas spontaneously circulates along the arrow (or along the opposite direction of the arrow) under a certain condition.

Figure 2

Single-particle energy band structure and density distributions (sketch) in each band. We assume that the bosons live only in lower three energy bands. The density ${\rho }_{\alpha }$ is defined in Eq. (6a, 6b, 6c).

Figure 3

(Color online) Predicted ground-state phase diagram of the model (1) under the condition (4). Quantum phase transitions are expected to occur around the black points. They, however, might be interpreted as a crossover phenomenon rather than a transition, since boson densities of higher bands would be nonzero even when the chemical potential $\mu$ is smaller than $\hslash {\omega }_0$. When $\mu$ is larger than $n\hslash {\omega }_0(n>2)$, the bosons on higher bands ${ϵ}_{n>2,l}$ would also possess an angular momentum. It is therefore expected that, for example, the bosons on ${ϵ}_{2,l}$ rotate clockwise, while those on ${ϵ}_{3,l}$ do anticlockwise. Namely, the almost complete or partial cancellation of the angular momentum can occur.