Quantum phases of a Feshbach-resonant atomic Bose gas in one dimension

We study an atomic Bose gas with an $s$-wave Feshbach resonance in one dimension. Most of the parameter region is occupied by a phase in which the superfluid fluctuations of atoms and molecules are the predominant ones, due to the phase fluctuations of atoms and molecules being locked by a Josephson coupling between them. When the density difference between atoms and molecules is close to zero, two additional phases may exist: the two-component Luttinger liquid, which contains two branches of gapless excitations, and the interchannel charge density wave where the relative density fluctuations between atoms and molecules are frozen at low energy.

Figure 1

(Color online) Schematic phase diagram of a FR atomic Bose gas in 1D: (a) on-site interactions only and (b) Tonks limit with nearest-neighbor repulsions between atoms (molecules) $V_a\left(V_m\right)$. $\delta ={\rho }_a-{\rho }_b$. The critical values for the occurrence of the I-CDW may be very small at $\mid \delta \mid =0$. [See Eq. (17).]

Figure 2

(Color online) Phase diagram for $\delta =0$: (a) suggested by tree-level scaling analysis and (b) predicted by the one-loop RG equations. $D_1<2$ and $D_2>2$ in region I, $D_1>2$ and $D_2<2$ in region II, $D_{1,2}>2$ in region III, and $D_{1,2}<2$ in region IV. Point $A$ and $B$ in (b) are tricritical points.

Figure 3

(Color online) Phase diagram for $a_0\mid \delta \mid$ being some rational number. For $a_0\mid \delta \mid$ being close to $k∕l$ where $k=1,2,\dots$, $l=2,3,\dots$, and $k$ and $l$ are coprime with one another, within the tree-level scaling, the ASF phase occurs for $D_1<2$, while the I-CDW phase exists in the region with $D_2<2∕l$. The region between them corresponds to the $2$-LL.