Bethe-ansatz analysis of near-resonant two-component systems in one dimension

We introduce a type of model of two-component systems in one dimension that can be analyzed by the Bethe-ansatz approach. The interspecies interactions are tunable via Feshbach resonant interactions, and the Yang-Baxter equation for exact solvability is fulfilled by fine-tuning the resonant energies. Although the strict exact solvability beyond the level of two-body scatterings would require extra singular counterterms, the models introduced here can still be well described by the Bethe ansatz for relatively small densities. The resulting systems can be described by introducing intraspecies repulsive and interspecies attractive couplings $c_1$ and $c_2$. This kind of system admits two types of interesting solutions: In the regime with $c_1>c_2$, the ground state is a Fermi sea of two-strings, where the Fermi momentum $Q$ is constrained to be smaller than a certain value $Q^{*}$, and it provides an ideal scenario to realize the BCS-BEC crossover (from weakly attractive atoms to weakly repulsive molecules) in one dimension. In the opposite regime with $c_1<c_2$, the ground state is a single bright soliton even for fermionic atoms, which reveals itself as an embedded string solution.

Figure 1

Condensation of levels for weakly interacting bosons in one dimension. The left panel is drawn for strong coupling and the right panel is drawn for weak coupling.

Figure 2

The critical value ${\lambda }^{*}$ as a function of $\xi =c_2/c_1$. The dots are obtained numerically and the thick line is the curve of the fitting function in Eq. (3.17). We can see that as $\xi \to 1$, ${\lambda }^{*}\to \infty$, which means that $Q^{*}$ tends to zero; this implies the instability of the system.

Figure 3

The solution $\sigma \left(x\right)$ to Eq. (3.11) for $\lambda \to {\lambda }^{*}\left(\xi \right)+0$. $\sigma \left(x\right)$ can be separated into the regular part ${\sigma }_{\text{reg}}\left(x\right)$ and the singular part ${\sigma }^0\left(x\right)$, where the latter grows with increasing $n$ and is seen in the figure as the part near $x=0$.

Figure 4

Density of states $\sigma \left(x\right)$ with fixed ${\gamma }_1$ and varying ${\gamma }_2$, where ${\gamma }_{1,2}\equiv c_{1,2}/n$. By tuning ${\gamma }_2$ toward ${\gamma }_1$, the behavior of the system changes from weakly attractive fermions (nearly flat at the center and sharp increase near the boundary) to fermionic super Tonks-Girardeau gas and finally to weakly interacting bosons (condensation at the center).

Figure 5

Numerical result for compressibility, where we have made it dimensionless by multiplying by $n^3$. Upper panel is plotted for the present model with ${\gamma }_1=5$, where $\kappa n^3$ changes with $\xi =c_2/c_1$. Lower panel is plotted for the Yang-Gaudin model by varying dimensionless coupling constant $\gamma =c_F/n$. The difference is apparent: for the present model, there is divergence for compressibility, while for the Yang-Gaudin model, the compressibility saturates at a finite value.

Figure 6

A typical spectrum for the $S=0$ excitations. There are two branches, one for hole type and one for particle type. At long wavelength, they both reduce to the phonon branch. The maximum of the hole branch is fixed at $k_{\text{max}}=k_F=\pi n/2$ and there is a periodicity of $2k_F=\pi n$ in the hole branch.

Figure 7

A typical spectrum for the $S=1/2$ excitations. It is similar to that in the Yang-Gaudin model. The minimum energy is fixed at $k_{\text{min}}=k_F=\pi n/2$.

Figure 8

Numerical result for sound velocity scaled with $v_F$. Upper panel is plotted for the present model with ${\gamma }_1=5$, where $v_c/v_F$ changes with $\xi =c_2/c_1$. Lower panel is plotted for the Yang-Gaudin model by varying dimensionless coupling constant $\gamma =c_F/n$. They are different in the following aspect: The present model has vanishing sound velocity at $\xi \to 1$, while the Yang-Gaudin model has a finite lower bound for the sound velocity: $v_c/v_F>0.5$.

Figure 9

The schematic picture for the distribution of the roots $k$. For two-strings with center momentum ${\mathrm{\Lambda }}_{\alpha }$, the two roots $k_{\alpha ,1}$ and $k_{\alpha ,2}$ are separated by a distance $c_2$ along the imaginary axis. For unpaired atoms, the roots lie on the real axis. We can see that when a single two-string or a single unpaired atom is added into the system, the $J_{\alpha }$ in Eq. (3.3) will be shifted by half of unity.

Figure 10

Typical spectra for the $S=1/2$ excitations, where the momentum is shifted by $k_F$. Three different choices of $\xi$ are shown. The lower panel is obtained from the upper panel by offsetting the spin gap. For very small $\xi$, the dispersion curve has three parts: a narrow quadratic region near the minimum, an intermediate linear region, and finally a quadratic region at large energy. For $\xi$ close to 1, the dispersion is purely quadratic.

Figure 11

Numerical result for spin gap scaled with ${\epsilon }_F$. The upper panel is plotted for the present model with ${\gamma }_1=5$, and the lower panel is plotted for the Yang-Gaudin model. They appear practically the same, but with the following difference: the present model terminates at $\xi =1$, while the Yang-Gaudin model will continue the logarithmic behavior with ever-growing $\gamma =c_F/n$.

Figure 12

Phase diagram in the h-n space at zero temperature, where $n$ and $h$ are scaled by ${\epsilon }_b=c_2^2/4$. The upper panel is plotted for $\xi =0.1$, and the lower panel is plotted for $\xi =0.8$. The case with $\xi =0.1$ is essentially the same as that in the Yang-Gaudin model, while in the case with $\xi =0.8$, the mixed-phase region (PP) is reduced. The phase boundary P-PP actually has an asymptote corresponding to the critical magnetic field $h=h_c$.

Figure 13

The embedded string solution. The two-string (the one in the enlarged circle) with inter-root separation $c_2$ is embedded in the $M$-string with inter-root separation $c_3=c_2-c_1$.