Self-bound many-body states of quasi-one-dimensional dipolar Fermi gases: Exploiting Bose-Fermi mappings for generalized contact interactions

Using a combination of results from exact mappings and from mean-field theory we explore the phase diagram of quasi-one-dimensional systems of identical fermions with attractive dipolar interactions. We demonstrate that at low density these systems provide a realization of a single-component one-dimensional Fermi gas with a generalized contact interaction. Using an exact duality between one-dimensional Fermi and Bose gases, we show that when the dipole moment is strong enough, bound many-body states exist, and we calculate the critical coupling strength for the emergence of these states. At higher densities, the Hartree–Fock approximation is accurate, and by combining the two approaches we determine the structure of the phase diagram. The many-body bound states should be accessible in future experiments with ultracold polar molecules.

Figure 1

Top: Plot showing the dependence of the energy $E$ of the lowest two-body state on the strength of the attractive interaction $l_{\mathrm{dd}}/l_{\perp }$, for two box lengths: $L/l_{\perp }=400$ [dotted (green) line] and $L/l_{\perp }=800$ [solid (red) line]. To bring out the absence of binding for $l_{\mathrm{dd}}<l_{\mathrm{crit}}\approx 1.44l_{\perp }$ and the expected ${(l_{\mathrm{dd}}-l_{\mathrm{crit}})}^2$ behavior of the binding energy near threshold, we plot the quantity $-\sqrt{-E/\left(\hslash {\omega }_{\perp }\right)}$. The dashed line is a fit of the numerical results to a linear function $\propto$$l_{\mathrm{dd}}-l_{\mathrm{crit}}$, and this holds over a wider range of couplings than might naively be anticipated. The rounding of the plots for $l_{\mathrm{dd}}\approx l_{\mathrm{crit}}$ is due to the finite size of the box. Bottom: Relative wave function for $l_{\mathrm{dd}}/l_{\perp }=1.3$ (left) and $l_{\mathrm{dd}}/l_{\perp }=1.8$ (right) and box length 200$l_{\perp }$.

Figure 2

Many-body ground-state energy per particle $E/N$ of quasi-1D fermionic dipoles versus density $n$ for three strengths of the attractive interaction, $l_{\mathrm{dd}}/l_{\perp }=1.44$ [thin (red) lines], $l_{\mathrm{dd}}/l_{\perp }=1.77$ [thick (light-blue) lines], and $l_{\mathrm{dd}}/l_{\perp }=1.9$ [thick (black) lines]. Dashed lines depict the low-density limiting behavior obtained from a mapping to bosons with a contact interaction. Dash-dotted lines represent the Hartree-Fock approximation, which is reliable at high densities. Solid curves interpolate between the low- and the high-density results.

Figure 3

Phase diagram of quasi-1D fermions with attractive dipolar interactions. The solid (black) $E=0$ line interpolates between the critical point $(l_{\mathrm{dd}},n)=(1.44l_{\perp },0)$ and the dash-dotted (black) Hartree-Fock $E=0$ line (see text). Above and to the left of the $E=0$ line, matter is unbound and can expand inde-finitely if not confined. Below the line, matter is bound and unconfined matter will oscillate about the equilibrium density. The solid (red) curve is a sketch of the equilibrium density, which again interpolates between the critical point $(l_{\mathrm{dd}},n)=(1.44l_{\perp },0)$ and the Hartree-Fock result [dash-dotted (red) line]. The onset of occupation of the first excited transverse level is shown by the dotted (black) line.