Few strongly interacting ultracold fermions in one-dimensional traps of different shapes

The ground-state properties of a few spin-1/2 fermions with different masses and interacting via short-range contact forces are studied within an exact diagonalization approach. It is shown that, depending on the shape of the external confinement, different scenarios of the spatial separation between components manifested by specific shapes of the density profiles can be obtained in the strong interaction limit. We find that the ground-state of the system undergoes a specific transition between orderings when the confinement is changed adiabatically from a uniform box to a harmonic oscillator shape. We study the properties of this transition in the framework of the finite-size scaling method adopted to few-body systems.

Figure 1

The shape of the potentials $V_{\sigma }(x,\lambda )$ for different values of the parameter $\lambda$ in natural units of a given flavor. For $\lambda =0$ a uniform box potential is restored. For increasing $\lambda$, the confinement transforms to the standard harmonic oscillator.

Figure 2

Spectra of the system consisting of $N_{\downarrow }=3$ and $N_{\uparrow }=1$ fermions as a function of the dimensionless interaction strength $g$. The top row corresponds to a harmonic oscillator potential and the bottom row to a box trap potential. Quasidegenerate energy bands seen in the left column split up when mass imbalance $\mu \ne 1$ is introduced (right column). The energy is given in the natural units of harmonic oscillator, $\hslash \omega$.

Figure 3

Single-particle densities ${\rho }_{\uparrow }\left(x\right)$ (thick blue line, heavier flavour) and ${\rho }_{\downarrow }\left(x\right)$ (thin red line, lighter flavor) calculated in the ground-state of the system for different numbers of fermions with $\mu =40/6$ and strong interaction $g=4$ confined in a box trap. The black vertical lines correspond to the walls of the box trap. In contrast to the separation induced by the mass difference for a harmonic potential [42], in this case the separation always occurs in the heavier fraction, independently of the way the fermions are distributed between flavors. In particular, the separation is present also for an equal number of fermions $N_{\uparrow }=N_{\downarrow }$. The positions and the densities are measured in units of $\sqrt{\hslash /\left(m_{\downarrow }\omega \right)}$ and $\sqrt{m_{\downarrow }\omega /\hslash }$, respectively.

Figure 4

Comparison of different separation scenarios for fermions with different mass $\left(\mu =40/6\right)$ driven by different shapes of the confinement, in the limit of strong interaction $g=4$: in the uniform box (left panels) and cropped harmonic oscillator (right panels). The thick blue and thin red lines represent the single-particle density profiles for heavy and light components, respectively. Note that, independently on the number of particles in a given flavor, the separation is always present in the heavier (for the uniform box) or the lighter (for the harmonic oscillator) component. The positions and the densities are measured in units of $\sqrt{\hslash /\left(m_{\downarrow }\omega \right)}$ and $\sqrt{m_{\downarrow }\omega /\hslash }$, respectively.

Figure 5

Single-particle densities ${\rho }_{\uparrow }\left(x\right)$ (thick blue line, heavier flavor) and ${\rho }_{\downarrow }\left(x\right)$ (thin red line, lighter flavor) calculated in the ground-state of the system of $N_{\uparrow }=3$ and $N_{\downarrow }=2$ in the limit of strong interaction, $g=4$. The positions and the densities are measured in units of $\sqrt{\hslash /\left(m_{\downarrow }\omega \right)}$ and $\sqrt{m_{\downarrow }\omega /\hslash }$, respectively.

Figure 6

The second moment $\sigma$ of the magnetization distribution Eq. (15) as a function of the shape of the confinement for different interaction strengths (from $g=4$ to $g=5$). Each plot corresponds to given numbers of particles in both flavors. Note that in extreme confinements, $\sigma$ saturates to a well defined value, while it changes rapidly in the vicinity of the transition point. The second moment $\sigma$ is given in the natural units of a harmonic oscillator, $\hslash /\left(m_{\downarrow }\omega \right)$.

Figure 7

Scaling properties of a few-body system. Left panels: Susceptibility $\chi$ as a function of the shape of the confinement $\lambda$ for different values of interactions and different number of particles. A characteristic peak of the susceptibility, whose height increases with $g$, is clearly visible. The vertical red line corresponds to the critical value ${\lambda }_c$ obtained after extrapolation of the results to infinite repulsion. Middle panels: Rescaled susceptibility as a function of a rescaled confinement shape parameter obtained after adopting the data-collapse method. Note that all data points collapse to one well-defined curve. Right panels: The second moment of the distribution $\sigma$ when the same scaling procedure is performed for the trap shape parameter. The susceptibility $\chi$ is given in the natural units of a harmonic oscillator, $\sqrt{\hslash /\left(m_{\downarrow }\omega \right)}$.