Density profile of interacting fermions in a one-dimensional optical trap

The density distribution of the one-dimensional Hubbard model in a harmonic trapping potential is investigated in order to study the effect of the confining trap. Strong superimposed oscillations are always present on top of a uniform density cloud, which show universal scaling behavior as a function of increasing interactions. An analytical formula is proposed on the basis of bosonization, which describes the density oscillations for all interaction strengths. The wavelength of the dominant oscillation changes with interaction, which indicates the crossover to a spin-incoherent regime. Using the Bethe ansatz the shape of the uniform fermion cloud is analyzed in detail, which can be described by a universal scaling form.

Figure 1

DMRG data for the fermion density $n\left(x\right)$ in a harmonic trap with $N=30$ and ${\omega }^2=4\times {10}^{-5}$ (points) compared to the analytical approximation in Eq. (12) (solid lines).

Figure 2

The width $L_{\text{F}}\left(U\right)$ of the fermion cloud in the trap as determined from the local Bethe ansatz density as a function of the scaling variable $U/\sqrt{N\omega }$. The different symbols (colors) correspond to many different choices of $U\in [0,20]$, ${\omega }^2\in [{10}^{-5},16\times {10}^{-5}]$, and $N\in [10,70]$. Inset: Effective exponent $\delta$ determined from a fit of $n_0^{\delta }\left(x\right)$ to the local Bethe ansatz density.

Figure 3

Density oscillations around the slowly varying part $n_0\left(x\right)$ for $N=30$ and ${\omega }^2=4\times {10}^{-5}$ compared to the analytical form in Eq. (12).

Figure 4

Amplitudes $A_{1,2}$ (top) and exponents ${\alpha }_{1,2}$ (bottom) as determined by fitting Eq. (12) to DMRG data for different choices of $U$, $\omega$, and $N$.