Few interacting fermions in a one-dimensional harmonic trap

We study spin-1/2 fermions, interacting via a two-body contact potential, in a one-dimensional harmonic trap. Applying exact diagonalization, we investigate their behavior at finite interaction strength and discuss the role of the ground-state degeneracy which occurs for sufficiently strong repulsive interaction. Even low temperature or a completely depolarizing channel may then dramatically influence the system's behavior. We calculate level occupation numbers as signatures of thermalization, and we discuss the mechanisms to break the degeneracy.

Figure 1

We plot the energy $E-E_{\mathrm{F}}$ as a function of the dimensionless interaction strength $g$ for different combinations of spin-up and spin-down particles. The energy offset $E_{\mathrm{F}}$ is the Fermi energy of the noninteracting system. We find ground-state degeneracies in the limit of strong interactions. We plot first 20 eigenenergies of the Hamiltonian in Hilbert spaces with fixed $S_z=N_{\uparrow }-N_{\downarrow }$, but without fixing the total spin.

Figure 2

(a) The probabilities $P_{\downarrow }$ and $P_{\uparrow }$ for finding $\uparrow$ and $\downarrow$ particles in different orbitals ($n=0,1,2,3$) as a function of the dimensionless interaction strength $g$. We consider the effect of temperature using a Boltzman distribution: the solid black and dashed blue lines denote the temperatures $kT/\hslash \omega =0$ and $0.3$, respectively. (b) Temperature dependence of the cumulative distribution function for $g=9$ (left) and $g=12$ (right). This function describes the probability of finding the $\downarrow$ particle above the $n$th harmonic oscillator level, where $n=0$ is shown by the thick solid line, $n=1$ by the dashed line, $n=2$ by the thin solid line, and $n=3$ by the dash-dotted line. In all plots $N=5$ and $N_{\downarrow }=1$.

Figure 3

(a) Influence of a symmetric anharmonicity of the trap to the spectrum of the Hamiltonian. Anharmonicity shifts eigenenergies of the Hamiltonian (black lines) relative to the eigenenergies of the Hamiltonian with the harmonic trap (gray lines). It does not lift the degeneracies at large $g$ and provides no mixing between different states in each manifold. (b) Energies as a function of $g$ in the presence of a Zeeman term. No degeneracies occur in the strongly interacting limit. (c) Zeeman breaking of the spin symmetry. For $N=4$ and $N_{\uparrow }=1$, the ground state at $g=12$ is time-evolved after switching on a Zeeman term, $\delta =0.05$. The probability for finding the state in one of the four possible spin configurations is plotted. The thick black, dashed blue, dotted red, and thin black lines correspond to the states ordered by energy (increasing).