Energy, contact, and density profiles of one-dimensional fermions in a harmonic trap via nonuniform-lattice Monte Carlo calculations

We determine the ground-state energy and Tan's contact of attractively interacting few-fermion systems in a one-dimensional harmonic trap, for a range of couplings and particle numbers. Complementing those results, we show the corresponding density profiles. The calculations were performed with a lattice Monte Carlo approach based on a nonuniform discretization of space, defined via Gauss-Hermite quadrature points and weights. This particular coordinate basis is natural for systems in harmonic traps, and can be generalized to traps of other shapes. In all cases, it yields a position-dependent coupling and a corresponding nonuniform Hubbard-Stratonovich transformation. The resulting path integral is performed with hybrid Monte Carlo as a proof of principle for calculations at finite temperature and in higher dimensions. We present results for $N=4,...,20$ particles (although the method can be extended beyond that) to cover the range from few- to many-particle systems. This method is exact up to statistical and systematic uncertainties, which we account for—and thus also represents an ab initio calculation of this system, providing a benchmark for other methods and a prediction for ultracold-atom experiments.

Figure 1

Abscissas and weights for Gauss-Hermite integration with $N_x^{}=10,20,40,80,160$ points. The $x$ axis is scaled by $1/\sqrt{N_x^{}}$ for display purposes. Note that $g\left(x_i^{}\right)=g{w}_i^{}{e}^{x_i^2}$ is the position-dependent coupling constant (see main text).

Figure 2

Tuning of the bare lattice coupling to match the exact ground-state energy of the two-body problem. The solid line shows the exact solution for the total ground-state energy of the two-body problem (including center-of-mass motion) in units of $\hslash \omega$, from Ref. [21]. For that line, $g_{\text{continuum}}=2a_{\text{HO}}^{}/a_0^{}$. The data shown with error bars represent our Monte Carlo results for different lattice sizes. For the latter, the horizontal axis is the bare lattice coupling $g$ multiplied by $5/N_x^{}$, which yields the correct renormalization factor at weak couplings.

Figure 3

Large-$\beta \omega$ extrapolation example for the energy of four fermions on a Gauss-Hermite lattice of $N_x^{}=10$ points. The unexpectedly large oscillations in the data at large couplings exemplify the numerical difficulties in computing in that regime. The horizontal dashed lines show fits to the asymptotic value.

Figure 4

Top panel: Ground-state energy per particle for $N=4$, 6, 8, 10, 12, 16, and 20 particles (from bottom to top) as a function of the coupling, for $N_x^{}=10$, 20, 40, and 80 lattice points. For two particles, the exact solution is reproduced, per our renormalization condition. Bottom panel: Ground-state energy $E_{\text{GS}}^{}$ in units of the noninteracting ground-state energy $E_{\text{GS,0}}^{}$, as a function of the coupling (as in the main plot), for $N=4$, 8, 12, 16, and 20 particles (from bottom to top), showing the approach to the thermodynamic limit $N\to \infty$.

Figure 5

Contact per particle for $N$ = 2, 4, 6, 8, 10, 12, 16, and 20 particles, as a function of the coupling, for $N_x^{}=80$. For two particles, the exact solution is also shown as a solid line.

Figure 6

Virial theorem test for $N_x^{}=40,80$ lattice points and $N=4$ particles. “E-2V” and “Contact” denote the left- and right-hand sides of Eq. (8), respectively, divided by the particle number. The theorem is exactly satisfied for the noninteracting case $a_0^{-1}=0$.

Figure 7

Density profile of unpolarized, spin-$1/2$ fermions for several particle numbers $N=4$, 8, 12, 16, 20. Top: Noninteracting case. Center: $2a_{\text{HO}}^{}/a_0^{}=1.67$. Bottom: $2a_{\text{HO}}^{}/a_0^{}=5.16$. See Supplemental Material [25] for the data plotted here.

Figure 8

Density profile of unpolarized, spin-$1/2$ fermions, for $2a_{\text{HO}}^{}/a_0^{}=0.64,1.67,2.71,5.16$. Top: $N=4$. Center: $N=8$. Bottom: $N=16$. See Supplemental Material [25] for the data plotted here.

Figure 9

Lattice-size dependence of the ground-state energy of $N$ = 12, 16, and 20 unpolarized spin-$1/2$ fermions, for $N_x^{}=20,40,80$. Top panel: $2a_{\text{HO}}^{}/a_0^{}=2.0$. Bottom panel: $2a_{\text{HO}}^{}/a_0^{}=3.3$. The error bars are purely statistical and show an estimated 3% error for the specific data points shown. Note the change of scale in the energy axes relative to that of the top panel of Fig. 4: the present plots are a zoom-in by a factor of $\simeq 9$.

Figure 10

Temporal lattice spacing $\left(\tau \right)$ dependence of the ground-state energy of unpolarized spin-$1/2$ fermions on an $N_x^{}=10$ nonuniform lattice, for several values of the coupling $2a_{\text{HO}}^{}/a_0^{}$, and for several particle numbers. The smoothness of the curves upon reducing $\tau$ by a factor of two shows that these effects are extremely small (see text for further details).