Coordinate space representation of a one-dimensional odd-parity pseudopotential

Abstract

We propose a discrete-space representation of a one-dimensional zero-range odd-parity pseudopotential. The proposed representation is validated by applying it to the analytically solvable case of two fermions in a harmonic trap and successfully recovering the exact energy spectrum and eigenfunctions. Furthermore, we use the square-well and modified Pöschl--Teller potentials as finite-range representations of the odd-parity interaction and study their convergence to the contact interaction when the range tends to zero. Finally, we perform natural orbital analysis and compute the eigenvalues of the one-body density matrix for different particle numbers, examining their dependence on the one-dimensional scattering length and identifying distinct physical regimes.

Figures (5)

• Figure 1: Discrete and continuous representations of the odd-parity interaction as a function of the relative coordinate between fermions, $x$. The discrete representation is shown as red crosses, with no specific point defined at $x=0$. The square-well potential, $V_{SW}$, is depicted by a dashed blue line while the modified Pöschl--Teller potential, $V_{PT}$, is shown by a dash-dotted green line.
• Figure 2: Energy spectrum of the relative motion of two fermions in a $1D$ harmonic trap as a function of the interaction parameter. The solid gray line shows the odd solutions in the zero-range model, as given by Eq. (\ref{['eq:energy_2_fermions']}), while the dotted black line corresponds to the energy obtained using the discrete representation of the potential Eq. (\ref{['Eq:V:discrete']}) with a grid spacing of $\Delta x = 0.01\,a_\mathrm{ho}$ over the interval $x \in [-10, 10]\,a_\mathrm{ho}$. The two curves coincide perfectly. Results for two models of finite-range continuous potentials are shown for potential range equal to $R=0.2,\,0.5,\,1.0\, a_\mathrm{ho}$ are shown in blue, red, and green, respectively. Dashed lines correspond to the calculations with the square-well and the dash-dotted lines to the modified Pöschl--Teller potentials.
• Figure 3: Relative wave functions of two fermions in a $1D$ harmonic trap. Panel (a) shows the three lowest relative wave functions for $a_{1D} = -a_\mathrm{ho}$, computed analytically and with the discrete representation. Panel (b) compares the ground state relative wave function for $a_{1D} = -a_\mathrm{ho}$ using the discrete and continuous representations.
• Figure 4: Eigenvalues of the OBDM for the ground state of a system of $N$ fermions confined in a harmonic trap. Panels (a) and (b) show the ten (for $N=2$) and seven (for $N=3$) largest eigenvalues, respectively, as a function of the interaction parameter. Numerical results using the discrete representation of the odd-parity potential are depicted as solid lines. The results obtained using Eq. (\ref{['Eq:wf:a>0']}) are shown with dotted lines. The analytical results for the two limiting cases of zero and infinite scattering length Francesc_OBDM_eigenvalues are included for comparison. Specifically, the values in the FTG limit ($-a_{ho}/a_{1D}=0$) are represented by crosses, while the asymptotic values in the non-interacting limit ($-a_{ho}/a_{1D}\to\infty$) are indicated by arrows. In panel (a), all eigenvalues are doubly degenerate, whereas in panel (b), the largest eigenvalue is non-degenerate and the rest are doubly degenerate. Panel (c) shows the OBDM eigenvalues for $N=2,\,3,\,4,\,5$ at an intermediate scattering length, $a_{1D} = -a_\mathrm{ho}$. Cross markers represent results obtained using the ground state wave function obtained using the discrete representation of the odd-parity potential function Eq. (\ref{['Eq:V:discrete']}) for $N=2,\,3$, while circular markers with error bars correspond to Monte Carlo results.
• Figure 5: Absolute value of the energy difference $\Delta E \equiv|E_{\mathrm{Anl.}}-E_{\mathrm{Num}}|/(\hbar\omega)$ between the zero-range analytical result, Eq. (\ref{['eq:energy_2_fermions']}), and the numerical results obtained using discrete and continuous representations, as a function of the grid spacing $\Delta x$ with $x \in [-10, 10]\,a_\mathrm{ho}$. Green squares, blue stars, and red circles correspond to results obtained using the discrete representation and the continuous representation with a square-well and a modified Pöschl–Teller potential, respectively. For the continuous representations, the interaction range is fixed to $R = 50\Delta x$. The convergence is analyzed for four scattering lengths: $a_{1D}/a_\mathrm{ho} = -5,\,-1,\,1,\,5$, shown in panels (a), (b), (c), and (d), respectively.