Eliminating the wave-function singularity for ultracold atoms by a similarity transformation

A hyperbolic singularity in the wave function of $s$-wave interacting atoms is the root problem for any accurate numerical simulation. Here, we apply the transcorrelated method, whereby the wave-function singularity is explicitly described by a two-body Jastrow factor, and then folded into the Hamiltonian via a similarity transformation. The resulting nonsingular eigenfunctions are approximated by stochastic Fock-space diagonalization with energy errors scaling with $1/M$ in the number $M$ of single-particle basis functions. The performance of the transcorrelated method is demonstrated on the example of strongly correlated fermions with unitary interactions. The current method provides the most accurate ground-state energies so far for three and four fermions in a rectangular box with periodic boundary conditions.

Figure 1

The Jastrow factor $e^{u\left(r\right)}$ as a function of particle separation $r$ with unitary interaction at different values of the momentum cutoff. $L$ is the size of the computational box.

Figure 2

Ground-state energy of two particles with unitary interactions ($1/a_s=0$) vs inverse size of single-particle basis $1/M$ from the transcorrelated method (circles) and with renormalized Dirac delta (crosses). Horizontal green line: reference energy $E_{\mathrm{ex}}=-3.786005{\hslash }^2/m{L}^2$ [57]. Inset: Difference to the reference energy on a log-log plot indicating power-law scaling $\propto 1/M^{1/3}$ for renormalization and $\propto 1/M$ for the transcorrelated approach.

Figure 3

Energy of zero-momentum ground state of two spin-up and one spin-down fermions with unitary interactions vs inverse size of single-particle basis $1/M$. Transcorrelated (“TrCorr”) are compared with semianalytical results from Ref. [60] (“scattering theory”) and AFQMC (“Endres AFQMC”) [20]. The horizontal yellow band marks the standard error of the extrapolated AFQMC results. Renormalized lattice calculations with FCIQMC using different single-particle dispersions: “Hubbard,” “quadratic,” and “quartic” as in Ref. [19] and the “magic” dispersion from Eqs. (122) and (124) in Ref. [18]. $E_0=4{\pi }^2{\hslash }^2/m{L}^2$ is the noninteracting energy.

Figure 4

Detail from Fig. 3 at enlarged scale. The error bars of the transcorrelated results show the stochastic errors from the FCIQMC method. The reference value from Ref. [60] (“scattering theory”) is marked with a horizontal line and error band in green, extrapolated result from Endres AFQMC [20] as a dashed yellow line (error not shown). The diagonal red and purple lines and bands indicate the linear fits and $1\sigma$ confidence bands obtained from ${\chi }^2$ fitting of the transcorrelated FCIQMC (TrCorr) results (four largest $M$ values), respectively. For details, see Appendix pp6-s2.

Figure 5

Ground-state energy of four fermions extrapolated to infinite basis-set limit. The horizontal (purple) line (TrCorr) shows the transcorrelated result $E/E_0=0.208339\pm 0.000094$ with the error indicated by yellow band. Results from Ref. [59] with Hamiltonian lattice 1 (“Bour 1”) and 2 (“Bour 2”) and AFQMC with Euclidian lattice (“Bour 3”) are shown alongside AFQMC results from Ref. [20] with $\mathcal{O}\left(4\right)$ (“Endres 1”) and $\mathcal{O}\left(5\right)$ (“Endres 2”) scaling, explicitly correlated Gaussian (“Yin”) [33], and renormalized lattice calculations following Ref. [19] using “Hubbard” and “quadratic” dispersion relations. For the numerical values, see Table 3 in Appendix pp6.

Figure 6

Convergence of the energy of one spin-up and one spin-down particle at unitary interaction with $k_{\mathrm{int}}$. The maximal value of the momentum for the single-particle basis was $16\pi /L$. The transcorrelated cutoff is kept to $k_c=2\pi /L$. The inset shows a detail at enlarged scale. The extrapolation to $1/k_{\mathrm{int}}=0$ is determined with a linear fit (shown as straight line) to the last three data points.

Figure 7

(a) The total number of walkers and (b) the shift $S$ and projected energy $E_p$ during the FCIQMC simulation for the example of two spin-up and one spin-down particles with unitary interactions at $M={11}^3$ using the transcorrelated approach at $k_c=2\pi /L$. After the target walker number of ${10}^6$ is reached, the previously constant shift parameter is updated in order to control the walker number. $E_0=4{\pi }^2{\hslash }^2/m{L}^2$ is the noninteracting energy.

Figure 8

The ground-state energy of two spin-up and two spin-down fermions. The purple (upper) and red (lower) line and band show the linear fit with $1\sigma$ confidence band obtained from ${\chi }^2$ fitting for $k_c=2\pi /L$ ($M=9^3,{11}^3,{13}^3,{15}^3$), and for $k_c=4\pi /L$ ($M={11}^3,{13}^3,{15}^3,{17}^3$), respectively. $E_0=4{\pi }^2{\hslash }^2/m{L}^2$ is the lowest noninteracting energy in the zero momentum sector.

Figure 9

Fitting procedure for results obtained using the renormalized lattice method with different single-particle dispersions. The value obtained using the transcorrelated method is included for comparison as the horizontal red line, with the error being smaller than the linewidth in this plot. The quadratic dispersion leads to two sets of points, depending on whether $M$ is even or odd. Here, we only show the results for odd values, which are considerably closer to the final extrapolated result.

Figure 10

The lowest energy of two spin-up and one spin-down fermions in the zero-momentum sector with the full transcorrelated Hamiltonian and with the approximated three-body term as per Eq. (F7). $E_0=4{\pi }^2{\hslash }^2/m{L}^2$ is the energy with zero interaction between the fermions.

Figure 11

The ground-state energy of two spin-up and two spin-down fermions with the full transcorrelated Hamiltonian and with the approximated three-body term as per Eq. (F7). $E_0=4{\pi }^2{\hslash }^2/m{L}^2$ is the noninteracting energy.