Pseudopotentials for an ultracold dipolar gas

A gas of ultracold molecules interacting via the long-range dipolar potential offers a highly controlled environment in which to study strongly correlated phases. However, at particle coalescence the divergent $1/r^3$ dipolar potential and associated pathological wave function hinder computational analysis. For a dipolar gas constrained to two dimensions we overcome these numerical difficulties by proposing a pseudopotential that is explicitly smooth at particle coalescence, resulting in a 2000-times speedup in diffusion Monte Carlo calculations. The pseudopotential delivers the scattering phase shifts of the dipolar interaction with an accuracy of ${10}^{-5}$ and predicts the energy of a dipolar gas to an accuracy of ${10}^{-4}{E}_{\mathrm{F}}$ in a diffusion Monte Carlo calculation.

Figure 1

(a) The local energy $E_{\mathrm{L}}={\psi }^{-1}\widehat{H}\psi$ as a function of radius in the $\ell =1$ angular momentum channel, showing in orange the divergence as $r^{-3}$ when the dipolar potential is used with the noninteracting wave function ${\psi }_{\mathrm{non}-\mathrm{int},\ell =1}$. Also shown in magenta is the local energy divergence as $r^{-5/2}$ when the dipolar potential is used with a wave function with an exponential cusp correction ${\psi }_{\exp ,\ell =1}$, and in blue the exact solution in this channel, given by a Bessel function cusp correction ${\psi }_{K_2,\ell =1}$. In red and green are the local energies of Troullier-Martins and ultratransferable (UTP) pseudopotentials, respectively, with the noninteracting wave function, which outside of the radius $r_{\mathrm{c}}$ shown by a dashed gray line join smoothly onto the real dipolar potential. The inset shows the same curves on a logarithmic scale. (b) The local energy in the $\ell =3$ channel, demonstrating that the Bessel function cusp correction ${\psi }_{K_2,\ell =3}$ is not accurate in other channels.

Figure 2

The dipolar potential, and Troullier-Martins and UTP pseudopotentials. The gray vertical line indicates $r_{\mathrm{c}}$, the pseudopotential cutoff radius.

Figure 3

(a) The error in the scattering phase shift $|{\delta }_{\mathrm{pseudo},1}\left(E\right)-{\delta }_{\mathrm{dipole},1}\left(E\right)|$. The filled gray curve is the density of scattering states $g\left(E\right)$ in the two-body Fermi sea on a linear scale. (b) The root-mean-squared error in the scattering phase shift as a function of interaction strength.

Figure 4

The deviation of the energy of two particles in a harmonic trap as calculated using both Troullier-Martins and UTP pseudopotentials from that calculated using the exact dipolar potential, as a function of interaction strength.

Figure 5

(a) The variation of the energy per particle in the Fermi gas with pseudopotential cutoff radius, calculated using DMC. The red points are for the Troullier-Martins pseudopotential, the green a UTP pseudopotential, and the magenta point is the exact dipolar potential. Stochastic error bars are of order ${10}^{-5}{E}_{\mathrm{F}}$. The vertical dashed line denotes the recommended cutoff radius. (b) The variance in the individual local energy samples (as seen in Fig. 1) taken during a DMC calculation using the pseudopotentials. Also shown are results for the dipolar potential both with and without Kato-like cusp corrections applied.

Figure 6

(a) The variation of the energy per particle in the Fermi gas with time step $\tau$. The magenta points are using the exact dipolar potential, and the green points using a UTP pseudopotential. The error bars show DMC stochastic errors, and are of order ${10}^{-5}{E}_{\mathrm{F}}$. Fitted values of the linear error parameters $a$ (see main text) are also given. (b) The standard error $s_E$ in the energy per particle in the Fermi gas, for both the dipolar potential and UTP pseudopotential. Values of the fitting parameters $\sigma$ for a $1/\sqrt{\tau }$ fit are also given for each.

Figure 7

The equation of state of the 2D isotropic, homogeneous dipolar gas. The blue curves show the first- and second-order perturbation theory ($E^{\left(1\right)}$ and $E^{\left(2\right)}$) equations of state [42], and our DMC data are shown in magenta for the dipolar potential, red for the Troullier-Martins pseudopotential calibrated at $E_{\mathrm{F}}/4$, and green for a UTP. The latter three curves overlie each other to within the width of the plotted lines. Stochastic error bars are of order ${10}^{-5}{E}_{\mathrm{F}}$. The black circles show data from DMC calculations using the dipolar potential by Matveeva and Giorgini (MG) in Ref. [26].

Figure 8

The deviation of the equation of state as calculated using the pseudopotentials from that calculated using the exact dipolar potential. The dipolar potential is shown in magenta, with the Troullier-Martins pseudopotential in red, the UTP in green, and first- and second-order perturbation theory ($E^{\left(1\right)}$ and $E^{\left(2\right)}$) in blue. The gray box around the results using the dipolar potential shows the target $3\times {10}^{-4}{E}_{\mathrm{F}}$ accuracy level.

Figure 9

The dipolar potential $V(r,\varphi )$ in magenta, and the UTP $V_{\mathrm{UTP}}(r,\varphi )$ for the same tilt angle, in green. The potentials are cut through for $3\pi /2<\varphi <2\pi$ to contrast the radial variation of the dipolar potential along $\varphi =3\pi /2$ and $\varphi =2\pi$, and show the smooth join of the UTP onto the dipolar potential at $r=r_{\mathrm{c}}$.

Figure 10

The equation of state of the tilted dipolar gas system as a function of tilt angle $\theta$. Our DMC data using the dipolar potential and UTP overlie one another to within the width of the plotted lines, with stochastic error bars of order ${10}^{-5}{E}_{\mathrm{F}}$. First-order perturbation theory $E^{\left(1\right)}$ is shown in blue.

Figure 11

The deviation of the equation of state as calculated using the tilted pseudopotential from that calculated using the exact dipolar potential. Results using the dipolar potential are shown in magenta, with those using UTP pseudopotential in green. Similarly to Fig. 8, the gray box around the results using the dipolar potential shows the targeted $3\times {10}^{-4}{E}_{\mathrm{F}}$ accuracy level.

Figure 12

(a) The variation of the energy per particle in the Fermi gas of tilted dipoles with time step $\tau$, with the values of the linear error parameters $a$. (b) The standard error $s_{\mathrm{E}}$ in the energy per particle in the Fermi gas, again with fitted $1/\tau$ parameters given.