High-fidelity pseudopotentials for the contact interaction

The contact interaction is often used in modeling ultracold atomic gases, although it leads to pathological behavior arising from the divergence of the many-body wave function when two particles coalesce. This makes it difficult to use this model interaction in quantum Monte Carlo and other popular numerical methods. Researchers therefore model the contact interaction with pseudopotentials, such as the square well potential, whose scattering properties deviate markedly from those of the contact potential. In this article, we propose a family of pseudopotentials that reproduce the scattering phase shifts of the contact interaction up to a hundred times more accurately than the square well potential. Moreover, the pseudopotentials are smooth, resulting in significant improvements in efficiency when used in numerical calculations.

Figure 1

(a) Bound- and scattering-state wave functions for contact interactions at $k_{\mathrm{F}}a=0.5$, offset by their respective eigenvalues (dashed lines) as a function of interparticle separation. The bare contact potential (represented by the gray area) is strongly attractive and harbors a single bound state. The scattering states incident on the potential incur a positive phase shift with respect to the noninteracting scattering wave function (dotted line). $r_{\text{n}}$ denotes the first node of the scattering wave function and $r_{\text{c}}$ denotes the first antinode, which we use as the cutoff radius when constructing pseudopotentials, as described in Sec. 2a. (b) The pseudopotentials at $k_{\mathrm{F}}a=1/2$ on the repulsive branch. The potential labeled “Troullier” denotes the pseudopotential derived using the Troullier-Martins formalism. The line labeled “UTP” denotes the pseudopotential derived using the UTP formalism. These formalisms are described in Sec. 2. (c) The wave functions for the relative motion of two particles interacting with a contact potential, the hard sphere, Troullier-Martins pseudopotential, and UTP, at $k=k_{\mathrm{F}}$.

Figure 2

The errors in phase shifts $|{\delta }_0^{\mathrm{PP}}\left(k\right)-{\delta }_0^{\mathrm{cont}}\left(k\right)|$ for the repulsive branch at $k_{\mathrm{F}}a=1/2$. ${\delta }_0^{\mathrm{cont}}$ is the $s$-wave scattering phase shift for the contact interaction and ${\delta }_0^{\mathrm{PP}}$ is the scattering phase shift for each of the pseudopotentials. The Troullier-Martins formalism is approximately two orders of magnitude more accurate than the hard and soft-sphere pseudopotentials commonly used as approximations to the contact interaction. The UTP formalism offers an additional factor of 2 improvement. The dotted green line labeled UTP-max denotes the phase-shift error of a variant of the UTP developed by minimizing the peak phase-shift error.

Figure 3

(a) The errors in phase shift for the attractive branch at $k_{\mathrm{F}}a=-1/2$, for different pseudopotentials. The Troullier-Martins formalism yields phase shifts that are 10 times closer to those of the contact interaction than the square well approximation. For all pseudopotentials described here, the quality of the potential depends on the choice of spatial cutoff. The Troullier-Martins and UTP were constructed with cutoff $r_{\text{c}}=(1/2)k_{\mathrm{F}}$. By contrast, the square well potential was constructed with $r_{\text{c}}\simeq 0.03k_{\mathrm{F}}$. (b) Convergence of the phase shifts with decreasing pseudopotential radius for $k_{\mathrm{F}}a=-1/2$.

Figure 4

(a) Band diagram for two atoms in a harmonic trap, calculated following Ref. [36]. (b) Mean-squared error in total energy for two atoms in a harmonic trap, for all bands below $E_{\max }$ (solid lines), for repulsive interactions $\left(k_{\max }a>0\right)$. UTP denotes the ultratransferable pseudopotential. The dashed line denotes the error in the ground-state energy with the UTP. The labels $-d/a$ and $-1/k_{\max }a$ describe the $x$ axis, which can be interpreted as either a change in trap size for constant interaction strength (varying $d/a$ where $d=1/\sqrt{\omega }$), or a change in interaction strength for constant trap size (varying $1/k_{\max }a$, where $k_{\max }=\sqrt{E_{\max }}$). The horizontal solid black line shows the typical many-body accuracy goal of $0.01\%$. (c) The pseudopotential error for attractive interactions $\left(k_{\max }a<0\right)$. (d) The pseudopotential error in the bound-state energy.

Figure 5

(a) Deviation of the equation of state from that predicted by third-order perturbation theory, as given by Eq. (2). The gray region denotes the confidence intervals in $E^{\left(3\right)}$. The line denoted $E^{\left(2\right)}$ is the equation of state derived from second-order perturbation theory [59]. We note that the equation of state obtained by using a soft-sphere pseudopotential deviates significantly from the line predicted by the perturbation expansion. (b) Standard deviation of the local energy $E_{\mathrm{L}}$ for the ground state of the soft sphere and the UTP. The soft-sphere pseudopotential exhibits a much larger standard deviation, which can be explained by the abrupt changes in potential energy when particles overlap.