Title: "Accurate Numerical Calculations for Strongly Correlated Ultracold Few-Fermion Systems with the Transcorrelated Approach"

Abstract:

"In the description of the ultracold few-fermion systems, the exact diagonalization approach is frequently applied in order to achieve reliability and accuracy in theoretical calculations. In this approach, the energies and the wave functions are obtained by diagonalizing the Hamiltonian in a many-body Fock basis. As the size of Hilbert space combinatorially increases with the size of the system, most of the calculations are limited to the intermediate interaction strength. Therefore, understanding and improving convergence properties is crucial in order to make the approach more widely applicable.

The rate of convergence of physical observables with increasing basis size is determined, for the most part, by the nature of the particle-particle interaction itself. In ultracold atoms, the interaction potential is usually modeled by a zero-range pseudopotential, which introduces a singularity in the wave function at the particle-particle coalescence point. This singularity causes painfully slow convergence in one spatial dimension whereas in two or three dimensions it can lead to pathologic behavior.

We improve the exisiting exact diagonaliyzation approach with two distinguished steps. First, we apply the Full Configuration Interaction Quantum Monte Carlo, which introducing a stochastic sampling in the wave function significantly descreases the memory requirements and accelerates the numerical calculations compare to the exact diagonalization approach. Then, we also apply the transcorrelated approach, where the wave function is considered as a product of a Jastrow-type two-particle function and a linear combination of Fock basis states. The Jastrow factor, which contains the singularity of the wave function, is folded into the Hamiltonian by a similarity transformation. Thus the singularity is removed from the Fock-space expansion to leading order. The transformation thus smoothes out the singularity of the original zero-range pseudopotential, which significantly improves the convergence rate of the transcorrelated eigenfunction in the Fock basis states.

We will present numerical examples for the few-fermion sytems at strong interactions in one dimension and for few-fermion systems in three dimensions at unitarity. In one dimension the transcorrelated approach improves the convergence of the energy error from M^{−1} to M^{−3}, where M is the number of the single-particle basis functions. In three dimensions due to the pathological zero-range pseudopotential, the exact diagonalization of the original Hamiltonian is not possible. The transcorrelated transformation eliminates the pathological nature of the Hamiltonian and yields a M^{-1} convergence, which is a significant improvement compared to the standard renormalization approach, M^{-1/3}."