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arXiv:1703.08720 · Physical Review A

Four fermions in a one-dimensional harmonic trap: Accuracy of a variational-ansatz approach

Daniel Pęcak, Amin S. Dehkharghani, Nikolaj Thomas Zinner, Tomasz Sowiński

Published May 25, 2017• 25 citations
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Abstract

A detailed analysis of a system of four interacting ultracold fermions confined in a one-dimensional harmonic trap is performed. The analysis is done in the framework of a simple variational ansatz for the many-body ground state, and its predictions are compared with the results of numerically exact diagonalization of the many-body Hamiltonian. A short discussion of the role of the quantum statistics, i.e., Bose-Bose and Bose-Fermi mixtures, is also presented. It is concluded that the variational ansatz, although appearing to be oversimplified, gives surprisingly good predictions of many different quantities for mixtures of equal as well as different mass systems. The result may have some experimental importance since it gives a quite simple and validated method for describing experimental outputs.

DOI
10.1103/physreva.95.053632

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A detailed analysis of a system of four interacting ultracold fermions confined in a one-dimensional harmonic trap is performed. The analysis is done in the framework of a simple variational ansatz for the many-body ground state, and its predictions are compared with the results of numerically exact diagonalization of the many-body Hamiltonian. A short discussion of the role of the quantum statistics, i.e., Bose-Bose and Bose-Fermi mixtures, is also presented. It is concluded that the variational ansatz, although appearing to be oversimplified, gives surprisingly good predictions of many different quantities for mixtures of equal as well as different mass systems. The result may have some experimental importance since it gives a quite simple and validated method for describing experimental outputs.
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Abstract We propose a very accurate and efficient variational scheme for the ground state of the system of p-wave attractively interacting fermions confined in a one-dimensional harmonic trap. By the construction, the method takes the non-analytical part of interactions exactly into account and thus it approximates the true ground-state wave function in a whole range of interactions very accurately. Within the method, we determine different properties of the system for a different number of particles and different interactions. In this way, we explore how the system and its features transit from the ideal non-interacting Fermi gas to the system of infinitely strong attractions. Additionally, we demonstrate that the ansatz may also be used on a repulsive branch of interactions where other numerical methods break down. The presented method of including zero-range interactions is very universal and may be easily generalized to other one-dimensional confinements.
Abstract In this paper we study a mixed system of bosons and fermions with up to six particles in total. All particles are assumed to have the same mass. The two-body interactions are repulsive and are assumed to have equal strength in both the Bose–Bose and the Fermi–Boson channels. The particles are confined externally by a harmonic oscillator one-body potential. For the case of four particles, two identical fermions and two identical bosons, we focus on the strongly interacting regime and analyze the system using both an analytical approach and density matrix renormalization group calculations using a discrete version of the underlying continuum Hamiltonian. This provides us with insight into both the ground state and the manifold of excited states that are almost degenerate for large interaction strength. Our results show great variation in the density profiles for bosons and fermions in different states for strongly interacting mixtures. By moving to slightly larger systems, we find that the ground state of balanced mixtures of four to six particles tends to separate bosons and fermions for strong (repulsive) interactions. On the other hand, in imbalanced Bose–Fermi mixtures we find pronounced odd–even effects in systems of five particles. These few-body results suggest that question of phase separation in one-dimensional confined mixtures are very sensitive to system composition, both for the ground state and the excited states.
We investigate theoretically the behavior of a one-dimensional interacting Bose-Fermi mixture with equal masses and equal repulsive interactions between atoms at finite temperature in the scheme of thermodynamic Bethe ansatz. Combining the Yang-Yang thermodynamic formalism with local-density approximation in a harmonic trap, we calculate the density distribution of bosons and fermions numerically by treating the radially and axially excited states as discrete and continuous ones, respectively. Our result from exactly solvable solutions may be used as a touchstone for one-dimensional interacting Bose-Fermi mixture for experimental data fitting where mean-field theoretical approaches fail.
Standard analytical construction of the many-body wave function of interacting particles in one dimension, beyond mean-field theory, is based on the Jastrow approach. The many-body interacting ground state is build up from the ground state of the non-interacting system and the product of solutions of the corresponding interacting two-body problem. However, this is possible only if the center-of-mass motion is decoupled from the mutual interactions. In our work, based on the general constraints given by contact nature of the atom-atom interactions, we present an alternative approach to the standard construction of the pair-correlation wave function. Within the proposed ansatz, we study the many-body properties of trapped bosons as well as fermionic mixtures and we compare these predictions with the exact diagonalization approach in a wide range of particle numbers, interaction strengths, and different trapping potentials.