Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics B Atomic Molecular and Optical Physics

An optimized description of a confined interacting Fermi system

Rocío Jáuregui, R. Paredes, L. Rosales-Zárate, G. Toledo Sánchez

Published Mar 2, 2010• 4 citations
Publisher

Abstract

We provide an efficient form to express the action of a many-body Hamiltonian of harmonically trapped interacting Fermi particles on wavefunctions built from paired states. The expression is suitable to numerically determine the ground state energy, regardless of the form of the two-body interaction. It takes advantage of the knowledge of the two-particle problem and the inherent properties of the matrix form of the many-body wavefunction. As an example, we evaluate the properties of a system composed of a balanced mixture of two families of fermions confined in a harmonic trap interacting through a short-range exponential potential. Numerical results for N ⩽ 10 and N = 35, 56, 84 and 165 particles of each family are reported. In the strong interacting regime corresponding to an infinite s-wave scattering length, our results give an upper bound to the Bertsch parameter for harmonically trapped systems ( E / E IFG ) 2 = 1 + β ⩽ 0.376 ± 0.008 with E the total energy and E IFG the energy for the analogous ideal Fermi gas. The influence of the harmonic trap and the interaction potential is exhibited in one and two-body correlation functions.

DOI
10.1088/0953-4075/43/6/065301

Related papers

Comparing numerical and analytical approaches to strongly interacting Fermi-Fermi and Bose-Fermi mixtures in one dimensional traps

F. F. Bellotti, Amin S. Dehkharghani, Nikolaj Thomas Zinner

We investigate one-dimensional harmonically trapped two-component systems for repulsive interaction strengths ranging from the non-interacting to the strongly interacting regime for Fermi-Fermi and Bose-Fermi mixtures. Energies, density profiles and correlation functions are obtained both numerically (DMRG and Exact diagonalization) and analytically. In order to facilitate a direct comparison with DMRG, we introduce a mapping of the energies calculated from a discrete lattice Hamiltonian to that of a continuous Hamiltonian, and show that our DMRG calculations can match the ground state energy for the two-component fermion systems even in the strongly interacting limit. The same is not true for the other features of the ground state and the DMRG method we employ produces unrealistic results when the interaction strength becomes large. Since DMRG results do not converge as the interaction strength is increased, analytical solutions are used as a benchmark to identify the point where the strongly interacting limit is reached numerically in which reasonable agreement between DMRG and analytical results can be obtained. In the case of Bose-Fermi mixtures, we find that the ground state of balanced mixtures of four or six particles tends to separate bosons and fermions, while in imbalanced Bose-Fermi mixtures we find pronounced odd-even effects in systems of four or five particles. These few-body results suggest that question of phase separation in one-dimensional confined mixtures are very sensitive to system composition.

•Jun 30, 2016•0 citations

Energetics and structural properties of trapped two-component Fermi gases

J. von Stecher, Chris H. Greene, D. Blume

Using two different numerical methods, we study the behavior of two-component Fermi gases interacting through short-range $s$-wave interactions in a harmonic trap. A correlated Gaussian basis-set expansion technique is used to determine the energies and structural properties, i.e., the radial one-body densities and pair distribution functions, for small systems with either even or odd $N$, as functions of the $s$-wave scattering length and the mass ratio $\ensuremath{\kappa}$ of the two species. Particular emphasis is put on a discussion of the angular momentum of the system in the BEC-BCS crossover regime. At unitarity, the excitation spectrum of the four-particle system with total angular momentum $L=0$ is calculated as a function of the mass ratio $\ensuremath{\kappa}$. The results are analyzed from a hyperspherical perspective, which offers unique insight into the problem. Additionally, fixed-node diffusion Monte Carlo calculations are performed for equal-mass Fermi gases with up to $N=30$ atoms. We focus on the odd-even oscillations of the ground-state energy of the equal-mass unitary system having up to $N=30$ particles, which are related to the excitation gap of the system. Furthermore, we present a detailed analysis of the structural properties of these systems.

arXiv:0801.2747•Physical Review A•Apr 16, 2008•84 citations

Variational ansatz for p-wave fermions confined in a one-dimensional harmonic trap

Przemysław Kościk, Tomasz Sowiński

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.

arXiv:2004.06610•New Journal of Physics•Aug 28, 2020•9 citations

SU( N ) fermions in a one-dimensional harmonic trap

Emma Laird, Zhe-Yu Shi, Meera M. Parish +1 more

We conduct a theoretical study of $\mathrm{SU}(N$) fermions confined by a one-dimensional harmonic potential. First, we introduce a numerical approach for solving the trapped interacting few-body problem, by which one may obtain accurate energy spectra across the full range of interaction strengths. In the strong-coupling limit, we map the $\mathrm{SU}(N$) Hamiltonian to a spin-chain model. We then show that an existing, extremely accurate ansatz---derived for a Heisenberg SU(2) spin chain---is extendable to these $N$-component systems. Lastly, we consider balanced $\mathrm{SU}(N$) Fermi gases that have an equal number of particles in each spin state for $N=2,\phantom{\rule{0.16em}{0ex}}3,\phantom{\rule{0.16em}{0ex}}4$. In the weak- and strong-coupling regimes, we find that the ground-state energies rapidly converge to their expected values in the thermodynamic limit with increasing atom number. This suggests that the many-body energetics of $N$-component fermions may be accurately inferred from the corresponding few-body systems of $N$ distinguishable particles.

arXiv:1707.07781•Physical Review A•Sep 1, 2017•30 citations

Energetic and structural properties of two-dimensional trapped mesoscopic Fermi gases

Emma Laird, Brendan C. Mulkerin, Jia Wang +1 more

We theoretically investigate equal-mass spin-balanced two-component Fermi gases in which pairs of atoms with opposite spins interact via a short-range isotropic model potential. We probe the distinction between two-dimensional and quasi-two-dimensional harmonic confinement by tuning the effective range parameter within two-dimensional scattering theory. Our approach, which yields numerically exact energetic and structural properties, combines a correlated Gaussian basis-set expansion with the stochastic variational method. For systems containing up to six particles, we: 1) Present the ground- and excited-state energy spectra; 2) Study non-local correlations by analysing the one- and two-body density matrices, extracting from these the occupation numbers of natural orbitals, the momentum distributions of atoms and pairs, and the molecular `condensate fraction’; 3) Study local correlations by computing the radial and pair distribution functions. This paper extends current theoretical knowledge on the properties of trapped few-fermion systems as realised in state-of-the-art cold-atom experiments.

arXiv:2506.20891•SciPost Physics Core•Apr 9, 2026•0 citations

Cooper Pairs in 2D Trapped Atoms Interacting Through Finite-Range Potentials

Erick Manuel Pineda-Ríos, R. Paredes

This work deals with the key constituent behind the existence of superfluid states in ultracold fermionic gases confined in a harmonic trap in 2D, namely, the formation of Cooper pairs in the presence of a Fermi sea in inhomogeneous confinement. For a set of finite-range models representing particle–particle interaction, we first ascertain the simultaneity of the emergence of bound states and the divergence of the s-wave scattering length in 2D as a function of the interaction potential parameters in free space. Then, through the analysis of two particles interacting in 2D harmonic confinement, we evaluate the energy shift with respect to the discrete harmonic oscillator levels for both repulsive and attractive cases. All of these results are the basis for determining the energy gaps of Cooper pairs arising from two particles interacting in the presence of a Fermi sea consisting of particles immersed in a 2D harmonic trap.

Atoms•Jan 7, 2025•0 citations
We provide an efficient form to express the action of a many-body Hamiltonian of harmonically trapped interacting Fermi particles on wavefunctions built from paired states. The expression is suitable to numerically determine the ground state energy, regardless of the form of the two-body interaction. It takes advantage of the knowledge of the two-particle problem and the inherent properties of the matrix form of the many-body wavefunction. As an example, we evaluate the properties of a system composed of a balanced mixture of two families of fermions confined in a harmonic trap interacting through a short-range exponential potential. Numerical results for N ⩽ 10 and N = 35, 56, 84 and 165 particles of each family are reported. In the strong interacting regime corresponding to an infinite s-wave scattering length, our results give an upper bound to the Bertsch parameter for harmonically trapped systems ( E / E IFG ) 2 = 1 + β ⩽ 0.376 ± 0.008 with E the total energy and E IFG the energy for the analogous ideal Fermi gas. The influence of the harmonic trap and the interaction potential is exhibited in one and two-body correlation functions.
We investigate one-dimensional harmonically trapped two-component systems for repulsive interaction strengths ranging from the non-interacting to the strongly interacting regime for Fermi-Fermi and Bose-Fermi mixtures. Energies, density profiles and correlation functions are obtained both numerically (DMRG and Exact diagonalization) and analytically. In order to facilitate a direct comparison with DMRG, we introduce a mapping of the energies calculated from a discrete lattice Hamiltonian to that of a continuous Hamiltonian, and show that our DMRG calculations can match the ground state energy for the two-component fermion systems even in the strongly interacting limit. The same is not true for the other features of the ground state and the DMRG method we employ produces unrealistic results when the interaction strength becomes large. Since DMRG results do not converge as the interaction strength is increased, analytical solutions are used as a benchmark to identify the point where the strongly interacting limit is reached numerically in which reasonable agreement between DMRG and analytical results can be obtained. In the case of Bose-Fermi mixtures, we find that the ground state of balanced mixtures of four or six particles tends to separate bosons and fermions, while in imbalanced Bose-Fermi mixtures we find pronounced odd-even effects in systems of four or five particles. These few-body results suggest that question of phase separation in one-dimensional confined mixtures are very sensitive to system composition.
Using two different numerical methods, we study the behavior of two-component Fermi gases interacting through short-range sss-wave interactions in a harmonic trap. A correlated Gaussian basis-set expansion technique is used to determine the energies and structural properties, i.e., the radial one-body densities and pair distribution functions, for small systems with either even or odd NNN, as functions of the sss-wave scattering length and the mass ratio $\ensuremath{\kappa}$ of the two species. Particular emphasis is put on a discussion of the angular momentum of the system in the BEC-BCS crossover regime. At unitarity, the excitation spectrum of the four-particle system with total angular momentum L=0L=0L=0 is calculated as a function of the mass ratio $\ensuremath{\kappa}$. The results are analyzed from a hyperspherical perspective, which offers unique insight into the problem. Additionally, fixed-node diffusion Monte Carlo calculations are performed for equal-mass Fermi gases with up to N=30N=30N=30 atoms. We focus on the odd-even oscillations of the ground-state energy of the equal-mass unitary system having up to N=30N=30N=30 particles, which are related to the excitation gap of the system. Furthermore, we present a detailed analysis of the structural properties of these systems.
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.
We conduct a theoretical study of SU(N\mathrm{SU}(NSU(N) fermions confined by a one-dimensional harmonic potential. First, we introduce a numerical approach for solving the trapped interacting few-body problem, by which one may obtain accurate energy spectra across the full range of interaction strengths. In the strong-coupling limit, we map the SU(N\mathrm{SU}(NSU(N) Hamiltonian to a spin-chain model. We then show that an existing, extremely accurate ansatz---derived for a Heisenberg SU(2) spin chain---is extendable to these NNN-component systems. Lastly, we consider balanced SU(N\mathrm{SU}(NSU(N) Fermi gases that have an equal number of particles in each spin state for N=2,3,4N=2,\phantom{\rule{0.16em}{0ex}}3,\phantom{\rule{0.16em}{0ex}}4N=2,3,4. In the weak- and strong-coupling regimes, we find that the ground-state energies rapidly converge to their expected values in the thermodynamic limit with increasing atom number. This suggests that the many-body energetics of NNN-component fermions may be accurately inferred from the corresponding few-body systems of NNN distinguishable particles.
We theoretically investigate equal-mass spin-balanced two-component Fermi gases in which pairs of atoms with opposite spins interact via a short-range isotropic model potential. We probe the distinction between two-dimensional and quasi-two-dimensional harmonic confinement by tuning the effective range parameter within two-dimensional scattering theory. Our approach, which yields numerically exact energetic and structural properties, combines a correlated Gaussian basis-set expansion with the stochastic variational method. For systems containing up to six particles, we: 1) Present the ground- and excited-state energy spectra; 2) Study non-local correlations by analysing the one- and two-body density matrices, extracting from these the occupation numbers of natural orbitals, the momentum distributions of atoms and pairs, and the molecular `condensate fraction’; 3) Study local correlations by computing the radial and pair distribution functions. This paper extends current theoretical knowledge on the properties of trapped few-fermion systems as realised in state-of-the-art cold-atom experiments.
This work deals with the key constituent behind the existence of superfluid states in ultracold fermionic gases confined in a harmonic trap in 2D, namely, the formation of Cooper pairs in the presence of a Fermi sea in inhomogeneous confinement. For a set of finite-range models representing particle–particle interaction, we first ascertain the simultaneity of the emergence of bound states and the divergence of the s-wave scattering length in 2D as a function of the interaction potential parameters in free space. Then, through the analysis of two particles interacting in 2D harmonic confinement, we evaluate the energy shift with respect to the discrete harmonic oscillator levels for both repulsive and attractive cases. All of these results are the basis for determining the energy gaps of Cooper pairs arising from two particles interacting in the presence of a Fermi sea consisting of particles immersed in a 2D harmonic trap.