[Paper Review] Configuration interaction approach to the few-body problem in a two-dimensional harmonic trap with contact interaction

Massimo Rontani, Sarah Reimann|arXiv (Cornell University)|Oct 23, 2008

Cold Atom Physics and Bose-Einstein Condensates33 references3 citations

TL;DR

This paper proposes a renormalized configuration interaction (CI) method to overcome convergence issues in few-body quantum systems with contact interactions in 2D harmonic traps. By diagonalizing a strength-renormalized contact potential in a truncated Hilbert space and calibrating against exact two-body solutions, the method yields cut-off-independent physical observables and quantifies CI calculation errors, enabling reliable few-body calculations despite the pathological nature of the Dirac delta potential.

ABSTRACT

The configuration interaction (CI) method for calculating the exact eigenstates of a quantum-mechanical few-body system is problematic when applied to contact interactions between the particles. In two and three dimensions, the approach fails due to the pathology of the Dirac delta-potential, making it impossible to reach convergence by gradually increasing the size of the Hilbert space. However, for practical applications this problem may be cured in a rather simple manner, by renormalizing the STRENGTH of the contact potential, which must be diagonalized in a TRUNCATED Hilbert space. The procedure relies on the comparison of CI energies and wave functions with those obtained by the exact solution of the two-body Schrodinger equation for the regularized contact interaction. The rather simple scheme, while keeping the numerical procedures still elementary, nevertheless provides both cut-off-independent few-body physical observables and an estimate of the error of the CI calculation.

Motivation & Objective

• To address the failure of standard configuration interaction (CI) methods in converging for contact interactions in two-dimensional quantum systems.
• To resolve the pathological behavior of the Dirac delta potential in 2D, which prevents convergence as the Hilbert space is enlarged.
• To develop a practical, numerically simple scheme that restores convergence and yields physically meaningful observables.
• To provide a systematic error estimate for CI calculations in few-body systems with contact interactions.

Proposed method

• The method introduces a strength-renormalized contact interaction to replace the singular Dirac delta potential.
• The renormalized contact potential is diagonalized within a truncated Hilbert space to compute eigenstates and energies.
• The strength of the contact interaction is calibrated by comparing CI results with the exact solution of the two-body Schrödinger equation for a regularized contact interaction.
• The procedure ensures that physical observables become independent of the momentum-space cut-off used in the regularization.
• The method relies on the comparison of CI energies and wave functions with exact two-body solutions to validate and tune the approach.
• The resulting scheme maintains computational simplicity while achieving reliable, cut-off-independent results.

Experimental results

Research questions

• RQ1 Can the configuration interaction method be made convergent for few-body systems with contact interactions in two dimensions?
• RQ2 How can the pathological divergence caused by the Dirac delta potential be remedied in a numerically tractable way?
• RQ3 What is the role of contact interaction strength renormalization in achieving cut-off-independent physical observables?
• RQ4 To what extent can the error in CI calculations be quantified using exact two-body solutions as a reference?
• RQ5 Can a simple, practical scheme be developed that preserves accuracy without requiring complex many-body techniques?

Key findings

• The renormalized CI method successfully achieves convergence for few-body systems with contact interactions in two-dimensional harmonic traps.
• Physical observables computed with the method are independent of the momentum-space cut-off, indicating robustness and reliability.
• The comparison with exact two-body solutions enables accurate calibration of the contact interaction strength.
• The method provides a quantitative estimate of the error inherent in the CI calculation, improving its predictive power.
• The approach maintains computational simplicity while overcoming the fundamental limitations of standard CI for contact interactions.