[Paper Review] Correlations in many-body systems with the Stochastic Variational Method

Hans Henrik Brandenborg Sørensen|arXiv (Cornell University)|Feb 4, 2005

Cold Atom Physics and Bose-Einstein Condensates37 references3 citations

TL;DR

This thesis develops and applies the Stochastic Variational Method (SVM) to study many-body correlations in trapped Bose-Einstein condensates (BECs), particularly beyond mean-field approximations. By using explicitly correlated Gaussian basis functions and stochastic optimization, it quantifies the role of two-body and higher-order correlations in few-body systems (N=3,4,10), showing that correlation effects become significant when the diluteness condition $ n|a|^3 \ll 1 $ breaks down, especially for negative scattering lengths near Feshbach resonances.

ABSTRACT

Few-body correlations often express the distinguishing characteristic features of a many-body system. This thesis studies such correlations within dilute Bose-Einstein condensates in the case of arbitrary negative s-wave scattering length. The N-boson problem is solved by using an ab-initio approach based on correlated Gaussians that allows explicit inclusion of few-body correlations with a computational complexity that is independent of the number of particles. Calculations introducing all higher-order correlations are also done for small systems. In the weakly interacting regime, two-body correlations are not only the simplest but also the most important. By varying the scattering length and comparing the ground state energy for different explicitly correlated trial wave functions this assumption is investigated under both weakly and strongly interacting conditions.

Motivation & Objective

• To investigate the role of two-body and higher-order correlations in few-body trapped bosonic systems beyond mean-field approximations.
• To assess the validity of mean-field theories like Gross-Pitaevskii in regimes where $ n|a|^3 \gtrsim 10^{-3} $ or $ |a|N/b_t \gtrsim 0.67 $, especially near Feshbach resonances.
• To develop and implement a numerically robust framework using the Stochastic Variational Method (SVM) for solving N-body quantum systems with explicit correlations.
• To determine whether insights from small systems (N=3,4) can be generalized to larger many-body BEC systems.
• To provide a computational toolset for studying correlation effects in ultracold atomic gases with tunable interactions.

Proposed method

• Employs the Stochastic Variational Method (SVM) with a trial wave function based on correlated Gaussian or exponential basis functions to include explicit two-body and many-body correlations.
• Uses a stochastic 'trial and error' procedure with Gram-Schmidt orthogonalization and iterative refinement to optimize nonlinear parameters in the basis functions.
• Applies symmetrization techniques to ensure proper bosonic or fermionic exchange symmetry in the many-body wave function.
• Implements a generalized eigenvalue solver for the variational problem, using arbitrary-precision arithmetic to maintain numerical accuracy.
• Introduces a recycling strategy and root-finding algorithm to improve convergence and avoid linear dependence in the basis set.
• Uses a C++ implementation with options for different interaction types (Gaussian, Coulomb, zero-range), trap potentials, and scattering length tuning.

Experimental results

Research questions

• RQ1 To what extent do higher-order correlation effects influence the ground state energy and structure of three- and four-boson systems in a trap?
• RQ2 How do two-body and three-body correlations compete or coexist in systems with negative scattering lengths near Feshbach resonances?
• RQ3 Can the Stochastic Variational Method reliably describe the transition from BEC-type to molecular-type states in few-body bosonic systems?
• RQ4 What is the quantitative impact of correlation effects when the mean-field approximation breaks down, particularly for $ n|a|^3 \gtrsim 10^{-3} $ or $ |a|N/b_t \gtrsim 0.67 $?
• RQ5 Can the behavior of small systems (N=3,4) be used to infer general trends in larger many-body BEC systems?

Key findings

• For $ N=3 $ and $ -\infty < a < 0 $, the SVM reveals significant two-body correlations and the formation of molecular bound states, especially as $ |a| $ increases.
• For $ N=4 $, the system exhibits strong correlation effects, with energy shifts and structural changes indicating the onset of many-body correlations beyond mean-field predictions.
• In the $ N=10 $ system with $ -0.2 \, b_t < a < 0 $, the results indicate that higher-order correlations become increasingly important as the system approaches the molecular instability regime.
• The method successfully identifies the ground state of the helium atom and reproduces known results, validating its accuracy for few-body systems.
• The numerical implementation demonstrates that the SVM can handle systems with up to $ N=10 $ bosons, though computational cost remains high, limiting larger-scale studies.
• The program output confirms that the scattering length and energy are sensitive to the choice of basis and optimization parameters, highlighting the need for careful numerical control.