# Bosonic Particle-Correlated States: A Nonperturbative Treatment Beyond Mean Field

## TL;DR

This paper introduces particle-correlated states (PCS) for dilute Bose gases, providing a nonperturbative method that captures correlations beyond mean-field theory and efficiently models large systems.

## Contribution

The paper develops a detailed formulation and efficient algorithms for pair-correlated states (PCS), extending beyond mean-field approximations to include particle correlations nonperturbatively.

## Key findings

- PCS accurately predicts ground state energies within 10^{-5} for N=1000 bosons.
- The iterative algorithm for RDMs operates in linear time with respect to N.
- Numerical results indicate PCS's potential for modeling strongly interacting Bose gases.

## Abstract

Many useful properties of dilute Bose gases at ultra-low temperature are predicted precisely by the (mean-field) product-state Ansatz, in which all particles are in the same quantum state. Yet, in situations where particle-particle correlations become important, the product Ansatz fails. To include correlations nonperturbatively, we consider a new set of states: the particle-correlated state of $N=l\times n$ bosons is derived by symmetrizing the $n$-fold product of an $l$-particle quantum state. The particle-correlated states can be simulated efficiently for large $N$, because their parameter spaces, which depend on $l$, do not grow with $n$. Here we formulate and develop in great detail the pure-state case for $l=2$, where the many-body state is constructed from a two-particle pure state. These paired wave functions, which we call pair-correlated states (PCS), were introduced by A. J. Leggett [Rev. Mod. Phys. ${\bf 73}$, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves for the reduced (marginal) density matrices (RDMs), i.e., the correlation functions, associated with PCS in time $O(N)$. The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large-$N$ limit. To test the efficacy of PCS, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS~state, both in its exact form and in its large-$N$ approximate form, and comparing with the exact ground state. For $N=1\,000$, the relative errors of the ground-state energy for both cases are within $10^{-5}$ over the entire parameter region from a single condensate to a Mott insulator. We present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.