# Reproduction of exact solutions of Lipkin model by nonlinear random-phase approximation

## TL;DR

This paper demonstrates that a nonlinear extension of the random-phase approximation (RPA) can exactly reproduce solutions of the Lipkin model, suggesting its potential for accurate many-body problem calculations.

## Contribution

The authors show that nonlinear RPA reproduces exact solutions of the Lipkin model, establishing its equivalence to the Schrödinger equation for this system.

## Key findings

- Exact solutions for N=2 analytically derived
- Numerical validation for N=20
- Nonlinear RPA matches the Schrödinger equation solutions

## Abstract

It is shown that the random-phase approximation (RPA) method with its nonlinear generalization, which was previously considered as approximation, reproduces the exact solutions of the Lipkin model. The nonlinear RPA is based on an equation nonlinear on eigenvectors and includes many-particle-many-hole components in the creation operator of the excited states. We demonstrate the exact character of solutions analytically for the particle number $N$ = 2 and, numerically, for $N$ = 20. This finding indicates that the nonlinear RPA is equivalent to the exact Schr\"{o}dinger equation, which opens up new possibilities for realistic calculations in many-body problems.