# Efficient algorithms for the dynamics of large and infinite classical central spin models

## TL;DR

This paper develops efficient algorithms to analyze the long-time dynamics of large classical central spin models, enabling accurate predictions of spin correlations in quantum dots with many nuclear spins.

## Contribution

The authors introduce a set of algorithms using orthogonal polynomial representations to compute long-time spin correlations in large and infinite central spin models.

## Key findings

- Algorithms accurately predict long-time correlations up to 10^5ħ/J_Q.
- Reduced differential equations suffice for large bath sizes.
- Benchmarking confirms the algorithms' accuracy with small bath data.

## Abstract

We investigate the time dependence of correlation functions in the central spin model, which describes the electron or hole spin confined in a quantum dot, interacting with a bath of nuclear spins forming the Overhauser field. For large baths, a classical description of the model yields quantitatively correct results. We develop and apply various algorithms in order to capture the long-time limit of the central spin for bath sizes from 1000 to infinitely many bath spins. Representing the Overhauser field in terms of orthogonal polynomials, we show that a carefully reduced set of differential equations is sufficient to compute the spin correlations of the full problem up to very long times, for instance up to $10^5\hbar/J_\mathrm{Q}$ where $J_\mathrm{Q}$ is the natural energy unit of the system. This technical progress renders an analysis of the model with experimentally relevant parameters possible. We benchmark the results of the algorithms with exact data for a small number of bath spins and we predict how the long-time correlations behave for different effective numbers of bath spins.