# Quantum effects in muon spin spectroscopy within the stochastic self-consistent harmonic approximation

## TL;DR

This paper introduces a quantum variational approach using the stochastic self-consistent harmonic approximation to accurately model the anharmonic vibrational effects of muons in materials, improving agreement with experimental data.

## Contribution

It applies the stochastic self-consistent harmonic approximation to include anharmonic quantum effects in muon spin spectroscopy calculations, advancing beyond harmonic approximations.

## Key findings

- Muon vibrational frequencies are strongly renormalized by anharmonicity.
- Including quantum anharmonic fluctuations stabilizes the muon at the octahedral site in bcc Fe.
- The method significantly improves agreement with experimental hyperfine field measurements.

## Abstract

In muon spin rotation experiments the positive implanted muon vibrates with large zero point amplitude by virtue of its light mass. Quantum mechanical calculations of the host material usually treat the muon as a point impurity, ignoring this large vibrational amplitude. As a first order correction, the muon zero point motion is usually described within the harmonic approximation, despite the large anharmonicity of the crystal potential. Here we apply the stochastic self-consistent harmonic approximation, a quantum variational method devised to include strong anharmonic effects in total energy and vibrational frequency calculations, in order to overcome these limitations and provide an accurate ab initio description of the quantum nature of the muon. We applied this full quantum treatment to the calculation of the muon contact hyperfine field in textbook-case metallic systems, such as Fe, Ni, Co including MnSi and MnGe, significantly improving agreement with experiments. Our results show that muon vibrational frequencies are strongly renormalized by anharmonicity. Finally, in contrast to the harmonic approximation, we show that including quantum anharmonic fluctuations, the muon stabilizes at the octahedral site in bcc Fe.