# Bose - Einstein condensation of triplons with a weakly broken U(1) symmetry

## TL;DR

This paper develops a self-consistent mean-field approach to describe Bose-Einstein condensation of triplons in quantum magnets with weakly broken U(1) symmetry, successfully fitting experimental data and predicting significant shifts in critical temperature due to anisotropy.

## Contribution

It introduces a self-consistent MFA method accounting for broken U(1) symmetry, improving upon previous approaches and aligning with experimental observations.

## Key findings

- Magnetization curves and energy dispersion fit well with the new model.
- Weak anisotropy causes a notable increase in critical temperature.
- Predicted shift in Tc can be around 10% for small anisotropy parameters.

## Abstract

The low-temperature properties of certain quantum magnets can be described in terms of a Bose-Einstein condensation (BEC) of magnetic quasiparticles (triplons). Some mean-field approaches (MFA) to describe these systems, based on the standard grand canonical ensemble, do not take the anomalous density into account and leads to an internal inconsistency, as it has been shown by Hohenberg and Martin, and may therefore produce unphysical results. Moreover, an explicit breaking of the U(1) symmetry as observed, for example, in TlCuCl3 makes the application of MFA more complicated. In the present work, we develop a self-consistent MFA approach, similar to the Hartree-Fock-Bogolyubov approximation in the notion of representative statistical ensembles, including the effect of a weakly broken U(1) symmetry. We apply our results on experimental data of the quantum magnet TlCuCl3 and show that magnetization curves and the energy dispersion can be well described within this approximation assuming that the BEC scenario is still valid. We predict that the shift of the critical temperature Tc due to a finite exchange anisotropy is rather substantial even when the anisotropy parameter \gamma is small, e.g., \Delta T_c \approx 10%$ of Tc in H = 6T and for \gamma\approx 4 \mu eV.