Scaling behavior for a class of quantum phase transitions

TL;DR

This paper demonstrates that for certain quantum phase transitions with a single bosonic zero mode, key metric quantities like fidelity exhibit a universal scaling behavior depending only on the ratio of parameters relative to the critical point.

Contribution

It establishes a universal scaling law for metric quantities near quantum critical points with a single bosonic zero mode, applicable to models like the Dicke and Lipkin-Meshkov-Glick models.

Findings

Fidelity depends only on the ratio of parameter differences at criticality.

Scaling applies to time-dependent quantities like Loschmidt echo.

Results are valid for models with a single bosonic zero mode.

Abstract

We show that for quantum phase transitions with a single bosonic zero mode at the critical point, like the Dicke model and the Lipkin-Meshkov-Glick model, metric quantities such as fidelity, that is, the overlap between two ground states corresponding to two values $\lambda_1$ and $\lambda_2$ of the controlling parameter $\lambda$, only depend on the ratio $\eta=(\lambda_1-\lambda_c)/(\lambda_2-\lambda_c)$, where $\lambda=\lambda_c$ at the critical point. Such scaling property is valid also for time-dependent quantities such as the Loschmidt echo, provided time is measured in units of the inverse frequency of the critical mode.