# Pressure dependence of ferroelectric quantum critical fluctuations

## TL;DR

This study investigates how pressure influences ferroelectric quantum critical fluctuations in SrTiO$_3$, confirming mean-field behavior and observing phonon mode stiffening as the system moves away from criticality, with implications for understanding quantum phase transitions.

## Contribution

First experimental measurement of soft phonon mode stiffening under pressure away from quantum criticality, validating theoretical models without adjustable parameters.

## Key findings

- Mean-field approximation confirmed experimentally.
- Logarithmic corrections are observable very close to the quantum critical point.
- Soft mode stiffens with pressure, indicating a move away from criticality.

## Abstract

The effect of ferroelectric fluctuations on the temperature dependent dielectric constant of SrTiO$_3\,$(STO) has been long studied. Those fluctuations have been shown in recent years to be quantum critical and STO demonstrated to form the archetypal quantum critical paraelectric. The effect of those same fluctuations on the pressure and temperature dependence of the ferroelectric soft phonon mode as the system is tuned away from criticality are reported for the first time in this paper. We show that the mean field approximation is confirmed experimentally. Furthermore, using a self-consistent model of the quantum critical excitations including coupling to the volume strain and without adjustable parameters, we determine logarithmic corrections that would be observable only very close to the quantum critical point. Thus, the mean-field character of the pressure dependence is much more robust to the fluctuations than is the temperature dependence. We predict stronger corrections for lower dimensionalities however. The same calculation confirms that the Lydanne-Sachs-Teller relation is valid over the whole pressure and temperature range considered. Therefore, the measured dielectric constant can be used to extract the frequency of the soft mode down to 1.5 K and up to 20 kbar of applied pressure. The soft mode is observed to stiffen further, raising the low-temperature energy gap and returning towards the expected shallow temperature dependence of an optical mode. This behavior is consistent with the existence of a quantum critical point on the pressure-temperature phase diagram of STO, which applied pressure tunes the system away from. This work represents the first experimental measurement of the stiffening of a soft phonon mode as a system is tuned away from criticality, a potentially universal phenomenon across a variety of phase transitions and systems in condensed matter physics.