Evidence for Goldstone-like and Higgs-like structural modes in the model $\mathrm{Pb}{\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}{\mathrm{O}}_3$ relaxor ferroelectric

Effective Hamiltonian simulations are conducted to unveil the nature of the low-frequency polar modes in the prototype relaxor ferroelectric, $\mathrm{Pb}\left({\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}\right){\mathrm{O}}_3$. Above the so-called $T^{*}$ temperature, only a single soft-mode exists, with its frequency increasing under heating. On the other hand, for temperatures lower than the freezing temperature, this single soft-mode splits into two modes, with one mode slightly changing its low resonant frequency while the other exhibiting a resonant frequency sharply increasing under cooling, in agreement with previous measurements. More importantly, we present evidences that these two modes can be regarded as Goldstone-like and Higgs-like modes, inherent to the Mexican-hat-form of the atomic displacement potential we also reveal here, therefore extending the types of systems exhibiting Higgs-boson characteristics to relaxor ferroelectrics.

Figure 1

Imaginary part of the dielectric response of PMN as a function of frequency at (a) 700 and (b) 50 K, along with the imaginary part of the (c) “local Goldstone” and (d) “local Higgs” responses at 50 K. The symbols show raw MD data, while the solid line displays the fit by one DHO. The inset in panel b represents the difference in the imaginary part between the raw dielectric response and its fit by one DHO at 50 K. Note that we do not show the interval below 50 ${\mathrm{cm}}^{-1}$ in the inset of panel (b) and in panel (d) to focus only on the numerically reliable high-frequency response.

Figure 2

Computed resonant frequencies in PMN as a function of temperature, as extracted from the total dielectric response. Experimental measurements are presented via the following open symbols: star [27], circle [15], pentagon [17], hexagon [10], and triangle [13]. The inset shows temperature dependence of the damping constants.

Figure 3

Data related to potential for the temperature of 200 and 50 K. (a), (b) Dependence of $-Ln\rho \left(u_x,u_y,u_z\right)$ on the amplitude of the local mode $\left|u\right|$ plotted along different crystallographic directions. The blue and yellow curves correspond to the local mode vectors being oriented along the pseudocubic $〈111〉$ and $〈100〉$ directions, respectively. Curves corresponding to other orientations of this local mode vector fill the shaded area. (c), (d) Isopotential lines of the effective potential in a (001) plane; and (e), (f) Effective potential $-Ln\rho \left(u_x{,u}_y,0\right)$ as a function of the $x$ and $y$ components of the local modes. Note that we use in panels (c) and (d) orange and navy colors for marking the maximum and minimum of the potential correspondingly.