Manifestation of structural Higgs and Goldstone modes in the hexagonal manganites

Structural phase transitions described by Mexican hat potentials should in principle exhibit aspects of Higgs and Goldstone physics. Here, we investigate the relationship between the phonons that soften at such structural phase transitions and the Higgs- and Goldstone-boson analogs associated with the crystallographic Mexican hat potential. We show that, with the exception of systems containing only one atom type, the usual Higgs and Goldstone modes are represented by a combination of several phonon modes, with the lowest-energy phonons of the relevant symmetry having substantial contribution. Taking the hexagonal manganites as a model system, we identify these modes using Landau theory, and predict the temperature dependence of their frequencies using parameters obtained from density functional theory. Separately, we calculate the additional temperature dependence of all phonon mode frequencies arising from thermal expansion within the quasiharmonic approximation. We predict that Higgs-mode softening will dominate the low-frequency vibrational spectrum of ${\mathrm{InMnO}}_3$ between zero Kelvin and room temperature, whereas the behavior of ${\mathrm{ErMnO}}_3$ will be dominated by lattice expansion effects. We present temperature-dependent Raman scattering data that support our predictions, in particular confirming the existence of the Higgs mode in ${\mathrm{InMnO}}_3$.

Figure 1

The Mexican hat potential describing a broken $\text{U}\left(1\right)$ symmetry, with the Higgs and Goldstone modes indicated.

Figure 2

(a) Crystal structure of the hexagonal manganites, black arrows indicating the tilt amplitude $Q$ and orange circle indicating the tilt angle $\theta$ of the oxygen-manganese bipyramids. The oxygens are marked red, the manganese atoms are purple, and the $R$ atoms are green. (b) Mexican hat free-energy landscape showing the energy as a function of amplitude ($Q$) and phase ($\theta$) of the order parameter. The $\text{U}\left(1\right)$-like symmetry at small order parameter values, and the six discrete minima at large order parameter values can be clearly seen. The fluctuations in the amplitude $\delta q$ (Higgs) and in the phase $\delta \theta$ (Goldstone) are indicated with red and orange arrows.

Figure 3

Calculated evolution of phonon frequencies from the Landau theory approach for (a) ferroelectric ${\mathrm{InMnO}}_3$ and (b) ${\mathrm{ErMnO}}_3$. Modes changing due to freeze-in of the primary order parameter are marked red (Goldstone, B1) and blue (Higgs, A1), while the phonon modes related to the polar instability (A1) are marked green. Shaded areas mark the temperature range accessible by our Raman experiments.

Figure 4

(a) Lattice parameters for ${\mathrm{ErMnO}}_3$ calculated within the quasiharmonic approximation. (b) Temperature dependence of the phonon frequencies calculated for ${\mathrm{ErMnO}}_3$ within the quasiharmonic approximation. We show only the A1 (red) and E2 (blue) modes for comparison with the Raman spectroscopy measurements in the next section.

Figure 5

Raman spectra collected at 10 K on single crystals of (a) ${\mathrm{ErMnO}}_3$, (b) ${\mathrm{InMnO}}_3$ ($\mathrm{P}\overline{3}\mathrm{c}1$), and (c) ${\mathrm{InMnO}}_3$ (${\mathrm{P}6}_3\mathrm{cm}$). The red and black lines show the intensity of Raman scattering in the parallel and perpendicular configurations, respectively. The polarization selection rule for the ${\mathrm{ErMnO}}_3$ sample ($P{6}_{3}cm$ symmetry) is clearly manifested.

Figure 6

Raman spectra at selected temperatures from 4 to 300 K for three hexagonal manganite samples: (a) ${\mathrm{ErMnO}}_3$, (b) ${\mathrm{InMnO}}_3$ ($\mathrm{P}\overline{3}\mathrm{c}1$), and (c) ${\mathrm{InMnO}}_3$ (${\mathrm{P}6}_3\mathrm{cm}$). The colored triangles mark calculated values from our DFT calculations. (d)–(f) Temperature dependence of the mode frequencies for the three samples: (d) ${\mathrm{ErMnO}}_3$, (e) ${\mathrm{InMnO}}_3$ ($\mathrm{P}\overline{3}\mathrm{c}1$), and (f) ${\mathrm{InMnO}}_3$ (${\mathrm{P}6}_3\mathrm{cm}$). The black vertical bars indicate the peak widths.