Spontaneous Symmetry Breaking and Collective Higgs-Goldstone Dynamics in Solid-State Phononic Frequency Combs

TL;DR

This work addresses the generation of phononic frequency combs via nonlinear coupling between Higgs‑like and Goldstone‑like phonon modes in hexagonal InMnO$_3$, driven by a short, high‑field THz pulse. It develops a nonlinear phononics model with two coupled coordinates $x_1$ and $x_2$, driven by $E_0$, with envelope $E_0 \exp\left[-\frac{4\ln 2}{\tau^2} t^2\right]\cos(\omega_D t)$, and analyzes the system using an implicit Radau integrator to map how $\tau$, $E_0$, $\omega_D$, and damping $Q$ shape comb formation. The results show clear threshold behaviors for comb onset, tunable comb spacing and bandwidth, and eventual breakdown of coherence at high drive or weak damping, highlighting the critical role of Higgs–Goldstone coupling in enabling controllable phononic spectra. These findings offer a route to ultrafast, phonon‑based spectral control in symmetry‑broken solids and advance understanding of lattice dynamics under strong nonlinear driving.

Abstract

We investigate the generation of phononic frequency combs arising from nonlinear coupling between Higgs-like and Goldstone-like phonon modes in hexagonal InMnO3. The Higgs-like mode, an infrared-active optical phonon, is resonantly driven by a short, high-electric field terahertz pulse, while the optically inactive Goldstone-like mode is indirectly excited through intrinsic nonlinear mode coupling. Using a nonlinear phononics model, we numerically solve the coupled equations of motion governing the lattice dynamics and analyze the resulting time- and frequency-domain responses. By systematically varying key drive and material parameters-including electric field amplitude, pulse width, driving frequency, and mode damping-we identify the conditions under which stable phononic frequency combs emerge. Our results reveal clear threshold behaviors for comb formation, tunability of comb spacing and spectral bandwidth through external control parameters, and a breakdown of coherent comb structure at high drive strengths or weak damping. These findings demonstrate how nonlinear Higgs-Goldstone interactions enable controllable phononic frequency comb generation and provide insight into ultrafast lattice dynamics in symmetry-broken materials.

Figures (6)

• Figure 1: Time-domain dynamics of the coupled Higgs-like and Goldstone-like phonon modes in hexagonal InMnO$_3$. The figure shows the displacement $x_1(t)$ of the infrared-active Higgs mode driven by a terahertz pulse, $x_2(t)$ of the Goldstone-like mode excited through nonlinear coupling.(a) shows the complete time domain response spanning 100ps, (b) shows the steady state response, the last 20% of the time doamin response, refer TABLE II for the values used.
• Figure 2: Frequency-domain dynamics of the coupled Higgs-like and Goldstone-like phonon modes. (a) shows the $x_1(t)$ mode and (b) shows the $x_2(t)$ mode. Refer TABLE II for the values used.
• Figure 3: Contour plots showing the peak electric field amplitude $E_0$ dependence of the frequency-domain dynamics of the coupled Higgs-like and Goldstone-like phonon modes. Panel (a) shows the response of the Higgs-like mode $x_1$, while panel (b) shows the response of the Goldstone-like mode $x_2$. Only the peak electric field amplitude $E_0$ is varied within the range $\left[26.9507,\,46.6454\right]$, while all other parameters are fixed to their default values (see TABLE II).
• Figure 4: Contour plots showing the dependence of the frequency-domain dynamics of the coupled Higgs-like and Goldstone-like phonon modes on the pulse width $\tau$. Panel (a) shows the response of the Higgs-like mode $x_1$, while panel (b) shows the response of the Goldstone-like mode $x_2$. Only $\tau$ is varied within the range $[0.5,\,2.0]~\mathrm{ps}$, while all other parameters are fixed to their default values (see Table II).
• Figure 5: Contour plots showing the dependence of the frequency-domain dynamics of the coupled Higgs-like and Goldstone-like phonon modes on the driving frequency $f_D$. Panel (a) shows the response of the Higgs-like mode $x_1$, while panel (b) shows the response of the Goldstone-like mode $x_2$. Only $f_D$ is varied within the range $\left[3.4,\,4.6\right]~\mathrm{THz}$, while all other parameters are fixed to their default values (see Table II).