Dynamical phase transition in a strongly hybridized phonon-triplon chain

TL;DR

The paper analyzes a dimerized spin-1/2 chain with strong spin-phonon coupling under weak THz driving, focusing on nonequilibrium steady states formed via dissipation. Using a mean-field Lindblad framework and bond-operator formalism, it shows that strong phonon-triplon hybridization near the two-triplon continuum drives sharp dynamical first-order transitions in triplon emission, confirmed by Floquet analysis of harmonic modes. It maps a comprehensive steady-state phase diagram as a function of drive frequency, phonon frequency, and damping, and provides experimentally relevant parameters for observing the transitions in laser-driven CuGeO3. The results reveal a controllable, dissipation-enabled nonlinear dynamical regime in strongly hybridized spin-phonon systems with potential for tunable quantum functionalities.

Abstract

We study a dimerized spin-1/2 chain, such as CuGeO$_3$, hosting triplon excitations coupled to optical phonons under weak terahertz laser driving. Both phonons and triplons weakly lose energy into the surrounding baths, forming a non-equilibrium steady state. In the strong phonon-triplon coupling regime, phonons near the two-triplon continuum hybridize strongly with triplons. Using mean-field Lindblad dynamics, we show that this strong hybridization induces sharp first-order phase transitions -- either single or simultaneous double -- in the emission spectrum, mainly due to dissipation-induced nonlinearities. Using mean-field Floquet analysis of harmonic modes in both sectors, we analytically confirm the existence of these phase transitions. Furthermore, we map the complete steady-state phase diagram by varying key control parameters and provide experimentally relevant parameters for observing these transitions in laser-driven CuGeO$_3$.

Figures (5)

• Figure 1: Schematic illustration of a driven-dissipative phonon-triplon chain, exemplified by $\mathrm{CuGeO}_3$. The spin chain exhibits dimerization, with alternating spins labeled $S_1$ and $S_2$ to denote the two inequivalent sites. Triplon excitations (blue dimers) and exchange couplings $J$ and $J'$ are modulated by optical phonons (red and blue springs). An ultrafast terahertz laser field selectively drives infrared-active optical phonon modes, inducing nonequilibrium spin dynamics (see Sec. \ref{['S2A']}). Coupling to an external bath (Sec. \ref{['S2B']}) facilitates energy dissipation and allows the system to reach a steady state.
• Figure 2: Mean-field time evolution of the triplon emission power, $\mathcal{P}(t)$, across a range of driving frequencies $\Omega/J$, shown for different phonon frequencies $\omega_0/J$. Parameters: $g/J = 0.5$, $g'/J' = 0.3$, $\mathcal{E}/\omega_0 = 0.01$, $\gamma_p/\omega_0 = 0.05$, and $\gamma_s/J = 0.01$. (a) For $\omega_0/J = 1.25$, the system reaches a low-power steady state with persistent oscillations. (b) At $\omega_0/J = 1.3$, a sharp jump in steady-state power appears near $\Omega/J \approx 1.4$ (indicated by the arrow), signaling a dynamical first-order phase transition (see main text). (c) For $\omega_0/J = 1.35$, the system settles into a high-power steady state near $\Omega/J \approx 1.38$, again signaling the emergence of a dynamically distinct phase beyond the transient regime.
• Figure 3: Dynamical first-order phase transition in the mean-field triplon emission power of a driven-dissipative dimerized spin-$1/2$ chain. (a) Numerical time-averaged mean-field $\overline{\mathcal{P}}$ in the NESS as a function of the driving frequency $\Omega / J$ for various phonon frequencies $\omega_0 / J$. (b) Analytical mean-field Floquet predictions for the imaginary part of $\gamma_s\, \varepsilon_{k = 0} \overline{n}_{k=0}$ plotted versus the frequency $\omega / J$ for the same phonon parameters, demonstrating excellent agreement at the critical driving frequencies. Parameters: $g/J = 0.5$, $g'/J' = 0.3$, $\mathcal{E}/\omega_0 = 0.01$, $\gamma_p/\omega_0 = 0.05$, and $\gamma_s/J = 0.01$.
• Figure 4: Mean-field phase diagram showing the triplon emission power as a function of driving frequency $\Omega/J$ and phonon frequency $\omega_0/J$. The diagonal line reflects resonance effects at $\Omega \approx \omega_0$, with slight deviations due to strong SPC and hybridization. The diagram also captures signatures of the one- and two-triplon band edges over a wide parameter range. For $\omega_0/J < 1.27$, where the phonon energy lies well below the two-triplon continuum, only modest shifts in the emission edge occur as $\Omega$ varies. In contrast, for $\omega_0/J > 1.27$, sharp, step-like features appear in the emission power, signaling first-order phase transitions. As $\omega_0/J$ approaches $\sqrt{2}$, the critical driving frequency $\Omega_c$ decreases, reaching a minimum near $\omega_0/J \approx 1.37$, before rising again. This non-monotonic trend arises from an optimal overlap between phonon and laser resonances combined with an instability. The dotted red line traces the analytical Floquet prediction from Eqs. \ref{['eq_14']}--\ref{['eq_19']}, showing excellent agreement with the numerically determined transition points. Parameters: $g/J = 0.5$, $g'/J' = 0.3$, $\mathcal{E}/\omega_0 = 0.01$, $\gamma_p/\omega_0 = 0.05$, and $\gamma_s/J = 0.01$.
• Figure 5: Parameter sensitivity around a known $\Omega_c/J$ in the mean-field triplon emission power $\overline{\mathcal{P}}$ for phonon energy $\omega_0/J = 1.35$. Panel (a) shows the response versus laser amplitude $\mathcal{E}/J$; panels (b)–(c) illustrate the effects of SPC strengths $g/J$ and $g'/J'$, respectively; panels (d)–(e) depict the influence of phonon and spin damping rates $\gamma_p/J$ and $\gamma_s/J$. The vertical dashed line indicates the upper boundary of the one-triplon band. The nick, step-like changes in emission power signal first-order phase transitions, revealing strong nonlinear dynamics and parameter sensitivity in the driven-dissipative chain. In panel (a), we fix the SPCs at $g/J = 0.5$ and $g'/J' = 0.3$. In panel (b), we fix the laser amplitude to $\mathcal{E}/J = 0.0135$ and set $g'/J' = 0.3$, while in panel (c), the same laser amplitude is used with $g/J = 0.5$. For panels (a)–(c), the damping rates are fixed at $\gamma_p/J = 0.0675$ and $\gamma_s/J = 0.01$. In panel (d), we fix $\mathcal{E}/J = 0.0135$, $g/J = 0.5$, $g'/J' = 0.3$, and $\gamma_s/J = 0.01$, while in panel (e), the same values for $\mathcal{E}$, $g$, and $g'$ are used with $\gamma_p/J = 0.0675$.