Parametric resonance in a spin-1/2 chain: dynamical effects of nontrivial topology

TL;DR

This work links parametric resonance in a driven spin-1/2 chain to the bulk topology of the Kitaev chain by mapping to spinless fermions in the rotating frame. It shows that the system’s response to turning on the modulation—whether abruptly or adiabatically—reveals topology through distinct time-averaged observables, including vanishing long-range longitudinal correlations in the topological phase and characteristic scaling of nonadiabatic corrections near gap closures. The authors derive explicit expressions for fermionic occupations and spin correlators, connect them to the winding number, and demonstrate a robust, tunable transition controlled by the modulation frequency. They validate the analytical predictions with Matrix Product State simulations, observing finite-size effects and a pronounced hysteresis memory effect tied to topological dynamics. The results suggest bulk dynamical signatures of topology accessible in qubit and spin-chain platforms, with potential relevance to non-equilibrium topological magnonics and quantum simulators.

Abstract

Resonant parametric modulation is a major tool of studying magnetic systems. For a spin-1/2 chain in a strong magnetic field, the resulting excitations can be mapped on fermionic excitations in the Kitaev chain. We show that the response to the modulation turn-on allows one to reveal dynamical aspects of the nontrivial topology of the periodic chain. In the topological regime, depending on how fast the turn-on is, the system displays the absence of spatial magnetization correlations or their increase with the increasing detuning of the modulation from resonance. The transition between the topological and trivial regimes is controlled by the modulation frequency.

Figures (6)

• Figure 1: The time-averaged excitation density, $1-\braket{\sigma_n^z}$, and the nearest-neighbor correlator $\mathcal{Q}^z(1)=\braket{\sigma_n^z\sigma_{n+1}^z} - \braket{\sigma_n^z}^2$ in a modulated closed spin chain as functions of the detuning of the modulation frequency $\mu$ scaled by the hopping integral $J$. The upper and lower panels refer to the sudden and quasi-adiabatic modulation turn-on. The matrix-product-state simulations refer to the chain periods $N=20$ and $N=40$, respectively. The final modulation amplitude is $F(t_f)=J/2$. The slow turn-on regime in the simulations was implemented using $F(t) = F(t_f)t/\tau_\mathrm{sl}$ with $\tau_\mathrm{sl} = 10^3/J$.
• Figure 2: The time-averaged correlators $|\braket{\sigma_n^+\sigma_{n+1}^-}| = -\braket{\sigma_n^+\sigma_{n+1}^-}$ for slow and sudden turn-on of parametric modulation as functions of the scaled frequency detuning $\mu/J$. The results refer to the final modulation amplitude $F(t_f)/J= 0.5$. The simulations of the sudden and slow turn-on refer to $N=20$ and $N=40$, respectively. The slow turn-on was implemented using $F(t) = F(t_f)t/\tau_\mathrm{sl}$ with $\tau_\mathrm{sl} = 10^3/J$.
• Figure 3: The time-averaged correlators $|\braket{\sigma_n^+\sigma_{n+1}^-}| = -\braket{\sigma_n^+\sigma_{n+1}^-}$ obtained by slowly increasing the drive to $F(t_f)/J=0.5$ for a fixed $\mu$ ( line and circles 1) and by slowly turning on the drive to the same $F/J$ in the trivial regime and then slowly changing the modulation frequency to arrive at the same $\mu$ (line and circles 2). The simulations refer to $N=40$ and $F(t) = F(t_f)t/\tau_\mathrm{sl}$ with $\tau_\mathrm{sl} = 10^3/J$.
• Figure S1: The time-averaged correlators $1-\braket{\sigma_z}$ (left panel) and $\mathcal{Q}^z(1)$ (right panel) obtained by slowly increasing the drive to $F_f/J=0.5$ for a fixed $\mu$ (line 1 and blue dots) and by slowly turning on the drive to the same $F_f/J$ in the trivial regime and then slowly changing the modulation frequency to arrive at the same $\mu$ (line 2 and red dots). The simulations refer to $N=40$ and $F(t) = F_f t/\tau_\mathrm{sl}$ with $\tau_\mathrm{sl} = 10^3/J$.
• Figure S2: The time-averaged correlator $\mathcal{Q}^z(1)$ for a sudden turn-on of parametric modulation as a function of the scaled frequency detuning $\mu/J$. The final modulation amplitude is $F/J= 0.5$. The analytical result refers to the thermodynamic limit $N\to \infty$.