Enhanced coherence in the periodically driven two-dimensional XY model

Abstract

Strong optical drives have been shown to induce transient superconducting-like response in materials above their equilibrium $T_c$. Many of these materials already exhibit short-range superconducting correlations in equilibrium. This motivates the question: can external driving enhance coherence in systems with superconducting correlations but no long-range order? We explore this scenario in the two-dimensional XY model with a periodically modulated stiffness using overdamped Langevin dynamics. We find that, even though the modulation leaves the average coupling unchanged, the drive can markedly increase long-range, time-averaged correlations in systems well above the equilibrium Berezinskii-Kosterlitz-Thouless temperature. The outcome depends on the ratio of the drive frequency to the intrinsic relaxation rate: faster drives primarily heat the system, suppressing correlations and conductivity. For slower drives, the optical conductivity is modified so that the real part exhibits a prolonged effective Drude scattering time, while the imaginary part has a strengthened low-frequency $1/ω$ behavior. We map out these regimes across temperature, frequency, and amplitude, and rationalize them via simple analytics and vortex-thermalization arguments. Overall, we identify a generic nonequilibrium route to enhance coherence in XY-like systems, with potential relevance to experiments reporting light-induced superconductivity.

Figures (11)

• Figure 1: Schematic of the periodically driven XY model. (a) Square-lattice geometry of the XY model with time-modulation of the superfluid stiffness $J$ (indicated by the blue arrow). (b) Illustration of the results for for sufficiently slow driving (indicated by the blue curve). Over one drive period, when $J$ increases, long-range coherence is enhanced (shown as blue shading) and the instantaneous vortex number decreases; by contrast, when $J$ decreases, correlations are suppressed and the vortex number increases (vortices shown as green circles; antivortices as red circles). On average, the system can exhibit enhanced long-range correlations even if the time-averaged vortex density is increased compared to equilibrium.
• Figure 2: Representative results versus driving amplitude. (a) Time-averaged correlator $\langle G(r)\rangle_t$ for different driving amplitudes. Shaded areas indicate statistical errors on the average. The short-distance behavior of $\langle G(r)\rangle_t$ remains essentially unchanged from equilibrium, while long-range correlations grow markedly with $A$. (b) Time-averaged vortex-antivortex number $\langle N_v\rangle_t$ as a function of driving strength, showing a monotonic increase. Shaded areas indicate statistical errors on the average. The inset schematically shows the numerical protocol: equilibration followed by driving. The quantities in the main panels are computed only within the driving window, i.e., after equilibration at finite temperature starting from an infinite-temperature, fully random state Supplement. In both panels, the drive is ramped on over $n \approx 3$ periods and maintained at a steady amplitude for $\approx 10$ periods. Further parameters: $L = 64$, $T = 1.2$, $\Omega = 0.01$, $N = 10$.
• Figure 3: Numerics versus analytics. Ratio between the time-averaged correlator $\langle G(r)\rangle_t$ and its undriven counterpart $G_0(r)$ for different driving amplitudes. Dots indicate numerical results (shaded areas indicate statistical errors on the average); dashed lines are the analytical predictions discussed in the main text. At short distances, the ratio $\langle G(r)\rangle_t / G_0(r)$ remains close to $1$, indicating essentially unchanged correlations, while at large distances it grows markedly with $A$, reflecting enhanced correlations---a trend well captured by both numerics and analytics. Results are shown over a steady driving period. Parameters: $L = 64$, $T = 1.2$, $\Omega = 0.001$, $N = 10$.
• Figure 4: Representative electromagnetic response. (a) Real (diamonds) and imaginary (circles) parts of the conductivity versus probe frequency in the undriven case, showing the characteristic Drude-like response above $T_{\rm BKT}$. (b) Real (diamonds) and imaginary (circles) parts of the conductivity versus probe frequency under driving at $\Omega = 0.01$ and amplitude $A = 0.7$, illustrating the enhanced coherent character of the driven state. The drive is ramped on over $\approx 3$ periods and maintained at a steady amplitude for $\approx 10$ periods. (c) Log–log plot of the imaginary part of the conductivity versus probe frequency, highlighting the significantly enhanced $1/\omega$ behavior as the driving strength $A$ is increased (cf. dashed line: $1/\Omega_p$). In all panels, shaded areas indicate statistical errors on the average. Further parameters: $L = 64$, $T = 1.2$, $N = 10$, $E_{p,0} = 0.004$.
• Figure 5: Representative results versus driving frequency and amplitude. Time-averaged correlator $\langle G(r = 15)\rangle_t$, used as a proxy for the large-distance behavior, in the $(\Omega, A)$ plane. The enhancement (light green region) is strongest in the low-$\Omega$, large-$A$ corner of parameter space, indicating that an adiabatic, strongly driven regime is most favorable for the mechanism under discussion. Here, the drive is ramped on over $n \approx 3$ periods and maintained at a steady amplitude for $\approx 10$ periods. Further parameters: $L = 64$, $T = 1.2$, $N = 10$.