Diffusion and relaxation of topological excitations in layered spin liquids

TL;DR

The paper addresses how to detect topological order in layered spin liquids by out-of-equilibrium pump–probe protocols that probe the emergent dimensionality of topological excitations. It develops a layered diffusion model with intralayer diffusion, interlayer pair hopping, and pair annihilation, yielding a nonlinear diffusion equation whose solutions show distinct scaling depending on annihilation: without annihilation the excitation front propagates as $Z(t)\sim (18 D_\perp \rho_0)^{1/3} t^{1/3}$, while with annihilation the front grows as $Z(t)\sim \sqrt{D_\perp/\lambda}\,\log t$ and the total density decays as $n(t)\sim (\log t)^2/t$. A density–dependent scaling $\rho(z,t)/\rho_0=\tilde{\rho}(z,t\rho_0)$ governs the dynamics, and noise introduces marginal corrections that slightly modify long-time behavior; finite-slab geometry offers experimentally accessible signatures such as a slow approach to equipartition with $\Delta(t)\sim e^{-\alpha (\log t)^2}$. Together, these results provide a robust, density-controlled route to observing the coupled diffusion and annihilation of 2D topological excitations in 3D layered materials, offering a practical path to identifying emergent gauge structure and dimensionality in quantum spin liquids.

Abstract

Relaxation processes in topological phases such as quantum spin liquids are controlled by the dynamics and interaction of fractionalized excitations. In layered materials hosting two-dimensional topological phases, elementary quasiparticles can diffuse freely within the layer, whereas only pairs (or more) can hop between layers - a fundamental consequence of topological order. Using exact solutions of emergent nonlinear diffusion equations and particle-based stochastic simulations, we explore how pump-probe experiments can provide unique signatures of the presence of $2d$ topological excitations in a $3d$ material. Here we show that the characteristic time scale of such experiments is inversely proportional to the initial excitation density, set by the pump intensity. A uniform excitation density created on the surface of a sample spreads subdiffusively into the bulk with a mean depth $\bar z$ scaling as $\sim t^{1/3}$ when annihilation processes are absent. The propagation becomes logarithmic, $\bar z \sim \log t$, when pair-annihilation is allowed. Furthermore, pair-diffusion between layers leads to a new decay law for the total density, $n(t) \sim (\log^2 t)/t$ - slower than in a purely $2d$ system. We discuss possible experimental implications for pump-probe experiments in samples of finite width.

Figures (6)

• Figure 1: A layered material, e.g., $\alpha$-RuCl$_3$, hosting a $2d$ topological phase is uniformly excited from the top by a laser pulse of intensity $I_P$. Single excitations are topological (red spheres), e.g. visons in a Kitaev spin liquid, and can diffuse freely within the $2d$ layers, whereas inter-layer motion requires a pair of excitations due to the constraints imposed by topological order and the emergent gauge structures. Excitations are eliminated only via pair-annihilation into vacuum (with a rate $\lambda$), emitting low-energy trivial excitations such as phonons (blue wavy lines). This leads to a subdiffusive and logarithmic spreading of the excitations into the bulk for $\lambda=0$ and $\lambda\ne0$ respectively. All characteristic time scales, measured via a probe laser for example (not shown), scales inversely with the pump intensity, $\tau \propto 1/I_P$.
• Figure 2: Density evolution in the top and bottom layers of a sample with $w=10$ layers, where initially all excitations reside in the top layer (using particle-based simulation, model \ref{['eq:model']}). The noisy simulation results are plotted in solid lines while dashed lines in (a) are obtained by solving the noiseless diffusion equation numerically on a discrete lattice. The collapse of plots for different initial densities $\rho_0$ upon rescaling the time and density confirms the scaling predicted by Eq. \ref{['eq:scaling2']}. (a) $\Gamma_\perp=0.5,\Gamma_\lambda=0$ (b) $\Gamma_\perp=0.4, \Gamma_\lambda=0.05$, (c) $\Gamma_\perp=0.3, \Gamma_\lambda=0.15$. All plots are averaged over 6 simulations using a $500\times 500\times 10$ grid. Initial particle densities $\rho_0$ are $0.4$ and $0.2$, corresponding to 100.000 and 50.000 particles, respectively. Panel (d) shows, for two parameter sets with $\rho_0=0.2$, that the difference between particle densities in top and bottom layers decays approximately with $e^{-\alpha \log^2 t}$. Fit parameters: $\alpha=0.062,\beta=3.5$ for the blue points; $\alpha=0.07,\beta=2.7$ for green points.
• Figure 3: (a) Layer density of excitations along the $z$ direction plotted in rescaled coordinates. The blue curves show snapshots at various times. A scaling collapse happens for long time scales, consistent with the the analytical prediction of the noiseless diffusion model (red-dashed curve). (b) The average depth $\bar{z}$ traversed by the excitations into the bulk as a function of time (in log-log scale). The expected scaling of $\bar{z}\sim t^{1/3}$ is shown by the dashed line. Simulation parameters: $L=300,\rho_0=0.1,\Gamma_\perp=0.4,\Gamma_\|=1$. Fit (in (a)):$\rho_0=0.1,D_\perp =0.8$
• Figure 4: The mean depth $\bar{z}$ traversed by the excitations into the bulk when pair-annihilation processes are present, $\lambda\ne0$. We find the exact simulation results to be consistent with the predictions of Eq. \ref{['eq:crossover']}. At early times, the simulation data is roughly consistent with the sub-diffusive $t^{1/3}$ power-law (green dashed line), which crosses over to a logarithmic scaling at longer times (red dashed line, $t_0=20$). Inset: Data is plotted in log-linear scale to show $\bar{z}\sim \log t$ scaling at long times. Simulation parameters: $L=500,w=100,\rho_0=0.2,\Gamma_\|=1,\Gamma_\perp=0.3,\Gamma_\lambda=0.1$.
• Figure 5: (a). Total density $n$ (particles per area) as a function of time (shown in log-linear scale), for $\Gamma_\perp=0.3, \Gamma_\lambda = 0.1$, starting from an initial density $\rho_0=0.2$. The red curve is obtained by numerically solving the noiseless diffusion equation (mean-field) on a grid of size $100$. (b). The ratio between the solution $n_{\text{mf}}(t)$ obtained from the noiseless model and exact simulation $n_{\text{sim}}(t)$ are plotted, which shows an approximately $n_\text{sim}/n_\text{mf}\sim \log t$ behavior for $t\gtrsim 10^2$. Inset: Plot of $t\,n(t)$, showing that $n_\text{sim}\sim (\log t)^2/t$ while $n_\text{mf}\sim (\log t)/t$. Simulation parameters are identical to that of Fig. \ref{['fig:com_anni']}.