Diffusion and relaxation of topological excitations in layered spin liquids

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.

💡 Research Summary

The paper investigates the non‑equilibrium dynamics of fractionalized excitations in three‑dimensional crystals composed of weakly coupled two‑dimensional topological spin liquids. Because of the underlying Z₂ gauge structure, a single vison (or any other topological quasiparticle) can move freely within a layer, but motion in the third direction is only possible when two excitations form a pair. This kinematic constraint leads to anomalous diffusion that can be probed by pump‑probe experiments.

The authors formulate two complementary models. The first is a stochastic lattice model in which particles perform random walks with a hard‑core constraint, in‑plane diffusion rate Γ∥, inter‑layer pair‑hopping rate Γ⊥, and pair‑annihilation rate λ. The second is a coarse‑grained continuum description valid at low densities: a nonlinear diffusion equation for the particle density ρ(z, x, y, t),

∂ₜρ = D∥∇²_{xy}ρ + D⊥∂²_z(ρ²) – λρ² + noise,

with D∥ = Γ∥a², D⊥ = 2Γ⊥a², and λ = 2Γλ. The noise terms encode stochastic fluctuations of the diffusive motion and of annihilation events.

A scaling analysis shows that the only intrinsic time scale is set by the initial surface density ρ₀, which is directly proportional to the pump intensity Iₚ. By rescaling space and time as (x,y,z) → (x ρ₀^{−½}, y ρ₀^{−½}, z) and t → t ρ₀^{−1}, the equation becomes independent of ρ₀. Consequently, all observable relaxation times scale as τ ∝ 1/ρ₀ ∝ 1/Iₚ.

In the absence of annihilation (λ = 0) the nonlinear diffusion equation admits an exact self‑similar solution. The mean penetration depth ⟨z⟩ grows as t^{1/3}, a sub‑diffusive law that is slower than the usual t^{1/2} bulk diffusion. This reflects the fact that only pairs can transport particles vertically, effectively reducing the dimensionality of the transport.

When pair annihilation is allowed (λ > 0) the nonlinear term dominates the long‑time dynamics, and the penetration depth crosses over to a logarithmic growth, ⟨z⟩ ∝ log t. Physically, annihilation removes particles faster than they can be carried deeper, leading to a “dynamic bottleneck.”

If inter‑layer pair hopping is present (Γ⊥ > 0), the total particle number n(t) decays more slowly than the 1/t law typical of pure two‑dimensional diffusion. The authors find a novel decay law n(t) ∝ (log²t)/t. The logarithmic factor originates from the coupling between the quadratic inter‑layer diffusion term and the annihilation term.

Extensive kinetic Monte‑Carlo simulations of the lattice model confirm the analytical predictions. The simulations show that the scaling collapse of density profiles with respect to ρ₀ holds for a wide range of parameters, and that the choice of initial condition (random single particles versus pre‑formed pairs) only affects the very early time dynamics. Noise corrections introduce subleading logarithmic factors but do not alter the leading scaling exponents.

The authors also analyze finite‑slab geometries, relevant for real crystals with a limited number of layers. They demonstrate that a surface‑localized pump creates a density front that propagates inward, and that a probe laser positioned at various depths would record a signal decaying as e^{−α log²t}, consistent with the (log²t)/t law for the total density.

Finally, the paper discusses experimental feasibility, focusing on candidate materials such as α‑RuCl₃, where Kitaev‑type spin liquids are believed to exist. The authors argue that ultrafast optical or THz pulses can selectively excite vison pairs on the surface, and that time‑resolved reflectivity or Raman measurements can monitor the depth‑dependent relaxation. Observation of the predicted inverse‑pump‑intensity scaling, the t^{1/3} (or log t) penetration law, and the (log²t)/t decay would constitute a clear signature of topological fractionalization and the emergent dimensional constraint in layered spin liquids.