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 topological excitations. In layered materials hosting two-dimensional topological order, 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 classical diffusion equations and particle-based stochastic simulations, we explore how pump-probe experiments can provide unique signatures of the presence of 2d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> topological excitations in a layered 3d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mover> <mml:mi>z</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:math> scaling as \sim t^{1/3} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> when annihilation processes are absent. The propagation becomes logarithmic, \bar z \sim \log t <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mover> <mml:mi>z</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> <mml:mo>∼</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">log</mml:mi> <mml:mo>⁡</mml:mo> </mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>∼</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">log</mml:mi> <mml:mo>⁡</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>/</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> - slower than in a purely 2d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> system. We discuss possible experimental implications for pump-probe experiments in finite-size systems.