  We systematically study the low-temperature specific heats for the
two-dimensional kagome antiferromagnet, Cu$_{3}$Zn(OH)$_6$FBr. The specific
heat exhibits a $T^{1.7}$ dependence at low temperatures and a shoulder-like
feature above it. We construct a microscopic lattice model of $Z_2$ quantum
spin liquid and perform large-scale quantum Monte Carlo simulations to show
that the above behaviors come from the contributions from gapped anyons and
magnetic impurities. Surprisingly, we find the entropy associated with the
shoulder decreases quickly with grain size $d$, although the system is
paramagnetic to the lowest temperature. While this can be simply explained by a
core-shell picture in that the contribution from the interior state disappears
near the surface, the 5.9-nm shell width precludes any trivial explanations.
Such a large length scale signifies the coherence length of the nonlocality of
the quantum entangled excitations in quantum spin liquid candidate, similar to
Pippard's coherence length in superconductors. Our approach therefore offers a
new experimental probe of the intangible quantum state of matter with
topological order.
