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In the quasi-1D heavy-fermion system YbNi$_4$(P$_{1-x}$As$_x$)$_2$ thepresence of a ferromagnetic (FM) quantum critical point (QCP) at $x_c$ $\approx0.1$ with unconventional quantum critical exponents in the thermodynamicproperties has been recently reported. Here, we present muon-spin relaxation($\mu$SR) experiments on polycrystals of this series to study the magneticorder and the low energy 4$f$-electronic spin dynamics across the FM QCP. Thezero field $\mu$SR measurements on pure YbNi$_4$(P$_{2}$ proved static longrange magnetic order and suggested a strongly reduced ordered Yb moment ofabout 0.04$\mu_B$. With increasing As substitution the ordered moment isreduced by half at $x = 0.04$ and to less than 0.005 $\mu_B$ at $x=0.08$. Thedynamic behavior in the $\mu$SR response show that magnetism remainshomogeneous upon As substitution, without evidence for disorder effect. In theparamagnetic state across the FM QCP the dynamic muon-spin relaxation ratefollows 1/$T_{1}T\propto T^{-n}$ with $1.01 \pm 0.04 \leq n \leq 1.13 \pm0.06$. The critical fluctuations are very slow and are even becoming slowerwhen approaching the QCP.

Magnetic order and spin dynamics across a ferromagnetic quantum critical point: $\mu$SR investigations of YbNi$_4$(P$_{1-x}$As$_x$)$_2$