We study the surface scaling behavior of a semi-infinite $d$-dimensional $\text{O}\left(N\right)$ spin system in the presence of a quenched random field and random anisotropy disorders. It is known that above the lower critical dimension $d_{\mathrm{LC}}=4$ the infinite models undergo a paramagnetic-ferromagnetic transition for $N>N_c$ ($N_c=2.835$ for the random field and $N_c=9.441$ for random anisotropy). For $N<N_c$ and $d<d_{\mathrm{LC}}$ there exists a quasi-long-range-order phase with a zero order parameter and a power-law decay of spin correlations. Using a functional renormalization group, we derive the surface scaling laws that describe the ordinary surface transition for $d>d_{\mathrm{LC}}$ and the long-range behavior of spin correlations near the surface in the quasi-long-range-order phase for $d<d_{\mathrm{LC}}$. The corresponding surface exponents are calculated to one-loop order. The obtained results can be applied to the surface scaling of periodic elastic systems in disordered media, amorphous magnets, and ${}^3$He-$A$ in aerogel.

• Received 7 May 2012

DOI:https://doi.org/10.1103/PhysRevE.86.021131