Random Defect Lines in Conformal Minimal Models

We analyze the effect of adding quenched disorder along a defect line in the 2D conformal minimal models using replicas. The disorder is realized by a random applied magnetic field in the Ising model, by fluctuations in the ferromagnetic bond coupling in the Tricritical Ising model and Tricritical Three-state Potts model (the φ12 operator), etc.. We find that for the Ising model, the defect renormalizes to two decoupled half-planes without disorder, but that for all other models, the defect renormalizes to a disorder-dominated fixed point. Its critical properties are studied with an expansion in ǫ ∝ 1/m for the m Virasoro minimal model. The decay exponents XN = N 2 (1 − 9(3N−4) 4(m+1)2 + O( 3 m+1 )) of the N th moment of the two-point function of φ12 along the defect are obtained to 2-loop order, exhibiting multifractal behavior. This leads to a typical decay exponent Xtyp = 1 2 (1+ 9 (m+1)2 +O( 3 m+1 )). One-point functions are seen to have a non-self-averaging amplitude. The boundary entropy is larger than that of the pure system by order 1/m. As a byproduct of our calculations, we also obtain to 2-loop order the exponent X̃N = N(1− 2 9π2 (3N−4)(q−2)+O(q−2)) of the N th moment of the energy operator in the q-state Potts model with bulk bond disorder.