RANDOM TRANSVERSE ISING SPIN CHAIN AND RANDOM-WALKS

We study the critical and off-critical (Griffiths-McCoy) regions of th e random transverse-held Ising spin chain by analytical and numerical methods and by phenomenological scaling considerations. Here we extend previous investigations to surface quantities and to the ferromagneti c phase. The surface magnetization of the model is shown to be related to the surviving probability of an adsorbing walk and several critica l exponents are exactly calculated. Analyzing the structure of low-ene rgy excitations we present a phenomenological theory which explains bo th the scaling behavior at the critical point and the nature of Griffi ths-McCoy singularities in the off-critical regions. In the numerical part of the work we used the free-fermion representation of the model and calculated the critical magnetization profiles, which are found to follow very accurately the conformal predictions for different bounda ry conditions. In the off-critical regions we demonstrated that the Gr iffiths-McCoy singularities are characterized by a single, varying exp onent, the value of which is related through duality in the paramagnet ic and ferromagnetic phases.