Harris criterion on hierarchical lattices: rigorous inequalities and counterexamples in Ising systems.
Random bond Ising systems on a general hierarchical lattice are considered. The inequality between the specific heat exponent of the pure system, alpha(p), and the crossover exponent straight phi, alpha(p)</= phi, gives rise to the possibility of a negative alpha(p) along with a positive phi, leading to random criticality in disagreement with the Harris criterion. An explicit example where this really happens for an Ising system is presented and discussed. In addition to that, it is shown that in the presence of full long-range correlations the crossover exponent is larger than in the uncorrelated case.
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School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel.