Distribution of RKKY coupling value in 1D crystal with disorder. Specific heat in XY model

The presence of disorder in one-dimensional crystals leads to the localization of all charge carriers and the calculation of the indirect exchange interaction (Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction) cannot be performed perturbatively on disorder. In two and three-dimensional systems it makes sense to calculate the magnitude of RKKY interaction perturbatively treating nearly free carriers scattering on the random potential, and this approach results in a rather high magnitude of the exchange interaction due to interference effects similar to weak localization. We show that in one-dimensional systems the indirect exchange interaction should be described as a random value with heavy-tail distribution function, which is calculated in this work, on scales of carriers localization length. We also demonstrate that heavy tails and the absence of a characteristic value of RKKY interaction magnitude leads to a significant change in observables for these systems. We calculate a specific heat for the one-dimensional XY model taking into account the effect of disorder and assuming that typical distance between impurities exceeds the localization length. In contrast to an ideal system, where specific heat temperature dependence has a peak at a certain temperature proportional to exchange constant describing characteristic energy scale, disorder eliminates the peak as soon as there is no characteristic excitation energy in this case anymore.