Critical exponents of the random-field O(N) model

The critical behavior of the random-field Ising model has been a puzzle for a long time. Different theoretical methods predict that the critical exponents of the random-field ferromagnet in D dimensions are the same as in the pure (D-2)-dimensional ferromagnet with the same number of the magnetization components. This result contradicts the experiments and simulations. We calculate the critical exponents of the random-field O(N) model with the (4+\epsilon)-expansion and obtain values different from the critical exponents of the pure ferromagnet in 2+\epsilon dimensions. In contrast to the previous approaches we take into account an infinite set of relevant operators emerging in the problem. We demonstrate how these previously missed relevant operators lead to the breakdown of the (6-\epsilon)-expansion for the random-field Ising model.