Random-field spin models beyond 1 loop: A mechanism for decreasing the lower critical dimension

The functional renormalization group for the random-field and random-anisotropy O(N) sigma models is studied to 2 loop. The ferromagnetic-disordered (F-D) transition fixed point is found to next order in d=4+epsilon for N > N-c (N-c=2.834 740 8 for random field, N-c=9.441 21 for random anisotropy). For N < N-c the lower critical dimension d=d(lc) plunges below d(lc)=4: we find two fixed points, one describing the quasiordered phase, the other is novel and describes the F-D transition. d(lc) can be obtained in an (N-c-N) expansion. The theory is also analyzed at large N and a glassy regime is found.