We employ the functional renormalization group framework at the
second order in the derivative expansion to study the
O(N)O(N)
models continuously varying the number of field components
NN
and the spatial dimensionality dd.
We in particular address the Cardy-Hamber prediction concerning
nonanalytical behavior of the critical exponents
\nuν
and \etaη
across a line in the (d,N)(d,N)
plane, which passes through the point (2,2)(2,2).
By direct numerical evaluation of \eta(d,N)η(d,N)
and \nu^{-1}(d,N)ν−1(d,N)
as well as analysis of the functional fixed-point profiles, we find
clear indications of this line in the form of a crossover between two
regimes in the (d,N)(d,N)
plane, however no evidence of discontinuous or singular first and second
derivatives of these functions for d>2d>2.
The computed derivatives of \eta(d,N)η(d,N)
and \nu^{-1}(d,N)ν−1(d,N)
become increasingly large for d\to 2d→2
and N\to 2N→2
and it is only in this limit that \eta(d,N)η(d,N)
and \nu^{-1}(d,N)ν−1(d,N)
as obtained by us are evidently nonanalytical. By scanning the
dependence of the subleading eigenvalue of the RG transformation on
NN
for d>2d>2
we find no indication of its vanishing as anticipated by the
Cardy-Hamber scenario. For dimensionality dd
approaching 3 there are no signatures of the Cardy-Hamber line even as a
crossover and its existence in the form of a nonanalyticity of the
anticipated form is excluded.
Measuring Functional Renormalization Group Fixed-Point Functions for Pinned Manifolds