Analyticity of critical exponents of the $O(N)$ models from nonperturbative renormalization

Abstract

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.

The critical O(N) CFT: Methods and conformal data

“…In [77] it was proposed the existence of a curve starting at N = d = 2 across which critical exponents are non-analytic, denoted the "Cardy-Hamber line". However, recent studies using the non-perturbative RG indicate that that the exponents are smooth across the proposed curve [63].…”

“…For all R = S, T, A, the order ε 3 computation was performed in [104] and order ε 4 in [222]. In the large N expansion, the order 1/N correction was found for R = T, A in [104] 63 and for R = S in [105]. In principle, the order 1/N 2 OPE coefficients for the T and A irreps can be determined numerically by evaluating the computation in [105] to high precision.…”

Stability of the Fulde-Ferrell-Larkin-Ovchinnikov states in anisotropic systems and critical behavior at thermal $m$-axial Lifshitz points

“…For details on the numerical procedure, see Ref. 71. For m → 0 we obviously recover the value pertinent to the standard O(N) model.…”

Physical properties of the massive Schwinger model from the nonperturbative functional renormalization group

“…Whether or not topological excitations, in particular when they are responsible for symmetry restoration, can be captured by truncations like the DE is a very interesting issue in many models. There is no doubt that the DE, which yields very accurate estimates of the critical exponents [31,32], captures the topological excitations of the three-dimensional O(3) model in which the hedgehog singularities are known to be essential [48,49]. The situation is less clear in lower dimensions.…”