We present models where γ+ and γ−, the exponents of the susceptibility in the high and low temperature phases, are generically different. In these models, continuous symmetries are explicitly broken down by discrete anisotropies that are irrelevant in the renormalization-group sense. The Zq-invariant models are the simplest examples for two-component order parameters (N = 2) and the icosahedral symmetry for N = 3. We compute accurately γ+ − γ− as well as the ratio ν/ν of the exponents of the two correlation lengths present for T < Tc.PACS numbers: 05.70. Jk, 11.10.Hi, 05.10.Cc The question of the equality of the critical exponents on the two sides of a second order phase transition has, apparently, not been raised for decades. The general renormalization group (RG) argument "showing" their equality goes as follows: Correlation functions are regular in the presence of an external field, which is sufficient to go continuously from one phase to the other. Moreover, if these functions satisfy the same RG equations above and below the critical temperature T c , the same should hold true for the scaling behavior of quantities such as the susceptibility, the correlation length or the specific heat. Since the renormalization properties of a theory are identical in its symmetric and spontaneously broken phases, it follows that the critical exponents are identical in both phases (see for instance [1][2][3]). This is indeed what happens generically.To the best of our knowledge, Nelson [4] was the first to propose a counterexample based on the O(2) model in dimension d = 3, to which is added either a cubic (CA) [4][5][6][7][8][9][10][11][12] or hexagonal (HA) anisotropy [4][5][6]. These anisotropies are taken into account in the GinzburgLandau hamiltonian by terms of order 4 and 6, respectively, which are irrelevant in the RG sense at the transition. The corresponding fixed point is thus O(2) symmetric. However, Nelson argued [4] that they are dangerously irrelevant [6,13] in the low-temperature phase and that they, therefore, induce a modification of the exponent γ − of the susceptibility. A rather counterintuitive result is that the difference γ + − γ − is larger for HA than for CA, whereas HA is "more irrelevant" than CA. A detailed study of the literature shows that, up until now, this striking result has been completely ignored.Because of its relationship with either deconfined quantum critical points [14] or pyrochlore [15] and the possible existence of two distinct phase transitions [16], the threedimensional XY model with HA (and more generally the Z q -invariant models) has been studied again [17,18]. Although only one transition has been found, the Z q models were shown to exhibit two correlation lengths below T c , ξ and ξ , that scale with two different critical exponents, ν and ν . All authors agree that ν/ν depends on the scaling dimension of the irrelevant HA term, but there are no less than three different scaling relations predicting this ratio, as well as several values obtained by Monte Carlo simulati...

We study the "Higgs" amplitude mode in the relativistic quantum O(N ) model in two space dimensions. Using the nonperturbative renormalization group and the Blaizot-Méndez-Galain-Wschebor approximation (which we generalize to compute four-point correlation functions), we compute the O(N )-invariant scalar susceptibility at zero temperature in the vicinity of the quantum critical point. In the ordered phase, we find a well-defined Higgs resonance for N = 2 and 3 and determine its universal properties. No resonance is found for N 4. In the disordered phase, the spectral function exhibits a threshold behavior with no Higgs-like peak. We also show that for N = 2, the Higgs mode manifests itself as a very broad peak in the longitudinal susceptibility in spite of the infrared divergence of the latter. We compare our findings with results from quantum Monte Carlo simulations and = 4 − (d + 1) expansion near d = 3.