O(N)-Universality Classes and the Mermin-Wagner Theorem

Truncation effects in the functional renormalization group study of spontaneous symmetry breaking

“…To have qualitatively correct results in d = 2 valid for all N anomalous dimension effects as introduced in LPA have to be considered. This has been recently shown in [10], which shows how LPA is able to reproduce numerically the behaviour predicted by the Mermin-Wagner theorem for d → 2 and N ≥ 2 (with the anomalous dimension η → 0 and the correlation length exponent ν → ∞ [53]), and correctly predicting at the same time SSB and a finite anomalous dimension exponent for the Ising model. We extended these results showing that when the anomalous dimension vanishes then no SSB transition is possible in d ≤ 2 (as it happens for the O(N ≥ 2) models).…”

“…This gives a clear explanation in the FRG framework of the presence or the lack of SSB in O(N ) models in d = 2. Of course, the fact that there is no SSB for N = 2 does not imply the absence of the Berezinskii Kosterlitz-Thouless transition [59,60], as can be seen also in FRG treatments [10,[61][62][63].…”

“…In [10] the study of how O(N ) universality classes depends continuously on the dimension d (and as well on N ), in particular for 2 < d < 3, was recently presented. The Ising model, i.e.…”

“…A powerful method used to consider the phase structure of a model, and consequently to study the appearance of SSB, is the functional renormalization group (FRG) method [15][16][17][18][19][20][21][22][23]. The O(N ) model has extensively studied using FRG approaches: as relevant for our purposes, we mention it was used to study as a function of dimension critical exponents of O(N ) models [10,24,25] and to investigate truncation effects and the regulator-dependence of the FRG equation [26][27][28][29][30][31][32][33][34][35][36][37], while a FRG study of the critical exponents of the Ising model for d < 2 was presented in [24]. The study of single-particle quantum mechanics can be seen as a "low-dimensional" statistical mechanics model: FRG studies addressed double well potential and quantum tunneling [38,39] and quartic anharmonic oscillators [40].…”

“…To have a complete picture of the d = 2 case one should go beyond LPA: as discussed in [10], in LPA' the limit of the anomalous dimension η for d → 2 is vanishing provided that N ≥ 2, and non-vanishing for N = 1. This gives a clear explanation in the FRG framework of the presence or the lack of SSB in O(N ) models in d = 2.…”