Why Might the Standard Large N Analysis Fail in the O(N) Model: The Role of Cusps in Fixed Point Potentials

Abstract

The large N expansion plays a fundamental role in quantum and statistical field theory. We show on the example of the O(N ) model that at N = ∞, its standard implementation misses some fixed points of the renormalization group in all dimensions smaller than four. These new fixed points show singularities under the form of cusps at N = ∞ in their effective potential that become a boundary layer at finite N . We show that they have a physical impact on the multicritical physics of the O(N ) model at finite N . We also show that the mechanism at play holds also for the O(N )⊗O(2) model and is thus probably generic.The 1/N expansion is one of the most important tools in field theory. It has played a prominent role in QCD [1] as well as in statistical mechanics and condensed matter physics [2,3]. One of its key features is that it can yield reliable results even in strongly coupled models because it is nonperturbative in the coupling constant(s). It also has the enormous advantage of not being linked to a particular dimension, contrary to the usual perturbative expansions. This latter feature has often allowed us to make a bridge between the perturbative expansions performed around the upper and the lower critical dimensions of a model. For instance, at leading order in the 1/N expansion both the Mermin-Wagner theorem in two dimensions and the mean-field behavior at criticality in dimensions d ≥ 4 are retrieved, which is out of reach of both perturbative expansions in = 4 − d [4] and = d − 2 [5].The success of the large N analysis relies on (i) the possibility to extend the original model to arbitrary values of N and (ii) the fact that the model is soluble at N = ∞. This is the case not only for the O(N ) and the gauge SU(N ) models but also for a large class of statistical field theories.We show in this Letter that, surprisingly, even for the O(N ) model, which is the textbook example for the 1/N expansion, the situation is not as simple as it is widely believed. More precisely, we show on the examples of the O(N ) and O(N )⊗O(2) models that at N = ∞ some fixed points (FPs) that play an important role even at a qualitative level were missed by the usual large N approach [6,7]. The presence of these fixed points changes the finite N (multicritical) physics of these models.Over the years, the importance of renormalization group (RG) FPs showing cusps has been recognized. This occurs for FP functions such as thermodynamics potentials that are singular for a certain value of their argument. This is the case of the celebrated random field Ising model and is responsible for the breakdown of supersymmetry and dimensional reduction [8,9]. This is also the case out of equilibrium for some reaction-diffusion problems [10]. To the best of our knowledge, the occurrence of FPs with a cusp is known only in replica theory applied to disordered systems and in field theories describing out of equilibrium statistical models (see, however, Ref. [11]). We prove below that they also play an important role in simple field theories such as...

Finite N origin of the Bardeen-Moshe-Bander phenomenon and its extension at N=∞ by singular fixed points

“…A study at the next order of the derivative expansion shows that it improves as well as α c and that it can be further improved by studying the dependence of these numbers on the choice of regulator function R k ðqÞ [50][51][52][53][54]: This will be the subject of a forthcoming publication [55]. Another challenge is to follow all FPs in the whole ðd; NÞ plane and more precisely at moderate and small N. This study has been partly done in [25] and will be fully clarified in a forthcoming publication. It would also be interesting to know whether the same BMB phenomenon occurs for all multicritical FPs of the OðNÞ models around their respective upper critical dimensions and whether it exists generically for models different from the OðNÞ models [24].…”

“…Crucial to our study is the Wilsonian functional and nonperturbative renormalization group (NPRG). As we show below, it does not only provide an efficient means of computing FPs that are out of reach of usual perturbative or 1=N approaches but it also allows us to tackle problems that cannot even be formulated in the usual perturbative framework [24][25][26][27][28][29][30][31][32][33]. Among them are FPs whose potential shows a cusp or a boundary layer.…”

“…It is convenient to rescaleρ andṼðρÞ as usual:ρ ¼ρ=N,V ¼Ṽ=N. The FP equation forVðρÞ thus reads [8,25,45]…”

Platonic field theories

“…Since the recent renewed interest in the multi-critical O(N )-models [43,44], future perspectives regard the analogous analysis of the multi-critical behavior of Cubic theories. We also notice that the universality classes with d c < 3 may correspond to some novel unitary 2d CFTs with discrete global symmetry and of central charge c > 1; these theories are likely to be irrational CFTs and can be studied with numerical conformal bootstrap methods [45].…”

Asymptotic safety of scalar field theories

“…In ordinary O(N ) models, singularities and cusp-like patterns have also been observed [41,42,45]. For further studies of cusps in O(N ) models and the interplay of finite and infinite N approximations, see [41,42,45,108,109].…”