# The "dangerous" fixed points for Renormalization Group

1. What is the definition of dangerously irrelevant renormalization-group (RG) fixed point?

2. What are some examples of dangerously irrelevant RG fixed points?

3. Do we also have the use of dangerously relevant RG fixed points? Do we also have the use of dangerously marginal RG fixed points? What are the examples?

4. The term "irrelevant, marginal, relevant" are for the IR (infra-red) low energy long distant fixed-point perspective, commonly used in the low energy physics and in condensed matter physics. In high energy and particle physics, people also consider from the UV perspective, thus "irrelevant, marginal, relevant" become "non-renormalizable, renormalizable, super-renormalizable" at the UV (ultra-violet) high energy short distant perspective. So do we also have concepts of: $$\text{dangerously irrelevant} \leftrightarrow \text{dangerously non-renormalizable},$$ $$\text{dangerously marginal} \leftrightarrow \text{dangerously renormalizable},$$ $$\text{dangerously relevant} \leftrightarrow \text{dangerously super-renormalizable},$$

See also a possible example here: Are irrelevant terms in the Kahler potential always irrelevant, even at strong coupling?. Partial answers and partial comments are all welcome. You need not to provide the full answer to write an answer.

• For first subquestion (v1), see Wikipedia. Commented Nov 14, 2016 at 19:23
• For 2. : an example is the quartic interaction in the ϕ4-theory for dimensions larger than four dimensions. The reason is that the scaling functions carry a singular dependence on the irrelevant coupling. One cannot simply take the irrelevant coupling to vanish in the effective treatment. Consequently, I don't think there is something is dangerously relevant couplings as relevant couplings are always taken into account. Commented Jun 17, 2017 at 10:55
• To add, I think, the terminology refers to an operator, not a fixed point. The idea being an operator that is irrelevant by power counting, but is important for stabilizing the correct ground state.
– vik
Commented May 19, 2019 at 19:55

Before going to dangerous irrelevance, it helps to briefly recapitulate what irrelevance under RG means in itself. When thinking of an RG fixed point, the scaling behaviour at low energies/long wavelengths is typically controlled only by a handful of relevant operators which dominate the physics, while all irrelevant terms progressively get smaller and eventually vanish on large scales. This is after all the crux of universality, in that a large number of microscopic details don't matter as they simply change irrelevant terms, which can be neglected in the low energy description of a physical system.

Now, an operator or term is said to be dangerously irrelevant when under RG, although the term scales to zero at large distances, it nonetheless cannot be forgotten and neglected. In other words, dangerously irrelevant operators can affect universal observable features associated to an RG fixed point, such as scaling exponents in critical phenomena, unlike "regular" irrelevant operators that can be safely neglected.

The canonical example of a dangerously irrelevant term is the quartic coupling $$\lambda$$ in the standard $$\phi^4$$ theory $$F=\int{\rm d}^dr\left[\dfrac{1}{2}(\nabla\phi)^2+\dfrac{m^2}{2}\phi^2+\dfrac{\lambda}{4}\phi^4\right]\;.$$ Above the upper critical dimension (i.e., $$d>4$$), the Gaussian fixed point with $$\lambda=0$$ is stable and the quartic coupling is irrelevant, scaling as $$\lambda(\ell)=\lambda_0\ell^{4-d}\to 0$$ with the RG scale $$\ell$$. One may naively think that the irrelevance of the nonlinear term permits us to simply set it to zero at the outset and work just with the Gaussian bit of the free energy/action. This is alright when $$m^2>0$$ (disordered or symmetry unbroken phase), but for $$m^2<0$$ (ordered or symmetry broken phase), we find that $$F$$ would be unbounded below if we had no $$\lambda$$ term. In particular, $$\lambda$$ sets the amplitude of the field/order parameter VEV through $$\phi_0=\pm\sqrt{-m^2/\lambda}$$, which depends sensitively on $$\lambda$$. Hence, although $$\lambda\to 0$$ on large scales, it can't be set to zero entirely, so $$\lambda$$ behaves as a dangerously irrelevant variable here. Keeping track of the irrelevant scaling of $$\lambda$$ in $$d>4$$ is essential to recovering the mean field scaling exponents within the RG setting, see standard texts like Goldenfeld's "Lectures on Phase Transitions and the Renormalization Group" for a discussion on this.

There are many more examples of dangerously irrelevant operators that we know of. One situation they are commonly encountered in is when the irrelevant term breaks a symmetry that is preserved at the fixed point. This occurs when the low energy effective theory has an emergent symmetry that isn't present in the microscopic/UV description. An example would be an Heisenberg ferromagnet, whose order-disorder transition is controlled by an $$O(N)$$ symmetric Wilson-Fisher fixed point, even though on the microscopic/UV scale, there may be crystalline lattice anisotropies that don't respect the full $$O(N)$$ symmetry. A classic example is one of cubic anisotropy, which is marginal at $$d=4$$ and turns out to be dangerously irrelevant in the calculation of certain observables. See Amit and Peliti, "On dangerous irrelevant operators." Annals of Physics 140.2 (1982): 207-231., for a further discussion on this. In this paper, they do distinguish two kinds of dangerously irrelevant variables as weak and strong, which is perhaps what you were referring to in terms of generalizations of the dangerous variable concept. Strictly marginal terms (i.e., with no RG flow whatsoever) are unusual, though they do appear in some CFTs and field theories in low dimensions. Note that the idea of operators being dangerous is only useful for irrelevant terms, as those are the ones we usually neglect and the dangerous prefix then reminds us that can't do that. Marginal and relevant variables are already known to be important and have to be accounted for at the fixed point, so there is no need for a "dangerously relevant" operator, atleast not one that I'm aware of.

Another related example of a "dangerous" symmetry breaking term is one for hexagonal symmetry in an $$XY$$ magnet. Even near $$d=4$$, operators that break continuous $$O(2)$$ spin-rotational symmetry down to a six-fold discrete rotation, are strongly irrelevant, yet their irrelevance is dangerous as they suppress the Goldstone modes upon symmetry breaking and lead to additional singularities in susceptibilities along the coexistence curve. See Nelson, "Coexistence-curve singularities in isotropic ferromagnets." Physical Review B 13.5 (1976): 2222. for a description of this scenario. The same mechanism involving discrete symmetry breaking provides a natural way to have different scaling exponents on either side of a phase transition, see Léonard and Delamotte, "Critical exponents can be different on the two sides of a transition: A generic mechanism." Physical review letters 115.20 (2015): 200601.

There are of course many more examples possible, both in the quantum and classical realm, and also in nonequilibrium and driven-dissipative systems, but the general idea remains the same.

• Great answer! I had thought about writing something similar, including the reference of Delamotte (whom I know personally). Glad someone had the will of writing it. +1 :)
The gaussian fixed point of a $$\phi^4$$ theory (describing phase transitions in dimension $$d\geq 4$$) has one relevant coupling (corresponding to the mass/correlation length) and an infinity of irrelevant couplings (one marginally so in $$d=4$$). Among them, the coupling corresponding to $$\phi^4$$ (call it $$g$$) is dangerously irrelevant.
Another example (more useful in our three-dimensional world) is the Gaussian fixed point of the (non-relativistic) dilute Bose and Fermi gases in $$d\geq 2$$, which dominates the physics at low density and temperature in cold atoms experiments (in particular, the logarithmic vanishing of the scattering T-matrix in two dimensions can be interpreted as a manifestation of the marginal irrelevance of the interaction). See Sachdev's book "Quantum Phase Transitions".