Are there any known models with limit cycles in their RG flow?

Question

The text-book presentation of the renormalization group (RG) leaves one with the impression that all systems will eventually flow to a fixed point. This is somewhat enforced by the phenomenological scaling hypothesis in the sense that we empirically see robust scaling near critical points in many physical systems.

For dynamical systems in 2D, aside from flowing to fixed points and running off to infinity, limit cycles are also possible. In 3D we can even have Lorenz attractors.

Limit cycle RG flows seem to admit an interesting physical interpretation: a theory that looks the same at energy scales $\Lambda,\,\frac{\Lambda}{L},\, \frac{\Lambda}{L^2},\cdots$.

Are such systems impossible? If yes, does that mean that there are constraints on RG flow equations that limits the form of the differential equations governing the flow?

Answer 1

Such systems are quite possible, modelled copiously, the focus of a cottage industry, and have numerous applications. Beyond the Bulycheva & Gorsky review arXiv:1402.2431 that @Buzz links above, in his references you'd find particularly instructive papers. Foremost, in my mind, are LeClair et al.'s "Russian doll spin models":

• A. LeClair, J. M. Román, and G. Sierra, Russian Doll Renormalization Group and Superconductivity, Phys. Rev. B69 20505 (2004) arXiv:cond-mat/0211338; Russian Doll Renormalization Group, Kosterlitz-Thouless Flows, and the Cyclic sine-Gordon model, Nucl. Phys. B675 584-606 (2003) arXiv:hep-th/0301042; Log-periodic behavior of finite size effects in field theories with RG limit cycles, Nucl. Phys. B700 407-435 (2004) arXiv:hep-th/0312141; A. LeClair and G. Sierra Renormalization group limit-cycles and field theories for elliptic S-matrices, J. Stat. Mech. 0408:P004 (2004) arXiv:hep-th/0403178.

• S. D. Glazek and K. G. Wilson, Limit Cycles in Quantum Theories, Phys. Rev. Lett. 89, 230401 (2002); Erratum, 92, 139901 (2004).

• E Braaten and H-W Hammer, Universality in Few-body Systems with Large Scattering Length, Phys. Rept. 428 (2006) 259-390 arXiv:cond-mat/0410417 [cond-mat.other].

• T L Curtright, X Jin, C K Zachos, RG flows, cycles, and c-theorem folklore, Phys Rev Lett. 108.131601 , arXiv:1111.2649 [hep-th] and T L Curtright and C K Zachos, Renormalization Group Functional Equations, Phys. Rev. D83 (2011) 065019. arXiv:1010.5174 [hep-th], whose section IV gives you the minimal cartoon of it, below.

The renormalization of the dimensionless couplings $g$ and $h$ under a change in system (Russian doll Hamiltonian) size $L$ is given by $$ \frac{dg}{d\ln L}=g^{2}+h^{2}\ ,\ \ \ h=\text{constant} $$ with $h$ the time-reversal breaking parameter.

Assuming $h\neq0$, change variables to $u=g/h$ and $t=h\ln L$. Then $$ \beta (u)=\frac{du}{dt}=1+u^{2} $$
and direct integration yields $$ u\left( t\right) =\tan\left( t+\arctan u_{0}\right) . $$

Thus the physics of the model repeats itself cyclically as the logarithm of the system size is changed, in evident evocation of nested Russian dolls.

There are numerous applications in spin physics, nuclear physics, and HEP ("Efimov states").

Answer 2

In 3D we can even have Lorenz attractors.

Cosmas already gave a good answer on the limit cycles, and, with regard to the possibility of chaotic RG flows, the answer seems to also be yes.

For instance, according to the paper Can Renormalization Group Flow End in a Big Mess? (arXiv),

the couplings in a renormalizable field theory may also flow towards more general, even fractal attractors.

[...] chaotic renormalization group flows [...] have many common virtues with realistic field theory effective actions. We conclude that if chaotic renormalization group flows are to be excluded, conceptually novel no-go theorems must be developed.