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

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").