Self-Organized Criticality in a Relativistic Yukawa Theory with Luttinger Fermions

We propose and investigate a Yukawa model featuring a dynamical scalar field coupled to relativistic Luttinger fermions. Using the functional renormalization group (RG) as well as large-\(N_{\textrm{f}}\) or perturbative expansions, we observe the emergence of an infrared attractive partial fixed point in all interactions at which all couplings become RG irrelevant. At the partial fixed point, the scalar mass parameter is RG marginal, featuring a slow logarithmic running towards the regime of spontaneous symmetry breaking. The long-range behavior of the model is characterized by mass gap formation in the scalar and the fermionic sector independently of the initial conditions. Most importantly, a large scale separation between the low-energy scales and the microscopic scales, e.g., a high-energy cutoff scale, is naturally obtained for generic initial conditions without the need for any fine-tuning. We interpret the properties of our model as a relativistic version of self-organized criticality, a phenomenon observed in specific statistical or dynamical systems. This entails natural scale separation and universal long-range observables. We determine nonperturbative estimates for the latter including the scalar and fermionic mass gaps.