Fluctuations of loop current order

We study the quantum critical theory of the fluctuations of loop current order in an effective model, developed based on our previous work arXiv:2402.09577. The study begins by deriving the effective theory of fluctuations for two phases: one with translationally invariant bond current order and another with a stripy bond current pattern. We then apply a modified Yukawa-SYK model to describe the system near the onset of loop current order, where fermionic and bosonic fields interact via random couplings. By performing disorder-averaging and deriving an effective action, we solve the corresponding saddle point equations and obtain the self-energies of fermionic and bosonic sectors. Additionally, we examine the fluctuation contributions around the saddle point, which provides insights into conductivity and current-current correlation functions in systems exhibiting loop current order.