Finite-size scaling of Landau-Ginzburg model for fractal time processes

The universality of critical phenomena and finite-size scaling are effective methods for measuring critical exponents in experiments and inferring the intrinsic interactions within materials. Here, we establish the finite-size scaling form of the Landau-Ginzburg model for fractal time processes and quantitatively calculate the critical exponents at the upper critical dimension. Interestingly, contrary to the traditional conception that critical exponents are independent of dynamic processes and proportional to correlation length, we find that fractal time processes can not only change critical exponents but also yield a scaling form of size dependent on fractional order and spatial dimension. These theoretical results provide a reasonable method to determine and measure the existence of fractal time processes and their associated critical exponents. The simulations of the Landau-Ginzburg model with fractional temporal derivatives and the Ising model with long-range temporal interactions not only reveal critical exponents distinct from those of standard models but also exhibit unique size effects characteristic of fractal time processes. These results validate the emergence of a new universality class and confirm the predictions of the finite-size scaling theory for fractal time processes.