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By analyzing the dynamical equations of dissipative diffusion systems with the help of Feynman diagrams, we explicitly show that Goldstone modes exist in these systems and how these Goldstone modes are related to the origin of self-organized criticality. The analysis gives a clear statistical field theory picture of self-organized criticality.

Considering the relations between interaction terms in microscopic Hamiltonian and terms in macroscopic Landau's free-energy-function (FEF) of a phase transition system, we propose a modified Landau's phenomenological FEF to deal with critical phenomena. According to scaling symmetry, it is indeed true that, the dimension of interaction space is fractal at critical points for a phase transition system. Following these analyses, we obtain all the critical indices reasonably classified in the classes denoted by , which satisfy all the scaling laws and coincide with experiments. A cogent example is given.

Presenting a modified quantum model of spin glasses, in which the self-interaction was included, and considering the perfect disorder state, we have proved that the spin glasses system is equivalent with a three-dimensional oscillator system including non-linear coupling. Neglecting some parts of non-linear coupling terms, we obtained the exact solution on energy partition function for spin glasses system. On this basis, calculating the entropy, free energy, specific heat and susceptibility, we obtained the spin glasses characteristics well.

The paper shows how the characteristic time exponent z in slowing down, and the fractal dimension of the order parameter field, are uniquely determined by distinct forms of special thermodynamic functions that describe the interacting systems with some terms of n. It is also shown that for the purpose of obtaining critical exponents, the renormalization group transformation is needed just to perform to the powers of the order parameter and not to their coefficients in thermodynamic functions. This advantage leads to results which are analytically exact.