The Development And Applications Of Renormalization Group Flow Equation Method

To solve the renormalized charge problem, the concept of renormalization was presented and later the corresponding renormalization group approach was developed. In 1971, Wilson applied renormalization group method to study critical phenomena in stastical physics, and the critical exponents obtained by renormalization group tally with those got by experiments. As science develops, different connotations of renormalization group come into being.In the first chapter of this thesis, the development process of renormalization group approaches and our main work are introduced. In the second chapter we introduce the application of renormalization group method in critical phenomena, which seems to avoid partition function but study the transformations which make partition function unchanged. These transformations are made up of renormalization group. Then the fixed points of transformations can be found, among which those unstable ones are critical points of phase transition. In the third chapter, we introduce another renormalization group method based on functional integral, which acts on action, integrating out fast modes but leaving the free field action S₀ invariant. At first we give an example of the renormalization group transformation of a scalar field in four dimensions. Then we apply this method and mean-field theory to study the stability of spinless fermions in one dimension at half filling to weak interactions. Itis indicated that the renormalization-group method correctly yields results in harmony with exact solution, while mean-field theory gives wrong predictions. In the fourth chapter, the newest progress of renomaTization group approach, namely renormalization group approach(also called continuous unitary transformation method) is introduced. Founded on this method, Hamiltonian is subjected to a series of infinitesimal continuous unitary transformations, then differential equations about parameters in Hamiltonian can be obtained. Based on boson-fermion model, we illustrate how to get the flow equations and how to solve spectral function of this model. The last chapter contains summary and outlook.