Research On The Critical Behavior Of Hexangular Lattice With The Method Of Real-Space Renormalization Group

Renormalization group method which calculates fractal dimension theoretically is the most useful tool when we study critical phenomena of phase transition. This way seems to avoid partition function conceptively, but study the transformation which makes partition function unchanged. These transformations are made up of renormalization group. Then the fixed point of transformation can be found, among which those unstable ones are critical points of phase transition. The gist of this method is that the behavior which contributes to critical phenomena is not small scaling one, but big scaling one (i.e. the correlative length is infinite) when phase transition happens. Thus some microscopic details (small scaling behavior) become unimportant. The real-space (or position-space) renormalization group method is close to fractal and is widely used in geometric phase transition systems without Hamilton, for example, seepage, lattice animal and random walk.In this paper, beginning with the characteristic of space structure of graphite, we explain its layer structure and choose lattice in a layer which is called hexangular lattice as our studied object. At first, the fractal property of hexangular lattice is illustrated from two aspects which are the definition of fractal and the form of hexangular lattice. Then the structure unit of hexangular lattice is found according to the principle of invariable symmetry. Renormalization transformation is processed when we regard the structure unit and the growth model as graphs before and after transformation respectively. After choosing the11thermodynamic function fugacity as parameter, we can write out the partition functions before and after transformation and the formula of renormalization transformation. At the same time, we also work out the fixed points of transformation. At last, the critical exponent and fractal dimension of correlative length are calculated when phase transition happens. Using the scaling law, we know the critical exponent of specific heat too. The peak-valley path method in computing partition function is originally used in chemistry to study Kekulean structure of benzenoid hydrocarbon. We introduce it firstly. The behavior of particles on peak-valley path can be regarded as a random walk in some restrictive conditions, the track of which is a kind of random fractal. Thus peak-valley path method is connected with fractal. Though we know the forms of partition function before and after transformation, we will not study directly the property of partition function itself but the transformation which makes it unchanged with the idea of renormalization group.Otherwise, in this paper, the random walk of particles is processed on lattices of graphite in a layer i.e. hexangular lattices, which makes our study have some physical background. Thus our results have some instructional function theoretically on the study of graphite and benzenoid hydrocarbon.