Topics in Statistic Mechanics

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Adv. Sta. Phy. Homework 5 XY Model Li,Zimeng PB06203182 and so H= (1.2) There are two parts in the above equation, the first part is irrelevant to spin, and we will omit it later. The second part can be transformed into continuous form by the following steps: Define , therefore 2.XY Model is frequently used to describe the universal behavior of Mott-insulator and superfluidity phase transition. Why it is possible? 2.1 Mott-insulator The first successful theory to describe the metals,insulators and transition between them is the noninteracting electron theory. This theory, or band theory, says that insulators have their highest filling band fully filled, and metals are just partially filled. But this theory failed to describe some metals and insulators that contradict the stated band filling. And later we found that electron-electron correlation, or the strong Coulomb repulsion of the filling electrons cannot be ignored, which leads to Mott's theory and the word - Mott insulator - to describe strong Coulomb repulsion state. The transition between metals and insulators is called MIT or Metal Insulator Transition. In Mott's opinion, a large Coulomb repulsion would split the band in two. The lower band consists of electrons that occupy empty sites with each site one electron, while the upper band consists of sites fully filled by electrons. Thus the lower band would be full and leads to an insulator. 2.2 Hubbard Model A prototype of theoretical understanding of MIT is the Hubbard model, which simplifies that electrons are only in a single band. Two most important parameters in Hubbard model is the correlation length U/t and the filling n, see Fig 2.2.1 below
Adv. Sta. Phy. Homework 5 XY Model Li,Zimeng PB06203182 Fig 2.2.1 There are two modes of the transition, the first one is filling-control MIT, or FC-MIT; And the second one, BC-MIT or band-control MIT is another case. We are not going to dip into the transitions here, but will take a look at how XY Model connects with the Mott insulator. U and t in the above figure will be defined below. The Hamiltonian of the Hubbard Model in the second-quantized form is given by (where we define two wave operators - on site i and on site j) , (2.2.1) N= So is the kinetic part and U is the Coulomb potential. From the above Hamiltonian,
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Topics in Statistic Mechanics

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