在這篇文章中，我們分析各種在Hofstadter-Hubbard模型裡有可能產生的低溫的相位。我們分別在實空間以及動量空間做平均場計算。在實空間裡，我們先找到有電荷密度波的相位，接著我們導出自洽方程式，藉此來算出電荷密度波的序參量。因為系統在半填滿時有SO(4)的對稱性，所以超導相與電荷密度波相在半填滿時共存。我們分析了序參量在不同的變數下的情況，如：溫度、電荷密度、交互作用的強度。最後，我們計算自由能來找出哪個相位是最穩定的。
我們利用泛函積分來導出有效的金茲堡－朗道自由能，藉此來得出超導與電荷密度波在二階微擾的係數。當系統遠離半填滿時，超導與電荷密度波的對稱性就會被破壞，我們可以找出哪一項破壞了這個對稱性。
此外，我們也在實空間發現不同的超導密度波相位。為了瞭解這些相位產生的機制，我們一樣用金茲堡－朗道自由能來分析並做到四階的微擾。
最後，我們得到在半填滿時，系統是超導相與電荷密度波相共存，遠離半填滿時，只有超導相存在，而超導密度波是自由能的局域最低點，因此不穩定。

In this thesis, we analyze possible low-temperature phases of the Hofstadter-Hubbard model. We use the mean field theory to calculate this system in the real space and momentum space separately. In real space, we need to find the wave vector in order to verify the charge density wave (CDW) phase. Next, we derive the self-consisted equations to obtain the order parameter. Because the system has SO(4) symmetry, we find out that the phase of CDW and the phase of the uniform s wave superconductor can be coexisted when system is at half-filling. We also analyze that the order parameter behavior depend on temperature, electron density and interaction strength. Finally, we calculate the free energy to find which phase can be stable.

We use functional integral to calculate the effectively Ginzburg Landau free energy to obtain the coefficient of CDW and uniform superconductivity in second order perturbation. When the system is away from half-filling, the symmetry between CDW and SC is broken. We can find that how the perturbation destroy the SO(4) symmetry.

Furthermore, there is pair density wave (PDW) phase with different wave vector in the Hofstadter-Hubbard model by the calculation of the real space. To understand the mechanism between the PDW, CDW, uniform phase of the superconductivity, we use Ginzburg Landau free energy to analyze them and do perturbation to fourth order.

Finally, the phase of the uniform superconductivity can be coexisted with CDW phase and lower the Free energy. We concluded that when system is at half-filling, the free energy of the coexistence of the CDW and uniform superconductivity is the lowest. Far from the half-filling, the uniform superconductivity is more stable than CDW phase. For PDW phase, the free energy between PDW and SC is the local minimum, and it is not the lowest free energy.

摘要
Abstract
Acknowledgements
Contents ------------------------------------------------------------------------------------- iii
List of Tables ----------------------------------------------------------------------------------------- iv
List of Figures ------------------------------------------------------------------------------------------- vi
1. Introduction ------------------------------------------------------------------------------------------ 1
2. The Mean Field Theory in the Real Space ---------------------------------------------------- 5
2.1 The Hofstadter model in the real space -------------------------------------------------- 5
2.2 The mean field theory ------------------------------------------------------------------------ 7
3. The Mean Field Theory in the Momentum Space ----------------------------------------- 14
3.1 The Hofstadter model in the momentum space -------------------------------------- 14
3.2 Group theory analysis for Hofstadter-Hubbard model at half-filling ------------ 18
4. Ginzburg Landau Free Energy ------------------------------------------------------------------ 21
4.1 Path integral formulism for partition function ---------------------------------------- 21
4.2 The Hofstadter-Hubbard model at half-filling ----------------------------------------- 25
4.3 When system is away from half-filling -------------------------------------------------- 27
5. PDW Phase in the Hofstadter-Hubbard Model -------------------------------------------- 30
5.1 The coefficient of PDW phase of free energy ----------------------------------------- 30
5.2 Coexistence of PDW and uniform superconductivity ------------------------------- 31
5.3 The fourth order perturbation for free energy expansion ------------------------- 33
5.4 Effective free energy of PDW phase ----------------------------------------------------- 34
5.5 The calculation for other PDW phase with different wave vector Q ------------- 36
6. Conclusion and Discussion ---------------------------------------------------------------------- 39
Reference ------------------------------------------------------------------------------------------- 41

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