在這篇論文中，我們探討了在銅酸鹽中電荷密度波與超導態的競爭關係。這裡的超導態包含了d-波的超導態與電子對密度波。我們用t−t′−J模型與Ginzburg-Landau的理論建構Ginzburg-Landau自由能並用這個方法去分析這些態之間競爭關係的相圖。我們首先用電子的格林函數去表示展開成冪次函數自由能內部的係數。這些格林函數接著用重整化平均場理論下的t−t′−J模型來計算。有了這些係數，我們就可以計算在有均勻磁場下的Ginzburg-Landau自由能並且去找相圖。我們發現在低磁場的環境下由d-波超導態主導而電子對密度波態會在中強度磁場且低溫的環境取代d-波超導態成為主要的相。最後，我們發現電子對密度波態會因電荷密度波而穩定且最後與電荷密度波同時出現。

In this thesis, we investigate the competition between the charge density wave and the superconducting states in cuprates. The superconducting states include the uniform d-wave superconducting state and the pair density wave states. To analyze the phase diagram resulting from the competition between these states, we resort to the Ginzburg-Landau theory by constructing the Ginzburg-Landau free energy functional from the renormalized mean-field t − t′ − J model. Specifically, we first express all relevant coefficients in the polynomial expansion of the Ginzburg-Landau free energy in terms of Green’s functions of electrons. The Green’s functions are then computed based on the renormalized mean-field results of the t − t′ − J model. With the computed coefficients, we compute the minimum of the Ginzburg-Landau free energy in the presence of a uniform magnetic field to find the phase diagram. We find that while the uniform d-wave superconducting state dominates in the low magnetic field regime, the pairing density wave state wins over for intermediate magnetic fields and lower temperatures. Furthermore, we find that the pair density wave state is stabilized by the charge density wave and emerges together with the charge density wave state.

Content IV

List of Figures VI

1 Introduction 1

2 Model 4
2.1 t-t'-J Model----------------------------------------------4
2.2 Slave Boson Method----------------------------------------5
2.3 t-t'-J Model in the Momentum space------------------------6
2.4 Numerical analysis of a simpler case----------------------7
2.5 Renormalized Mean-Field Theory----------------------------11
2.5.1 Simplified t−J Model------------------------------------11
2.5.2 t−t′−J Model--------------------------------------------13

3 Ginzburg-Landau Theory 16
3.1 Landau Theory---------------------------------------------16
3.2 Ginzburg-Landau Theory------------------------------------19
3.3 Free energy with Coexisting SC and PDW--------------------20
3.4 Self-consistent gap equations-----------------------------23

4 Coefficient Calculation 25
4.1 Feynman Diagram-------------------------------------------25
4.2 Green's Function from Feynman Diagram---------------------27
4.3 Summation over Matsubara Frequency------------------------29
4.3.1 Contour Integration in Many-Body Theory-----------------29
4.3.2 Matsubara Frequency Summation---------------------------29
4.4 Calculation of others αn----------------------------------30
4.4.1 α2------------------------------------------------------31
4.4.2 α3 and α4-----------------------------------------------32
4.5 Cubic term of γ1 and γ2-----------------------------------32
4.6 Biquadratic Term βn---------------------------------------33

5 Results 35
5.1 Coefficients----------------------------------------------35
5.2 Applying Magnetic Field-----------------------------------37

6 Conclusion and Discussion 40

Reference 42

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