本篇論文對二維鐵磁性的Ising模型進行了有限尺度標度分析。
臨界現象的發生需要系統在熱力學極限上，即系統大小為無限大。但在多數情況下，數值模擬只能計算有限大小系統的物理量。而有限尺度標度分析提供了一個能從有限大小的物理量去逼近無限大系統下的臨界現象。
在計算上，張量網絡的方法被用來計算有限大小系統的各種物理量。兩種實體空間的重整化群的方法被用來分析，即higher-order tensor renormalization group (HOTRG)和core-tensor renormalization group (CTRG).
透過有限尺度標度分析，我們可以很好地得到系統的臨界溫度、臨界指數，以及其共行結構。

In this paper, we investigate the finite size scaling of the 2D ferromagnetic Ising model.
Critical behavior only occurs at the thermodynamic limit, which is when the system size approaches infinity. However in most cases, numerical simulations can only compute quantities for finite size systems. Finite size scaling analysis provides us with a method to approach critical behavior using quantities in finite size systems.
For computation, we employ the tensor network method to calculate various quantities for finite size systems. The data are analysed using two types of real space tensor renormalization group methods: higher-order tensor renormalization group (HOTRG) and core-tensor renormalization group (CTRG),.
We demonstrate that finite size analysis enables us to accurately extract critical temperature, critical exponents, and conformal structure of the system.

Abstract (Chinese) --------------------------------------------------------- I
Abstract ------------------------------------------------------------------ II
Contents ------------------------------------------------------------------III
List of Figures ------------------------------------------------------------ V
List of Tables ------------------------------------------------------- ---VIII

1 Introduction ------------------------------------------------------------- 1
1.1 Phase Transition ------------------------------------------------------- 1
1.2 Finite Size Scaling ---------------------------------------------------- 2
1.2.1 Crossing Point Analysis ---------------------------------------------- 3
1.2.2 Fitting at Critical Point -------------------------------------------- 4
1.3 2D Ising Model --------------------------------------------------------- 5
1.4 Tensor Network Method -------------------------------------------------- 6
1.5 Transfer Matrix -------------------------------------------------------- 8
1.6 Conformal Tower --------------------------------------------------------11
1.7 Entanglement Entropy ---------------------------------------------------14

2 Methodology --------------------------------------------------------------17
2.1 Higher-order Tensor Renormalization Group ------------------------------17
2.2 Core-tensor Renormalization Group --------------------------------------19
2.3 1-site Translation Operator in Tensor Network --------------------------22
2.4 Improvements for Tensor Network Renormalization Group ------------------24

3 Numerical Results --------------------------------------------------------26
3.1 Crossover Length Scale -------------------------------------------------27
3.2 Correlation Length Per Site --------------------------------------------30
3.3 Spontaneous Magnetization ----------------------------------------------33
3.4 Specific Heat Per Site -------------------------------------------------37
3.5 Conformal Tower --------------------------------------------------------38
3.6 Entanglement Entropy ---------------------------------------------------40
3.7 Eigenenergy ------------------------------------------------------------42

4 Conclusion ---------------------------------------------------------------45
5 Appendix -----------------------------------------------------------------47

5.1 Derivation for Crossing Point Analysis ---------------------------------47
Bibliography ---------------------------------------------------------------50

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