透過將原始希爾伯特空間降低到有效的維度空間，張量網路已經被廣泛發展並應用於模擬量子多體系統。在我們的研究中，我們利用時間演化區塊消除法（TEBD）作為數值工具進而研究具有橫向與縱向場中一維易辛模型的量子動力學。該模型是一個沒有解析解的不可積系統。
我們透過焠火系統參數來研究非平衡態動力學，並展示縱向磁場如何影響長時間的動力學，其中包括許多重要的性質，如糾纏、弛豫、熱化等。此外，我們可以透過可積和不可積系統中的焠火動力學直接探測量子相變。透過分析物理可觀測量（如橫向磁化強度）的時間演化，可以利用傅立葉轉換獲得傅立葉頻譜來辨識量子臨界點的位置。

The tensor network algorithm has been extensively developed and widely applied to efficiently simulate quantum many-body systems, reducing the original Hilbert space to a lower effective space. In our studies, we employ the Time-Evolving Block Decimation (TEBD) as a numerical tool to investigate the quantum dynamics of the 1D Ising Model with transverse and longitudinal fields. This model represents a non-integrable system without analytic solutions.
We explore the non-equilibrium dynamics by quenching the system parameters, demonstrating how the longitudinal magnetic field can influence long-time dynamics, which includes critical properties such as entanglement, relaxation, and thermalization, among others. Furthermore, quantum phase transitions can be directly probed through quench dynamics in both integrable and non-integrable systems. Analyzing the time evolution of physical observables (such as transverse magnetization), we can identify the location of quantum critical points using Fourier spectrum analysis with Fourier Transformation.

摘要
Abstract
Acknowledgements
1 Introduction 1
2 Tensor Network Algorithm 4
2.1 MPS and TEBD ............................ 4
2.2 Matrix Product State Formalism ................... 5
2.3 Canonical Form of MPS and Expectation Value . . . . . . . . . . . 7
2.4 Time-Evolving Block Decimation ................... 8
2.4.1 Time Evolution......................... 8
2.4.2 Suzuki-Trotter Decomposition................. 9
2.4.3 Infinte TEBD(iTEBD)..................... 10
2.4.4 Updating Process of TEBD .................. 10
2.4.5 The Truncation During SVD.................. 12
3 1D Transverse Field Ising Model 15
3.1 Analytical Solution ........................... 15
3.1.1 Diagonalizing the Hamiltonian................. 15
3.1.2 The Heisenberg Equation of Motion . . . . . . . . . . . . . 17 3.1.3 Physical Observables ...................... 18
3.2 The Quench Dynamics of Transverse Magnetization via the Fourier Analysis................................. 19
4 Numerical Results of Quantum Quench Dynamics 24
4.1 Transverse Magnetization ....................... 24
4.2 Entanglement Entropy ......................... 26
5 1D Transverse Field Ising Model in the Presence of a Longitudinal Field 29
5.1 Ground-State Phase Diagram through Imaginary Time Evolution . 30
5.2 Fourier Analysis in Identifying the Critical Points through Quench
Dynamics ................................ 34
5.3 Ground State Phase Diagram from Quench Dynamics: Analysis of
Experimentally Observables ...................... 41
6 Conclusions and Future Work 43
Bibliography 45

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