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研究生： 朱奕豪
Jhu, Yi-Hao
論文名稱： 一維淬火動力系統中的量子糾纏熵能譜
Entanglement spectrum in one-dimensional quench dynamics
指導教授： 陳柏中
Chen, Pochung
張明強
Chung, Ming-Chiang
口試委員: 牟中瑜
Mou, Chung-Yu
米格爾
Miguel A. Cazalilla
高英哲
Kao, Ying-Jer
學位類別： 博士
Doctor
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2016
畢業學年度： 104
語文別： 中文
論文頁數： 72
中文關鍵詞： 淬火動力學 、量子糾纏 、一維
外文關鍵詞： Quench dynamics, Entanglement, One dimension
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    • 本論文研究一維自由系統的量子糾纏熵能譜於量子淬火後的行為。 利用拓樸最大糾纏態來定義系統的非普通拓樸性質，我們發現系統於時間很長時所達到的平衡態，其拓樸性質可由一有效磁場 $\mathbf{S}_{\mbox{eff}}$ 來描述。 若 $\mathbf{S}_{\mbox{eff}}$ 具有非普通貝瑞相位，則系統處於非普通拓樸態，反之亦然。考慮具體的二元鏈模型和P波超導模型，我們發現系統必需要在同一個拓樸相之間淬火才能夠有拓樸平衡態，此外，我們也發現系統都是以冪次律收斂到其平衡態。 然而，我們發現二元鏈模型的平衡態比起P波超導模型的平衡態來得更具有普適性。 在二元鏈模型之中，我們發現其冪次律收斂的指數永遠為3/2且在同一個拓樸相之間淬火時，系統永遠會有拓樸平衡態。 但在P波超導模型中，我們發現其冪次律收斂的指數可被系統參數改變，且在同一個拓樸相之間淬火時，系統也並非永遠具有拓樸平衡態。

    In this thesis, we study the one particle entanglement spectrum (OPES) after a quantum sudden quench in the one dimensional and quadratic systems. Using the topological maximal entangled state (tMES) as a signature of non-trivial topology. We find the topology of the steady state at the long time limit can be captured by a pseudo-magnetic field $\mathbf{S}_{\mbox{eff}}$, where the non-trivial or trivial Berry phase of $\mathbf{S}_{\mbox{eff}}$ determines the steady state to be topological or not. Using the dimerized chain and p-wave superconductor as practical examples, we find the topological steady states can only exist for the quenches between the same topological phase. On the other hand, the convergences to the steady states are found to be power-law like. However, we find the steady states in the dimerized chain are more universal than the steady states in the p-wave superconductor. In dimerized chain, we find the exponents of the power-law convergences are always 3/2 and the topological steady states are always found for the quenches between the same topological phase. While, in the p-wave superconductor, the exponent of the power-law convergence can be changed by tuning the parameters and one can have trivial steady states even for the quenches between the same topological phase.

    1 Introduction . . . . . . . . . . . . . . . . . . . . 1 2 Systems and Methods . . . . . . . . . . . . . . . . 4 2.1 Systems . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Models . . . . . . . . . . . . . . . . . . . . .4 2.1.2 Setup of the Sudden Quench . . . . . . . . . . . 8 2.2 Methods . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 One-Particle Entanglement Spectrum . . . . . . . 9 2.2.2 Saddle Point Expansion . . . . . . . . . . . . . 12 3 Quench Dynamics of the OPES $\lambda_m(t)$ . . . . . 18 3.1 Dimerized Chain . . . . . . . . . . . . . . . . . 19 3.2 P-wave Superconductor . . . . . . . . . . . . . . 24 3.3 Summary . . . . . . . . . . . . . . . . . . . . . 29 4 Convergence to the Steady State . . . . . . . . . . .32 4.1 Dimerized Chain . . . . . . . . . . . . . . . . . 34 4.2 P-wave Superconductor . . . . . . . . . . . . . . 37 4.2.1 Class I: $|\frac{\mu'/2}{{\Delta'}^2 -1}|<1$ . . 37 4.2.2 Class II: $|\frac{\mu'/2}{{\Delta'}^2 -1}|=1$ . .42 4.2.3 Class III: $|\frac{\mu'/2}{{\Delta'}^2 -1}|>1$ . 42 4.3 Discussion and Summary . . . . . . . . . . . . . . 45 5 Summary and Outlook . . . . . . . . . . . . . . . . 48 A The Practical Forms of $G(t)$ and $G^{eff}$ . . . . .50 A.1 Dimerized Chain . . . . . . . . . . . . . . . . . 51 A.2 P-wave Superconductor . . . . . . . . . . . . . . 52 B Maximal Angular Velocity and Band Gap . . . . . . . .53 B.1 Extrema of the Angular Velocity $\omega(k)$ . . . .53 B.2 Extrema of the Dispersion $\epsilon(k)$ . . . . . .54 B.3 Relationship between $k_0$ and $k_g$ . . . . . . . 55 C Obtaining $k_j$, $n_j$ and $ \beta^q_j$ . . . . . . . 60 C.1 Dimerized Chain . . . . . . . . . . . . . . . . . 61 C.1.1 Finding saddle points $k_j$ and its order $n_j$ .61 C.1.2 De fining $\beta^q_j$ . . . . . . . . . . . . . . 62 C.2 P-wave Superconductor . . . . . . . . . . . . . . 63 C.2.1 Finding saddle points $k_j$ and its order $n_j$ .63 C.2.2 De fining $ \beta^q_j$ . . . . . . . . . . . . . . 65

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