對稱張量網路對自旋1/2海森堡模型在籠紋晶格之研究

Abstract

凝態物理學的其中一個核心目標是精確地生成實際交互作用系統中的量子相圖。而量子多體糾纏的研究為我們提供了許多關於量子態結構的關鍵洞見。尤其張量網路最近已成為一種強大的數值技術，能夠可靠地捕捉系統中的糾纏結構以及量子相關性，提供描述量子態的自然語言。最近的數值研究顯示，自旋-1/2 海森堡模型在籠紋晶格上（KAFH）的基態包含著自旋液體的有力證據；然而，這種新奇物質的性質還是未知的。

在本論文中，我們利用對稱投影糾纏單體態來分類可能的Z2自旋液體，並用數值方法獲得最佳基態。首先，我們利用投影糾纏單體態的投影對稱群概念，事先處理張量量值的等效性，將我們波函數中的局部張量進行分類，允許我們在獲得對稱投影糾纏單體態波函數的每個有限分類下，分別進行變分簡單更新模擬。接著我們使用角轉移矩陣重整群（CTMRG）演算法來獲得更精確的能量測量。因此，我們的方法使我們能夠在熱力學極限下了解KAFH模型基態的本質。此外，最小糾纏態的重疊提供了我們基態波函數中Z2拓撲序的證據。

One of the central goals in condensed matter physics is accurately generating quantum phase diagrams of realistically interacting systems. The study of quan- tum many-body entanglement has provided many key insights into the structure of quantum states of matter. Tensor networks (TNs) have recently emerged as powerful numerical techniques that reliably capture the entanglement structure and quantum correlations in the system, offering a natural language to describe quantum states. Recent numerical studies show that the ground state of the spin-1/2 Kagome antiferromagnetic Heisenberg (KAFH) model contains strong evidence of a spin liquid phase; however, the characteristics of this new exotic phase of matter are unknown. In this thesis, we utilize symmetric projected entangled simplex states (PESS) to classify possible Z2 spin liquid phases and numerically obtain the optimal ground state. In particular, we employ the concept of projective symmetry group (PSG) for PESS, allowing us to deal with tensor gauge equivalence beforehand to classify our local tensor in the TN wavefunction. The classification allows us to obtain a finite number of classes of symmetric PESS wavefunctions and perform a variational simple update simulation within each class separately. Additionally, we use the Corner Transfer Matrix Renormalization Group (CTMRG) algorithm to obtain more accurate energy measurement. Therefore, our approach enables us to understand the nature of the ground state of the KAFH model at the thermodynamic limit. Furthermore, the minimal entanglement states (MES) overlapping provides evidence of the Z2 topological order within our ground state wavefunction.