多體物理中的張量網路方法

量子多體系統由許多量子粒子相互作用構成,其集體行為與糾纏結構是現代凝聚態物理與量子資訊科學的核心議題。然而,這些系統的希爾伯特空間維度會隨粒子數呈指數增長,使得許多模型難以透過解析方法獲得解答。為了理解其基態性質、相變與動態行為,數值模擬成為一個重要的工具。張量網路提供了一種能有效捕捉量子糾纏結構並大幅降低計算複雜度的數值方法,使我們能在有限資源下處理原本難以處理的量子多體問題。 在課程後半也會介紹二維系統中的自旋液體以及拓樸序。在拓樸態研究中,張量網路能刻劃傳統序參數無法描述的長程糾纏,並能精確描述一些特定的拓樸態,使得張量網路成為一種描述多體量子態的新工具。 本課程將由基礎出發,逐步介紹與實作主要的張量網路架構,並以程式語言實做: Matrix Product State(MPS)/Density Matrix Renormalization Group(DMRG):一維量子鏈的基態模擬 Time-Evolving Block Decimation(TEBD):時間演化方法 Tensor Renormalization Group:張量重整化群 ---------------------------------------- Quantum many-body systems are composed of numerous interacting quantum particles, whose collective behavior and entanglement structures are central topics in modern condensed matter physics and quantum information science. However, the Hilbert space dimension of these systems grows exponentially with the number of particles, making many models difficult to solve analytically. To understand their ground-state properties, phase transitions, and dynamical behaviors, numerical simulation has become an essential tool. Tensor networks provide an efficient numerical method that captures quantum entanglement structures and significantly reduces computational complexity, allowing us to address otherwise intractable quantum many-body problems with limited resources. In the latter part of the course, we will also introduce spin liquids and topological order in two-dimensional systems. In the study of topological phases, tensor networks can characterize long-range entanglement that cannot be described by conventional order parameters, and they can precisely describe certain topological states. This makes tensor networks a new and powerful tool for describing many-body quantum states. This course will start from the fundamentals and gradually introduce and implement the main tensor network architectures, with practical programming implementations: Matrix Product State (MPS) / Density Matrix Renormalization Group (DMRG): ground-state simulation of one-dimensional quantum chains Time-Evolving Block Decimation (TEBD): time evolution methods Tensor Renormalization Group (TRG): tensor renormalization methods

量子力學Quantum Mechanics, 基礎的程式語言Basic programming skill(例如Python)

作業與出席(Homework and participation): 70 % 期末專題(Final project): 30 %

Review articles U. Schollwöck – The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011). R. Orús – A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014). Glen Evenbly – A Practical Guide to the Numerical Implementation of Tensor Networks I: Contractions, Decompositions and Gauge Freedom, arXiv:2202.02138 (2022) Open Access Book Shi-Ju Ran – Tensor Network Contractions: Methods and Applications to Quantum Many-Body Systems, Springer Lecture Notes in Physics, Open Access Book (2020) Find more here: https://tensornetwork.org/reviews_resources.html Introductory Websites https://tensornetwork.org/, Author: Edwin Miles Stoudenmire https://www.tensors.net/, Author: Glen Evenbly

地點

SC405

時間

需預約By appointment

聯絡方式

cmchung@nycu.edu.tw