授業の目的 【日本語】
Goals of the Course(JPN) | | 講義のテーマ：物性物理学の数学的側面
物性物理学は，物質の固体相および液体相の性質を研究する量子力学の一分野である．本科目の目的は，関数解析，スペクトル理論，作用素環など，物性物理学の基礎となる数学的理論を概説し，それらが物理的性質や現象とどのように関連しているかを理解することである． |
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授業の目的 【英語】
Goals of the Course | | Course title: Mathematical aspects of condensed matter physics
Condensed matter physics is a branch of quantum mechanics that studies the microscopic and macroscopic properties of solid and liquid phases of matter. The goal of the course is to introduce the mathematical theories that underpin condensed matter physics, including functional analysis, spectral theory and operator algebras, and understand their relevance to physical properties and phenomena. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | | この科目を修了するまでに，学生は量子力学系および量子スピン系の格子模型に関する実践的な知識を習得します．また，ハミルトニアンのスペクトルを計算する手法を身につけ，作用素のスペクトル特性と物理的性質との関連性を理解します．さらに，英語による数学的な説明を聞き，数学的内容を英語で記述することについても，さらなる経験を積みます． |
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到達目標 【英語】
Objectives of the Course | | | By the end of the course, students will acquire a working knowledge of lattice models of single-particle quantum mechanical systems and quantum spin chains. Students will also obtain techniques for computing the spectrum of Hamiltonian models and understand the connection between spectral properties of operators with physical properties of a system. Finally students will gain further experience in listening to and writing mathematics in English. |
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授業の内容や構成
Course Content / Plan | | For this topic, we will look at the mathematical aspects of condensed matter physics in the setting of lattice systems. For the first part, we will consider single-particle dynamics (free fermions) and the interaction between the spectrum of an operator with physical properties of the system. For the second part, we will introduce and study the elementary properties of quantum spin systems.
Course outline (tentative):
Part 1: Quantum dynamics and the spectrum
- Introduction to single-particle quantum mechanics
- Locality and the Bloch decomposition/Fourier series
- Types of quantum motion and spectral type
- Linear response theory and the Kubo formula
- Topology and the integer quantum Hall effect
Part 2: Quantum spin systems
- Basics of quantum spin systems and the Heisenberg model
- Spontaneous symmetry breaking and long-range order
- Ground states and the Haldane conjecture
- Operator algebras and the infinite volume limit |
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履修条件
Course Prerequisites | | この講義を英語で行います． The course will be conducted in English.
Students are expected to have some familiarity with basic functional analysis, such as the definition of Hilbert spaces and their elementary properties.
Some knowledge of quantum mechanics will be useful but is not strictly required. |
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関連する科目
Related Courses | | | Any course on functional analysis, spectral theory and mathematical physics |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Grading will be based on 3-4 homework assignments that will be distributed throughout the semester. It is also expected that students actively participate in the lectures. |
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教科書・テキスト
Textbook | | Lecture notes will be provided.
Further references and materials will be distributed when necessary. |
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参考書
Reference Book | | Part 1:
Walter Rudin. Real and Complex Analysis. McGraw-Hill Education, 1986.
Michael Reed and Barry Simon. Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press Inc., 1980.
Part 2:
Hal Tasaki. Physics and Mathematics of Quantum Many-Body Systems. Springer, 2020. |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | | Students will be expected to independently review the lectures and consult the references for additional information. Students will also submit solutions to selected exercises. |
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注意事項
Notice for Students | | Information about the course will regularly updated on TACT and the following website.
https://sites.google.com/site/khomologyzone/teaching/masters-course-2026 |
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他学科聴講の可否
Propriety of Other department student's attendance | | | Any interested student (particularly from physics) is welcome to attend the course |
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他学科聴講の条件
Conditions of Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | | quantum mechanics, spectral theory, quantum spin chain |
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履修の際のアドバイス
Advice | | The course is a good opportunity to practice listening to and doing mathematics in English.
この科目は英語での数学を使う機会であり，学生さんが頑張ってほしいです． |
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授業開講形態等
Lecture format, etc. | | | The course will be in-person. If necessary, online and remote methods will also be implemented. |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | | If need arises, lectures will be recorded or held online. |
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