大阪大学での集中講義 （記録） / Lecture series at U Osaka

公開: 2015年7月29日 / 最終更新日: 2015年9月12日
阪大素粒子論研究室の公式ページへ

以下のような集中講義をおこないます。 題材は（広い意味での）磁性が中心ですが、（非相対論的な）大自由度量子系の非自明な物理を理解する営みの最良の到達点のいくつかを描き出すことを目指します。 興味をお持ちの方の参加を歓迎します（と、ぼくが書いていいのかどうかしらないけど、歓迎します）。
I will give a series of lectures at U Osaka. The topics are mainly from magnetism (in the broad sense), but my goal is to illustrate some of the best results in our attempt to understand nontrivial physics in (non-relativistic) many-body quantum systems. Anybody is welcome (at leas for me).

成績評価について

About grading

Prerequisites

• Elementary statistical mechanics and elementary quantum mechanics
• Basic picture about the ferromagnetic phase transition
• Basics about quantum mechanical spin operators
• Basics about creation and annihilation operators, both bosonic and fermionic

Plan of the lectures

DAY 1: Long-range order (LRO) and spontaneous symmetry breaking (SSB) in nonrelativistic quantum systems

• Some facts from the ferromagnetic phase transitions
• LRO and SSB in the ground state of quantum spin systems
  • Some elementary linear algebra
  • Quantum spin systems --- general definitions and properties
  • Ferromagnetic Heisenberg model
  • Antiferromagnetic (AF) Heisenberg model (Marshall-Lieb-Mattis theorem)
  • LRO in the ground state of the Heisenberg AF in \(d\ge2\) (Dyson-Lieb-Simon theorem and extensions)
  • From LRO to SSB --- Ising model under transverse magnetic field
  • From LRO to SSB (Kaplan-Horsch-von der Linden theorem)
  • From LRO to SSB in systems with a continuous symmetry (Koma-Tasaki theorem)
  • Very short remarks about the ground states for the infinite system
  • Very short remarks about phase transition, LRO and SSB in the AF Heisenberg model at nonzero temperatures (Dyson-Lieb-Simon and Koma-Tasaki theorems)
• LRO and SSB associated with the Bose-Einstein condensation
  • Lattice boson system
  • Off-diagonal LRO
  • Low-lying states with explicit symmetry breaking
  • Physical "SSB" in a coupled system

DAY 2: "Quantum spin liquid" in the ground states of low dimensional quantum spin systems

• Haldane conjecture and related results
  • Haldane conjecture
  • Theorem which rules out "unique ground state + gap" (Lieb-Scultz-Mattis thoerem, and its extensions by Affleck-Lieb)
  • Initial (personal) thoughts --- Semi-classical approach
• AKLT (Affleck-Kennedy-Lieb-Tasaki) model and the VBS picture
  • AKLT model for \(S=1\)
  • VBS (valence-bond-solid) state
  • VBS state in the standard basis --- hidden antiferromagnetic order
    (quick introduction to MPS (matrix product sates))
  • VBS states on open chains --- edge states
  • VBS picture
• Haldane phase
  • Haldane conjecture for the \(S=1\) Heisenberg antiferromagnetic chain
  • The model with anisotropy
  • Peculiar features of the Haldane phase
  • Non-local unitary transformation and the hidden \(\mathbb{Z}_2\times\mathbb{Z}_2\) symmetry breaking (Kennedy, Tasaki)
• Some related issues (very briefly!)
  • Stability of the Haldane phase (symmetry protected "topological" order)
  • VBS states in higher dimensions
  • remarks about "states vs. Hamiltonian"

DAY 3: The origin of magnetism and the Hubbard model --- constructive condensed matter physics

• Hubbard model
  • Operators and states
  • Hopping Hamiltonian
  • Interaction Hamiltonian
  • Hubbard model
• Half filled system
  • Limitting cases
  • Perturbation
  • Lieb's theorem
• Towards ferromagnetism
  • Toy model with three sites
  • Nagaoka-Thouless ferromagnetism (briefly, if possible)
• Flat-band ferromagnetism
  • Model and main theorem
  • Special properties of the model
  • Proof
  • Some remarks
• Ferromagnetism in a non-singular Hubbard model
• Metallic ferromagnetism

Useful references available (to anyone!) on the web

DAY 2
I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets
田崎晴明『量子スピン系の理論：Haldane Gap, Disordered Ground States, Quantum Spin Liquid and All That in Quantum Spin Systems』

DAY 3
H. Tasaki, From Nagaoka's Ferromagnetism to Flat-Band Ferromagnetism and Beyond: An Introduction to Ferromagnetism in the Hubbard Model