# wł̏Wu iL^j / Lecture series at U Osaka

J: 2015N729 / ŏIXV: 2015N912
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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).

c萰 / Hal Tasaki

2015 N 9 9 ijA10 i؁jA11 ij13:00  ܂Łixe͐j
September 9, 10, and 11, from 13:00 till the end (with some intermissions)

wiLLpXjw D D303 D501 iύX܂j
the University of Osaka (Toyonaka campus), Faculy of Science D303

@LƋL@̂܂Ƃi2015N828jOɊȒPɖڂʂĂĂj
@Summary of symbols and notations (Revised: Aug. 28, 2015) Please take a look before the lectures.

@um[gi菑j/ hand written lecture notes (pdf, 3-4 MB): DAY1, DAY2, DAY3
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I will discuss such typical many body systems as the Heisenberg model (and its variants), many-boson systems on a lattice (a model for cold atoms in a trap), and the Hubbard model, focusing on nontrivial physical phenomena including spontaneous symmetry breaking, quantum fluctuation in ground states, and the generation of ferromagnetism.

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The plan is that I speak in Japanese and write in English (on the blackboard). Questions/discussions in English are welcome.

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You are supposed to solve two or three from the problems that I will present during the class. The problems will be of elementary nature.

## 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 1
T. Koma and H. Tasaki, Symmetry Breaking and Finite Size Effects in Quantum Many-Body Systems

DAY 2
I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets
c萰wʎqXsn̗_FHaldane Gap, Disordered Ground States, Quantum Spin Liquid and All That in Quantum Spin Systemsx

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

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