Physics and Mathematics of Quantum Many-Body Systems

Preface
Contents
Symbols
1 Introduction
	1.1 Universality in Macroscopic Physics
	1.2 Overview of the Book
	References
2 Basics of Quantum Spin Systems
	2.1 Quantum Mechanics of a Single Spin
	2.2 Quantum Spin Systems
	2.3 Time-Reversal and the Kramers Degeneracy
	2.4 The Ferromagnetic Heisenberg Model
	2.5 The Antiferromagnetic Heisenberg Model
	References
Part I Long-Range Order and Spontaneous Symmetry Breaking
3 Long-Range Order and Spontaneous Symmetry Breaking in the Classical and Quantum Ising Models
	3.1 Motivation from the Heisenberg Antiferromagnet
	3.2 Classical Ising Model
	3.3 Quantum Ising Model
	3.4 General Theory of Low-Lying States and SSB
	References
4 Long-Range Order and Spontaneous Symmetry Breaking in the Antiferromagnetic Heisenberg Model
	4.1 Existence of Long-Range Order
		4.1.1 Main Results
		4.1.2 Proof of the Existence of LRO
	4.2 The ``Tower\'\' of Low-Lying States and Spontaneous Symmetry Breaking
		4.2.1 Main Results
		4.2.2 Proofs
	4.3 Ground States of the Infinite System
		4.3.1 Construction of Ground States
		4.3.2 Physical Versus Unphysical Ground States
	4.4 Equilibrium States of the Heisenberg Model
		4.4.1 Disorder at High-Temperature and in One-Dimension
		4.4.2 Berezinskii\'s Harmonic Approximation
		4.4.3 Absence of Order in Two Dimensions
		4.4.4 LRO and SSB in Three or Higher Dimensions
	References
5 Long-Range Order and ``Spontaneous Symmetry Breaking\'\' in Bose–Einstein Condensates
	5.1 The Model and the Equivalence to the XY Model
	5.2 Off-Diagonal Long-Range Order
	5.3 Symmetry Breaking ``Ground States\'\'
	5.4 Physical Ground States of Bose–Einstein Condensates
	5.5 ``SSB\'\' in Coupled Bose–Einstein Condensates
	References
Part II Haldane Phenomena and Beyond
6 Ground States of the Antiferromagnetic Heisenberg Chains
	6.1 Haldane ``Conjecture\'\'
	6.2 The Lieb–Schultz–Mattis Theorem
	6.3 Semi-classical Approach
	References
7 Affleck–Kennedy–Lieb–Tasaki Model
	7.1 The Model and Main Results
		7.1.1 The Hamiltonian and the Main Theorem
		7.1.2 The Exact Ground State
		7.1.3 The Uniqueness of the Ground State
		7.1.4 The Proof of the Existence of Gap
	7.2 Properties of the AKLT Model
		7.2.1 Hidden Antiferromagnetic Order
		7.2.2 Matrix Product Representation
		7.2.3 AKLT Model on Open Chains
	7.3 Extensions and Related Models
		7.3.1 Spin Chains with Higher S and the VBS Picture
		7.3.2 Higher Dimensional Models
		7.3.3 Briegel–Raussendorf State (Cluster State) and Its Generalizations
	References
8 Haldane Phase
	8.1 Characterization of the Haldane Phase
		8.1.1 Topological Phase Transition in Anisotropic Model
		8.1.2 Hidden Antiferromagnetic Order
		8.1.3 Emergence of Edge States
	8.2 Hidden mathbbZ2timesmathbbZ2 Symmetry Breaking
		8.2.1 Phase Diagram of the λ-D Model
		8.2.2 Nonlocal Unitary Transformation
		8.2.3 The Picture of Hidden mathbbZ2timesmathbbZ2 Symmetry Breaking
	8.3 Symmetry Protected Topological Phase
		8.3.1 More on the Picture of the Hidden mathbbZ2timesmathbbZ2 Symmetry Breaking
		8.3.2 Symmetry Protected Topological (SPT) Phase
		8.3.3 Entanglement and ``Topological\'\' Indices for SPT Phases
		8.3.4 ``Topological\'\' Indices for Matrix Product States
		8.3.5 Lieb–Schultz–Mattis Type Theorem Without Continuous Symmetry
		8.3.6 Rigorous Index Theorems for SPT Phases
	8.4 Topological Order in Kitaev\'s Toric Code Model
	References
Part III Hubbard Model and the Origin  of Magnetism
9 Introduction to the Hubbard Model
	9.1 What is the Hubbard Model?
	9.2 Tight-Binding Description of Electrons in a Solid
		9.2.1 Wave Functions for Electrons
		9.2.2 Creation and Annihilation Operators in the Wave Function Formalism
		9.2.3 The Fock Space Representation
	9.3 Definition of the Hubbard Model
		9.3.1 Single-Electron Schrödinger Equation
		9.3.2 Hamiltonian of the Hubbard Model
		9.3.3 Transformations and Symmetry of the Hubbard Hamiltonian
	9.4 Bosonic Hubbard Model
	References
10 Half-Filled Models: Lieb\'s Theorems and the Origin of Antiferromagnetism and Ferrimagnetism
	10.1 Formal Perturbation Theory for Strong Interaction
	10.2 Lieb\'s Theorems
		10.2.1 Lieb\'s Theorem for the Attractive Hubbard Model
		10.2.2 Lieb\'s Theorem for the Repulsive Hubbard Model
		10.2.3 Lieb\'s Ferrimagnetism
		10.2.4 Proofs of Theorems 10.2 and 10.3
		10.2.5 Extensions and Other Rigorous Results
	References
11 The Origin of Ferromagnetism
	11.1 Basic Properties of Ferromagnetism
		11.1.1 Definition and Basic Theorem
		11.1.2 Instability of Ferromagnetism
		11.1.3 Toy Model with Two Electrons
		11.1.4 Stoner Criterion
	11.2 Nagaoka\'s Ferromagnetism
		11.2.1 Weak Version of Nagaoka\'s Theorem
		11.2.2 Nagaoka\'s Theorem and the Connectivity Condition
		11.2.3 Instability of Nagaoka\'s Ferromagnetism
	11.3 Flat-Band Ferromagnetism
		11.3.1 Tasaki\'s Flat-Band Ferromagnetism
		11.3.2 Mielke\'s Flat-Band Ferromagnetism
		11.3.3 Construction of Tight-Binding Models with Flat-Bands
		11.3.4 General Theory of Flat-Band Ferromagnetism
	11.4 Ferromagnetism in Non-singular Hubbard Models
		11.4.1 Wannier State Perturbation Theory
		11.4.2 Local Stability and the Spin Wave Excitation
		11.4.3 Ferromagnetism in Non-singular Hubbard Models
	11.5 Toward Metallic Ferromagnetism
		11.5.1 Heuristic Arguments
		11.5.2 Rigorous Results
	References
Appendix A Mathematical Appendices
A.1 Dirac Notation
A.2  Useful Properties of Operators
A.2.1  Commutator and Operator Norm
A.2.2  Exponential of an Operator
A.2.3  Inequality Between Self-adjoint Operators and Nonnegative Operators
A.3  Quantum Mechanical Angular Momentum
A.3.1  Definition and Basic Properties
A.3.2  SU(2) Invariant Hamiltonian
A.3.3  Addition of Angular Momenta
A.4  Some Linear Algebra
A.4.1  Perron–Frobenius Theorem
A.4.2  Decomposition of Matrices
A.4.3  Antilinear Operators
A.5  Groups and Their Representations
A.6  Wigner\'s Theorem
A.6.1  Statement and the Proof
A.6.2  Applications
A.7  Operator Algebraic Formulation of Infinite Systems
Appendix  Solutions
Index