[Physics/Math]/Physics

# Quantum Field Theory of Many-body Systems by Xiao-Gang Wen

# Quantum Field Theory of Many-body Systems by Xiao-Gang Wen
Xiao-Gang Wen
Department of Physics, MIT
First published 2004
docuK format 테스트 해본다고 만들었던거 같은데... 아까워서 그냥 다시 좀 더 정리. Copyright 가 내꺼가 아니라, 어차피 책 전체를 docuK 화 했어도 공개 포스팅은 못했을듯. 그런데 이거 책 pdf 에서 복붙한건가??? 초반만 읽고 읽지도 않았던거 같은데 이책. 왜 이 책을 골랐지? =ㅇ=;;;;; QFT 공부해 볼 생각에 그랬나??? QFT 알 못. QM 도 잘 모르겠는데, QM 을 특수상대론하고 어찌 짬뽕한건지도 잘 이해가 안갔던... 시작이 Second Quantization 이었던거 같은데... 교수가 설명을 못하는거야? 내가 멍청해서 이해를 못하는거야? 교수님은 제대로 이해 하고 설명했는지도 의문이었음... 뿌웱.
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• 2015-12-17 : First posting. 정리하기 귀찮다. 방치하자.
##[.hiden.no-sec-N] PREFACE The quantum theory of condensed matter (i.e. solids and liquids) has been dominated by two main themes. The first one is band theory and perturbation theory. It is loosely based on Landau's Fermi liquid theory. The second theme is Landau's symmetry-breaking theory and renormalization group theory. Condensed matter theory is a very successful theory. It allows us to understand the properties of almost all forms of matter. One triumph of the first theme is the theory of semiconductors, which lays the theoretical foundation for electronic devices that make recent technological advances possible. The second theme is just as important. It allows us to understand states of matter and phase transitions between them. It is the theoretical foundation behind liquid crystal displays, magnetic recording, etc. As condensed matter theory has been so successful, one starts to get a feeling of completeness and a feeling of seeing the beginning of the end of condensed matter theory. However, this book tries to present a different picture. It advocates that what we have seen is just the end of the beginning. There is a whole new world ahead of us waiting to be explored. A peek into the new world is offered by the discovery of the fraction quantum Hall effect (Tsui et al, 1982). Another peek is offered by the discovery of high-Tc superconductors (Bednorz and Mueller, 1986). Both phenomena are completely beyond the two themes outlined above. In last twenty years, rapid and exciting developments in the fraction quantum Hall effect and in high-Tc superconductivity have resulted in many new ideas and new concepts. We are witnessing an emergence of a new theme in the many-body theory of condensed matter systems. This is an exciting time for condensed matter physics. The new paradigm may even have an impact on our understanding of fundamental questions of nature. It is with this background that I have written this book. The first half of this book covers the two old themes, which will be called traditional condensed matter theory. The second part of this book offers a peek into the emerging new theme, which will be called modern condensed matter theory. The materials covered in the second part are very new. Some of them are new results that appeared only a few months ago. The theory is still developing rapidly.
1 When I started to write this book in 1996, I planned to cover some new and exciting developments in quantum many-body theory. At that time it was not clear if those new developments would become a new theme in condensed matter theory. At the moment, after some recent progress, I myself believe that a new theme is emerging in condensed matter theory. However, the theory is still in the early stages of its development. Only time will tell if we really do get a new theme or not.
2 Some people may call the first theme traditional condensed matter theory and the second theme modern condensed matter theory.
After reading this book, I hope, instead of a feeling of completeness, readers will have a feeling of emptiness. After one-hundred years of condensed matter theory, which offers us so much, we still know so little about the richness of nature. However, instead of being disappointed, I hope that readers are excited by our incomplete understanding. It means that the interesting and exciting time of condensed matter theory is still ahead of us, rather than behind us. I also hope that readers will gain a feeling of confidence that there is no question that cannot be answered and no mystery that cannot be understood. Despite there being many mysteries which remain to be understood, we have understood many mysteries which initially seemed impossible to understand. We have understood some fundamental questions that, at the beginning, appeared to be too fundamental to even have an answer. The imagination of the human brain is also boundless.
3 I wonder which will come out as a 'winner', the richness of nature or the boundlessness of the human imagination.
However, to make revolutionary advances in physics, we cannot allow our imagination to be trapped by the formalism. We cannot allow the formalism to define the boundary of our imagination. The mathematical formalism is simply a tool or a language that allows us to describe and communicate our imagination. Sometimes, when you have a new idea or a new thought, you might find that you cannot say anything. Whatever you say is wrong because the proper mathematics or the proper language with which to describe the new idea or the new thought have yet to be invented. Indeed, really new physical ideas usually require a new mathematical formalism with which to describe them. This reminds me of a story about a tribe. The tribe only has four words for counting: one, two, three, and many-many. Imagine that a tribe member has an idea about two apples plus two apples and three apples plus three apples. He will have a hard time explaining his theory to other tribe members. This should be your feeling when you have a truly new idea. Although this book is entitled Quantum field theory of many-body systems, I hope that after reading the book the reader will see that quantum field theory is not everything. Nature's richness is not bounded by quantum field theory. I would like to thank Margaret O'Meara for her proof-reading of many chapters of the book. I would also like to thank Anthony Zee, Michael Levin, Bas Overbosch, Ying Ran, Tiago Ribeiro, and Fei-Lin Wang for their comments and suggestions. Last, but not least, I would like to thank the copy-editor Dr. Julie Harris for her efforts in editing and polishing this book. ## TOC ## Introduction ### 1.1 More is different The collective excitations of a many-body system can be viewed as particles. However, the properties of those particles can be very different from the properties of the particles that form the many-body system. Guessing is better than deriving. Limits of classical computing. Our vacuum is just a special material. A quantitative change can lead to a qualitative change. This philosophy is demonstrated over and over again in systems that contain many particles (or many degrees of freedom), such as solids and liquids. The physical principles that govern a system of a few particles can be very different from the physical principles that govern the collective motion of many-body systems. New physical concepts (such as the concepts of fermions and gauge bosons) and new physical laws and principles (such as the law of electromagnetisnl) can arise from the correlations of many particles (see Chapter 10). Condensed matter physics is a branch of physics which studies systems of many particles in the 'condensed' (i.e. solid or liquid) states. The starting-point of current condensed matter theory is the Schrodinger equation that governs the motion of a number of particles (such as electrons and nuclei). The Schrodinger equation is mathematically complete. In principle, we can obtain all of the properties of any many-body system by solving the corresponding Schrodinger equation. However, in practice, the required computing power is immense. In the 1980s, a workstation with 32 Mbyte RAM could solve a system of eleven interacting electrons. After twenty years the computing power has increased by 100-fold, which allows us to solve a system with merely two more electrons. The computing power required to solve a typical system of $10^{23}$ interacting electrons is beyond the imagination of the human brain. A classical computer made by all of the atoms in our universe would not be powerful enough to handle the problem/' Such an impossible computer could only solve the Schrodinger equation for merely about 100 particles.6 We see that an generic interacting many-body system is an extremely complex system. Practically, it is impossible to deduce all of its exact properties from the Schrodinger equation. So, even if the Schrodinger equation is the correct theory for condensed matter systems, it may not always be helpful for obtaining physical properties of an interacting many-body system. Even if we do get the exact solution of a generic interacting many-body system, very often the result is so complicated that it is almost impossible to understand it in full detail. To appreciate the complexity of the result, let us consider a tiny interacting system of 200 electrons. The energy eigenvalues of the system are distributed in a range of about 200 eV. The system has at least $2^{200}$ energy levels. The level spacing is about $200 \text{eV}/2^{200}$ = $10^{-60}$ eV. Had we spent a time equal to the age of the universe in measuring the energy, then, due to the energy-time uncertainty relation, we could only achieve an energy resolution of order $10^{-33}$ eV. We see that the exact result of the interacting many-body system can be so complicated that it is impossible to check its validity experimentally in full detail.7 To really understand a system, we need to understand the connection and the relationship between different phenomena of a system. Very often, the Schrodinger equation does not directly provide such an understanding. As we cannot generally directly use the Schrodinger equation to understand an interacting system, we have to start from the beginning when we are faced with a many-body system. We have to treat the many-body system as a black box, just as we treat our mysterious and unknown universe. We have to guess a low-energy effective theory that directly connects different experimental observations, instead of deducing it from the Schrodinger equation. We cannot assume that the theory that describes the low-energy excitations bears any resemblance to the theory that describes the underlying electrons and nuclei. This line of thinking is very similar to that of high-energy physics. Indeed, the study of strongly-correlated many-body systems and the study of high-energy physics share deep-rooted similarities. In both cases, one tries to find theories that connect one observed experimental fact to another. (Actually, connecting one observed experimental fact to another is almost the definition of a physical theory.) One major difference is that in high-energy physics we only have one 'material'(our vacuum) to study, while in condensed matter physics there are many different materials which may contain new phenomena not present in our vacuum (such as fractional statistics, non-abelian statistics, and gauge theories with all kinds of gauge groups). ### 1.2 'Elementary' particles and physics laws Emergence—the first principle of many-body systems. Origin of 'elementary' particles. Origin of the 'beauty' of physics laws. (Why nature behaves reasonably.) Historically, in our quest to understand nature, we have been misled by a fundamental (and incorrect) assumption that the vacuum is empty. We have (incorrectly) assumed that matter placed in a vacuum can always be divided into smaller parts. We have been dividing matter into smaller and smaller parts, trying to discover the smallest 'elementary' particles—the fundamental building block of our universe. We have been believing that the physics laws that govern the 'elementary'particles must be simple. The rich phenomena in nature come from these simple physics laws. However, many-body systems present a very different picture. At high energies (or high temperatures) and short distances, the properties of the many-body system are controlled by the interaction between the atoms/molecules that form the system. The interaction can be very complicated and specific. As we lower the temperature, depending on the form of the interaction between atoms, a crystal structure or a superfluid state is formed. In a crystal or a superfluid, the only low-energy excitations are collective motions of the atoms. Those excitations are the sound waves. In quantum theory, all of the waves correspond to particles, and the particle that corresponds to a sound wave is called a phonon.8 Therefore, at low temperatures, a new 'world' governed by a new kind of particle—phonons— emerges. The world of phonons is a simple and 'beautiful' world, which is very different from the original system of atoms/molecules. Let us explain what we mean by 'the world of phonons is simple and beautiful'. For simplicity, we will concentrate on a superfluid. Although the interaction between atoms in a gas can be complicated and specific, the properties of emergent phonons at low energies are simple and universal. For example, all of the phonons have an energy-independent velocity, regardless of the form of the interactions between the atoms. The phonons pass through each other with little interaction despite the strong interactions between the atoms. In addition to the phonons, the superfluid also has another excitation called rotons. The rotons can interact with each other by exchanging phonons, which leads to a dipolar interaction with a force proportional to 1/r4. We see that not only are the phonons emergent, but even the physics laws which govern the low-energy world of the phonons and rotons are emergent. The emergent physics laws (such as the law of the dipolar interaction and the law of non-interacting phonons) are simple and beautiful. ### 1.3 Corner-stones of condensed matter physics ### 1.4 Topological order and quantum order ### 1.5 Origin of light and fermions ### 1.6 Novelty is more important than correctness ### 1.7 Remarks: evolution of the concept of elementary particles ## Path integral formulation of quantum mechanics ###2.1 Semiclassical picture and path integral ###2.2 Linear responses and correlation functions ###2.3 Quantum spin, the Berry phase, and the path integral ###2.4 Applications of the path integral formulation ## Interacting boson systems ### 3.1 Free boson systems and second quantization ### 3.2 Mean-field theory of a superfluid ### 3.3 Path integral approach to interacting boson systems ### 3.4 Superfluid phase at finite temperatures ### 3.5 Renormalization group ### 3.6 Boson superfluid to Mott insulator transition ### 3.7 Superfluidity and superconductivity ### 3.8 Perturbative calculation of the thermal potential ## Free fermion systems ### 4.1 Many-fermion systems ### 4.2 Free fermion Green's function ### 4.3 Two-body correlation functions and linear responses ### 4.4 Quantized Hall conductance in insulators ## Interacting fermion systems ### 5.1 Orthogonality catastrophe and X-ray spectrum ### 5.2 Hartree-Fock approximation ### 5.3 Landau Fermi liquid theory ### 5.4 Perturbation theory and the validity of Fermi liquid theory ### 5.5 Symmetry-breaking phase and the spin-density-wave state ### 5.6 Nonlinear $\sigma$-model ## Quantum gauge theories ### 6.1 Simple gauge theories ### 6.2 $Z_2$ lattice gauge theory ### 6.3 $U(1)$ gauge theory and the XY-model in $1+2$ dimensions ### 6.4 The quantum $U(1)$ gauge theory on a lattice ### Theory of quantum Hall states ### 7.1 The Aharonov-Bohm effect and fractional statistics ### 7.2 The quantum Hall effect ### 7.3 Effective theory of fractional quantum Hall liquids ### 7.4 Edge excitations in fractional quantum Hall liquids ## Topological and quantum order ### 8.1 States of matter and the concept of order ### 8.2 Topological order in fractional quantum Hall states ### 8.3 Quantum orders ### 8.4 A new classification of orders ## Mean-field theory of spin liquids and quantum order ### 9.1 Projective construction of quantum spin-liquid states ### 9.2 The SU(2) projective construction ### 9.3 Topological orders in gapped spin-liquid states ### 9.4 Quantum orders in symmetric spin liquids ### 9.5 Continuous phase transitions without symmetry breaking ### 9.6 The zoo of symmetric spin liquids ### 9.7 Physical measurements of quantum orders ### 9.8 The phase diagram of the $J_1$--$J_2$ model in the large-N limit ### 9.9 Quantum order and the stability of mean-field spin liquids ### 9.10 Quantum order and gapless gauge bosons and fermions ## String condensation.an unification of light and fermions ### 10.1 Local bosonic models ### 10.2 An exactly soluble model from a projective construction ### 10.3 $Z_2$ spin liquids and string-net condensation ### 10.4 Classification of string-net condensations ### 10.5 Emergent fermions and string-net condensation ### 10.6 The quantum rotor model and $U(1)$ lattice gauge theory ### 10.7 Emergent light and electrons from an $SU(N_f)$ spin model ## RRA