A quantitative change can lead to a qualitative change. The system with many degrees of freedom can demonstrate qualitatively new phenomena. This introductory chapter summarizes the principle of emergence, which states that the properties of material are mainly determined by the organizations (or the orders) of atoms in the material. The different organizations of microscopic degrees of freedom not only give rise to traditional symmetry breaking orders (such as crystal, magnets, superfluids, etc), they also give rise to new topological/quantum orders (such as fractional quantum Hall states, superconductors, etc). Those different orders lead to the rich properties of materials and phenomena that we see every day. The collective excitations in the new topologically ordered states can even be gauge bosons and fermions that satisfy Coulomb's law. So the principle of emergence in many-body systems may explain the origin of light and electrons, as well as the beauty in laws of physics.
A quantitative change can lead to a qualitative change. The system with many degrees of freedom can demonstrate qualitatively new phenomena. This introductory chapter summarizes the principle of emergence, which states that the properties of material are mainly determined by the organizations (or the orders) of atoms in the material. The different organizations of microscopic degrees of freedom not only give rise to traditional symmetry breaking orders (such as crystal, magnets, superfluids, etc), they also give rise to new topological/quantum orders (such as fractional quantum Hall states, superconductors, etc). Those different orders lead to the rich properties of materials and phenomena that we see every day. The collective excitations in the new topologically ordered states can even be gauge bosons and fermions that satisfy Coulomb's law. So the principle of emergence in many-body systems may explain the origin of light and electrons, as well as the beauty in laws of physics.
According to the principle of emergence, the properties of material are mainly determined by how the atoms are organized in the material. Such organization is formally called order. The vast range of materials is a result of the rich variety of orders that atoms can have. For a long time, it has been believed that all orders are described as symmetry breaking. A comprehensive theory for phases and phase transitions is developed here based on the symmetry breaking picture. The existence of FQH states (and superconducting states) indicates that there are new states of matter that cannot be described as symmetry breaking. Completely new theory is needed to describe those new states of matter. This chapter outlines the theory of topological order and theories of quantum order for the new states of matter, such as FQH states. Many new concepts and new language, such as topology-dependent degeneracy, fractional statistics, edge states, etc, are introduced to describe new states of matter.
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe. This book is a pedagogical and systematic introduction to the new concepts and quantum field theoretical methods in condensed matter physics. It discusses many basic notions in theoretical physics which underlie physical phenomena in nature, including a notion that unifies light and electrons. Topics covered include dissipative quantum systems, boson condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, and spin liquid and string-net condensation. Methods discussed include the path integral, Green's functions, mean-field theory, effective theory, renormalization group, bosonization in one- and higher dimensions, non-linear sigma-model, quantum gauge theory, dualities, projective construction, and exactly soluble models beyond one-dimension.
I give a pedagogical introduction to some of the many particles and gauge fields that can emerge in correlated matter. The standard model of materials is built on Landau's foundational principles: adiabatic continuity and spontaneous symmetry breaking. These ideas lead to quasiparticles that inherit their quantum numbers from fundamental particles, Nambu-Goldstone bosons, the Anderson-Higgs mechanism, and topological defects in order parameters. I then describe the modern discovery of physics beyond the standard model. Here, quantum correlations (entanglement) and topology play key roles in defining the properties of matter. This can lead to fractionalised quasiparticles that carry only a fraction of the quantum numbers that define fundamental particles. These particles can have exotic properties: for example Majorana fermions are their own antiparticles, anyons have exchange statistics that are neither bosonic nor fermionic, and magnetic monopoles do not occur in the vacuum. Gauge fields emerge naturally in the description of highly correlated matter and can lead to gauge bosons. Relationships to the standard model of particle physics are discussed.
In general, we know that there are four states of matter solid, liquid, gas and plasma. But there are much more states of matter. For e. g. there are ferromagnetic states of matter as revealed by the phenomenon of magnetization and superfluid states defined by the phenomenon of zero viscosity. The various phases in our colorful world are so rich that it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. Topological phenomena define the topological order at macroscopic level. Topological order need new mathematical framework to describe it. More recently it is found that at microscopic level topological order is due to the long range quantum entanglement, just like the fermions fluid is due to the fermion-pair condensation. Long range quantum entanglement leads to many amazing emergent phenomena, such as fractional quantum numbers, non- Abelian statistics ad perfect conducting boundary channels. It can even provide a unified origin of light and electron i.e. gauge interactions and Fermi statistics. Light waves (gauge fields) are fluctuations of long range entanglement and electron (fermion) are defect of long range entanglements.The Himalayan Physics Vol. 6 & 7, April 2017 (108-111)
We review the progress in the last 20–30 years, during which we discovered that there are many new phases of matter that are beyond the traditional Landau symmetry breaking theory. We discuss new “topological” phenomena, such as topological degeneracy that reveals the existence of those new phases—topologically ordered phases. Just like zero viscosity defines the superfluid order, the new “topological” phenomena define the topological order at macroscopic level. More recently, we found that at the microscopical level, topological order is due to long-range quantum entanglements. Long-range quantum entanglements lead to many amazing emergent phenomena, such as fractional charges and fractional statistics. Long-range quantum entanglements can even provide a unified origin of light and electrons; light is a fluctuation of long-range entanglements, and electrons are defects in long-range entanglements.
The study of many-body physics has provided a scientic playground of surprise and continuing revolution over the past half century. The serendipitous discovery of new states and properties of matter, phenomena such as superfluidity, the Meissner, the Kondo and the fractional quantum hall eects, have driven the development of new conceptual frameworks for our understanding about collective behavior, the ramications of which have spread far beyond the connes of terres- trial condensed matter physics- to cosmology, nuclear and particle physics. Here I shall selectively review some of the developments in this eld, from the cold-war period, until the present day. I describe how, with the discovery of new classes of collective order, the unfolding puzzles of high temperature superconductivity and quantum criticality, the prospects for major conceptual discoveries remain as bright today as they were more than half a century ago. 1 Emergent Matter: a new Frontier Since the time of the Greeks, scholars have pondered over the principles that govern the universe on its tiniest and most vast scales. The icons that exemplify these frontiers are very well known - the swirling galaxy denoting the cosmos and the massive accelerators used to probe matter at successively smaller scales- from the atom down to the quark and beyond. These traditional frontiers of physics are largely concerned with reductionism: the notion that once we know the laws of nature that operate on the smallest possible scales, the mysteries of the universe will nally be revealed to us(1). Over the last century and a half, a period that stretches back to Darwin and Boltzmann- scientists have also become fascinated by another notion: the idea that to understand nature, one also needs to understand and study the princi- ples that govern collective behavior of vast assemblies of matter. For a wide range of purposes, we already know the microscopic laws that govern matter on the tiniest scales. For example, a gold atom can be completely understood with the Schrodinger equation and the laws of quantum mechanics established more than seventy years ago. Yet, a gold atom is spherical and featureless- quite unlike the lustrous malleable and conducting metal which human society so prizes. To under- stand how crystalline assemblies of gold atoms acquire the properties of metallic gold, we need new principles{ principles that describe the collective behavior of matter when humungous numbers of gold atoms congregate to form a metallic crystal. It is the search for these new principles that denes the frontiers of many- body physics in the realms of condensed matter physics and its closely related