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.
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.
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.
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.
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)
The Landau-Ginzburg theory has been immensely successful to model different states of matter. In this theory, order is associated with spontaneous symmetry breaking described using group theory. For many decades now, new states of matter have been experimentally produced (e.g. Hall fluids). In these liquids, matter is strongly correlated and has an internal structures, like in a solid, however the implied order cannot be characterized by symmetry breaking of local order parameters. In other words, to understand these states of matter, we have to go beyond the Landau-Ginzburg approach and, as sustain by Xiao- Gang Wen (2004), develop a new classification of orders, especially what he called topological orders. Moreover this notion of order should be introduced via quantities that can be measured. This talk sets out to achieve three primary objectives: 1. Evaluate Different Notions of Order: This involves a critical examination of various proposed notions of order to discern whether they represent an evolution or a transformation from the concept as envisaged in the Landau-Ginzburg theory. The development of topological order, particularly noted for its adherence to measurable quantities, provides an intriguing backdrop for this analysis. The theories of Nussinov and Ortiz (2009a, 2009b), which continue to honor symmetry principles, will be scrutinized in detail to facilitate this comparison. 2. Explore the Concept of Topological Transition: Inspired by Continentino (2017), this section will delve into the implications of topological order on our understanding of phase transitions. The inquiry will center on whether topological transitions introduce an entirely new paradigm or if they are fundamentally analogous to traditional notions of phase transition. 3. Investigate the Notion of Transition as Ontological Emergence: The final aim is to discuss how this novel perspective on transitions might exemplify a case of diachronic ontological emergence, as suggested by Humphreys (2016) and Guay and Sartenaer (2016). The hypothesis posits that quantum fluid transitions offer the most compelling evidence yet of such emergence.
Matter exists in many different phases, for example in solid state or in liquid phase. There are also phases in which the ordering of atoms is the same, but which differ in some other respect, for example ferromagnetic and paramagnetic states. According to Landau's symmetry breaking theory every phase transition is connected to a symmetry breaking process. A solid material has discrete translational symmetry, while liquid phase has continuous translational symmetry. But it has turned out that there also exist phase transitions that can occur without a symmetry breaking. This phenomenon is called topological order. In this thesis we consider one example of a theoretical model constructed on a two dimensional lattice in which one obtains topological order.
Abstract In primary school, we were told that there are four phases of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four phases of matter, such as hundreds of crystal phases, liquid crystal phases, ferromagnet, anti-ferromagnet, superfluid, etc. Those phases of matter are so rich, it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. However, there are even more interesting phases of matter that are beyond Landau symmetry breaking theory. In this paper, we review new ‘topological’ phenomena, such as topological degeneracy, that reveal the existence of those new zero-temperature phase—topologically ordered phases. Microscopically, topologically orders are originated from the patterns of long-range entanglement in the ground states. As a truly new type of order and a truly new kind of phenomena, topological order and long-range entanglement require a new language and a new mathematical framework, such as unitary fusion category and modular tensor category to describe them. In this paper, we will describe a simple mathematical framework based on measurable quantities of topological orders (S, T, c) proposed around 1989. The framework allows us to systematically describe all 2+1D bosonic topological orders (i.e. topological orders in local bosonic/spin/qubit systems).