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.
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.
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)
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).
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.
It has long been thought that all different phases of matter arise from symmetry breaking. Without symmetry breaking, there would be no pattern, and matter would be featureless. However, it is now clear that for quantum matter at zero temperature, even symmetric disordered liquids can have features, giving rise to topological phases of quantum matter. Some of the topological phases are highly entangled (that is, have topological order), whereas others are weakly entangled (that is, have symmetry-protected trivial order). This Review provides a brief summary of these zero-temperature states of matter and their emergent properties, as well as their importance in unifying some of the most basic concepts in nature.
The Landau Fermi liquid theory and the Ginzburg-Landau phase transition theory stand as two pivotal cornerstones in traditional condensed matter physics, achieving significant success in addressing crucial physical phenomena such as BCS superconductors and liquid helium superfluids. However, marked by the discoveries of the quantum Hall effect and high-temperature superconductivity in the 1980s, it gradually became evident that for a broad class of novel quantum states, such as fractional quantum Hall states and quantum spin liquids, their properties transcend the Landau Fermi liquid theory and Ginzburg-Landau phase transition theory. Topological order and its related concepts of long-range many-body quantum entanglement and fractionalized excitation have become the key concepts to understand these exotic quantum states. Designing and identifying topologically ordered states of matter in quantum materials and quantum simulation systems, and probing and manipulating their fractionalized excitations, are important research directions in modern condensed matter physics. In recent years, great progress has been made in the quantum simulation and manipulation of topological order on highly controllable quantum simulation platforms, such as Rydberg atomic systems, superconducting quantum processors, and two-dimensional moiré superlattices. This article provides a brief overview of recent research advances and challenges in the study of topological order in traditional condensed matter systems and quantum simulation experimental platforms. It also provides prospects for the future developments of this field.