Abstract
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).
Highlights
Before we present the result from the numerical calculation, let us discuss a stacking operation,[63] denoted by
It can be realized by the following filling-fraction ν = 3/2 bosonic fractional quantum Hall (FQH) wave function described by the following pattern of zeros: {nl} = {n0, n1, n2, · · · }: Ψ4B 9/5 : {nl} = 30|30|30| · · ·, (65) i.e. neven = 3 and nodd = 0
We review the discovery and development of topological order – a new kind of order beyond Landau symmetry breaking theory in many-body systems
Summary
Condensed matter physics is a branch of science that study various properties of all kinds of materials, such as mechanical properties, hydrodynamic properties, electric properties, magnetic properties, optical properties, thermal properties, etc. The common theme is the principle of emergence, which states that the properties of a material are mainly determined by how particles are organized in the material. Rapid and exciting developments in FQH effect and in high Tc superconductivity resulted in many new ideas and new concepts Looking back at those new developments, it becomes more and more clear that, in last 25 years, we were witnessing an emergence of a new theme in condensed matter physics. The new theme is associated with new kinds of orders, new states of matter and new class of materials beyond Landau’s symmetry breaking theory. This is an exciting time for condensed matter physics. The new paradigm may even have an impact in our understanding of fundamental questions of nature – the emergence of elementary particles and the four fundamental interactions.[8,9,10,11,12,13]
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