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. 1. Introduction 1.1. Phases of Matter and Landau Symmetry-Breaking Theory Although all matter is formed by only three kinds of particles: electrons, protons, and neutrons, matter can have many different properties and appear in many different forms, such as solid, liquid, conductor, insulator, superfluid, and magnet. According to the principle of emergence in condensed matter physics, the rich properties of materials originate from the rich ways in which the particles are organized in the materials. Those different organizations of the particles are formally called the orders in the materials. For example, particles have a random distribution in a liquid (see Figure 1(a)), so a liquid remains the same as we displace it by an arbitrary distance. We say that a liquid has a “continuous translation symmetry.” After a phase transition, a liquid can turn into a crystal. In a crystal, particles organize into a regular array (a lattice) (see Figure 1(b)). A lattice remains unchanged only when we displace it by a particular set of distances (integer times of lattice constant), so a crystal has only “discrete translation symmetry.” The phase transition between a liquid and a crystal is a transition that reduces the continuous translation symmetry of the liquid to the discrete symmetry of the crystal. Such a change in symmetry is called “spontaneous symmetry breaking.” We note that the equation of motions that governs the dynamics of the particles respects the continuous translation symmetry for both cases of liquid and crystal. However, in the case of crystal, the stronger interaction makes the particles to prefer being separated by a fixed distance and a fixed angle.
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