Abstract
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.
Highlights
Physicists used to believe that Landau symmetrybreaking theory describes all possible quantum phases of matter and all possible quantum phase transitions. (Quantum phase transitions are zero temperature phase transitions.) For example, the superfluid is described by a U(1) symmetry breaking
It was realized that the order parameter in the Ginzburg-Landau Chern-Simons is not gauge invariant and is not physical. This is consistent with the topological order understanding of fractional quantum Hall (FQH) states which suggests that FQH has no off-diagonal long-range order and cannot be described by local order parameters
We find that the new kind of waves and the emergent statistics are so profound, that they may change our view of universe
Summary
In the case of crystal, the stronger interaction makes the particles to prefer being separated by a fixed distance and a fixed angle This makes particles to break the continuous translation symmetry down to discrete translation symmetry “spontaneously” in order to choose a low-energy configuration (see Figure 2). Physicists believed that Landau symmetry-breaking theory describes all possible orders in materials and all possible (continuous) phase transitions. Conductor, insulator, superfluid, and magnets can exist at zero temperature and are examples of quantum phases of matter. Physicists used to believe that Landau symmetrybreaking theory describes all possible quantum phases of matter and all possible (continuous) quantum phase transitions. (Quantum phase transitions are zero temperature phase transitions.) For example, the superfluid is described by a U(1) symmetry breaking. A boson superfluid is labeled by (U(1), {1}), where U(1) is the symmetry group of the boson Hamiltonian which conserves the boson number and {1} is the trivial group that contains only identity
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