In this course, we will review the basic physics and recent progress in topological phases of matter from the perspectives of entanglement and quantum mechanics. Traditionally, topological phases have been studied through quantum field theories and associated field theoretic concepts. On the other hand, entanglement and various other concepts emergent from quantum information theories have been proven to be extremely useful in studies of topological phases. In this course, basing ourselves on the traditional field theoretic approach, we attempt to bridge the two approaches and explore the impact of `entanglement’ in studies of topological phases and other quantum matters.

We plan to use the minimal amounts of mathematics because several key ideas and concepts appearing in the recent research on topological phases can be understood without much mathematics and learned through examples. The goals of this course are to inspire the audience to learn more about topological phases and to help the audience to overcome ‘language barrier’ in learning the physics of modern topological phases.

• Classifications of topological phases: quantum Hall states and topological insulators

• “Phenomenology” of topological phases

• This part of the talk will not include much mathematics.

• Basic quantum field theories of topological phases

• Simple microscopic models and effective field theory

• Basic conformal field theories

• Anomaly of topological phases

• Basics of entanglement

• Entanglement entropy in topological phases

• Entanglement spectrum in topological phases

• Entanglement spectrum in topological phases (continued)

• Entanglement spectrum and entropy in other quantum matter

• We will summarize the course and present our own future directions on the subjects.

• Other perspectives of entanglement in quantum matter