Topological States on Interfaces Protected by Symmetry In this book the author theoretically studies two aspects of topological states. First novel states arising from hybridizing surface states of topological insulators are theoretically introduced. As a remarkable example the author shows the existence of gapless interface states at the interface between two different topological insulators which belong to the same topological phase. While such interface states are usually gapped due to hybridization the author proves that the interface states are in fact gapless when the two topological insulators have opposite chiralities. This is the first time that gapless topological novel interface states protected by mirror symmetry have been proposed. Second the author studies the Weyl semimetal phase in thin topological insulators subjected to a magnetic field. This Weyl semimetal phase possesses edge states showing abnormal dispersion which is not observed without mirror symmetry. The author explains that the edge states gain a finite velocity by a particular form of inversion symmetry breaking which makes it possible to observe the phenomenon by means of electric conductivity