TWO-DIMENSIONAL DIRAC FERMIONS IN Z² TOPOLOGICAL PHASES

In quantum vacuum, a free electron is governed by the Dirac equation [1] as is a wave equation differing from the Schrödinger equation. The equation is given by the Dirac Hamiltonian, a construction based on the compatibility made by Paul Dirac, between the single-particle quantum mechanics and the theory of special relativity. Consequently, the intrinsic properties of spin-1/2 and antiparticle (positron) of the electron are revealed in the solution. In the more involved quantum field theory, which combines the Dirac equation and the classical field theory, Dirac fermions can emerge as the quanta of a field [2]. In addition, there are also Weyl fermions [3] and Majorana fermions [4] proposed, from the Dirac equation or its variant, despite the fact that they have not been found in quantum vacuum yet. A door to modern particle physics has thus been opened along this line. On the other hand, the quantum field theory is also useful in the description of quasiparticles in condensed matter [5]. When quasiparticles near the single-particle ground state acquire an effective Hamiltonian mimicking the Dirac Hamiltonian, they behave as Dirac fermions and obey the Dirac equation. In such a connection, it is of importance and interest to see how one can get a convenient condensed matter playground for the tabletop test and the realization of the quantum field theory. Recently, both Weyl fermions [26] and Majorana fermions [7] have been discovered from condensed materials, which can be deemed as the greatest achievements in science.