Ground State Energy, Spin Gap and Hole Superconductivity on Square Lattice Heisenberg Antiferromagnet by Majorana Fermion

Abstract

The mechanism of high Tc superconductivity has not been solved completely yet. It was argued by Anderson that the properties of high Tc superconductivity are normal but the normal state properties are not normal. This seems to be true. We approach the subject of high Tc superconductivity viewing it from the normal state. First the ground state of the S=l/2 square lattice Heisenberg antiferromagnet is investigated using a BCS type variational method and a massless Majorana fermion representation for the spin 1/2 operator. Second when a hole is doped, we investigate how the spin gap appears as an elementary excitation of the spin fluctuation, where we use another Majorana fermion representation for an itinerant hole. Third we investigate the situation in which more holes are doped on the square lattice antifer­ romagnet and spin Majorana fermions are exchanged between itinerant holes. Here, the spin gap becomes unstable, and high Tc superconductivity appears instead of the spin gap, where we describe the superconductive gap by a massive Dirac fermion in the Nambu representation instead of a Majorana fermion. We also discuss the noncoexistence of the spin gap and the superconductive gap formed by the holes and seek to determine the critical concentration of holes, that concentration at which the spin gaps disappear and the superconductive gaps replace them. We also examine the phase diagram of the spin gap state and the supercon­ ductive state as a function of the hole concentration and the temperature. The method we adopt is that introduced by Tsvelik,l) who uses the Majorana fermion representation for spins and holes in the Kondo problem and the heavy fermion problem. A Majorana fermion is massless, and, using this representation, we can avoid the use of both the slave boson and fermion to represent the spin operator and their gauge fields. Then we can use the BCS-type variational method for the interaction of the Majorana fermions without using the Gutzwiller projection together with the restriction of the double occupancy of the opposite spins on the same site, as adopted in the modified variational method by Zhang, Gros, Rice and Shiba. 2 l