Symmetry properties of exciton states in one-dimensional systems

Abstract

We discuss a cyclic chain of N identical sites, each with two possible quantum states, whose Hamiltonian may be written in terms of the spin operators for N spins S = 1/2, and transformed into the Fermi site excitation representation. We show that in the Fermion wave representation the requirement that the states transform according to irreducible representations of the translational symmetry group demands that the quasiparticle momenta are drawn from the representations of the double group. They must be all integral or all half-integral when the number of quasiparticles is odd or even respectively. The requirement that a quantum-mechanical operator has a definite symmetry type leads to very complicated conditions unless the operator conserves the odd or even particle number. If it does the symmetry operator can be defined in terms of Fermi wave operators and projection operators. The time reversal transformation is developed in detail and applied to problems of degeneracy in McConnell's model of singlet excitons and in the anisotropic transverse Heisenberg antiferromagnet. In the Bogoliubov-Valatin transformation time reversal shifts the quasiparticle momenta by ± π, converting integral into half-integral momenta when N is odd. The energies of the excited states are doubly-periodic functions of the momentum.