Approximate second quantization and kinematic interaction

3 we have applied the second quantization for investigation of exciton states. The first step was to express the crystal Hamiltonian in terms of creation and annihilation Pauli operators Pjf and P/ of single-molecule excited states, where the index s indicates the lattice points where the molecule is placed, and / labels the molecular excited states. When taking into account only one fth excited molecular state, the operators P%f and P/ satisfy the following commutation rules (see eqn 3.28) 

Equations (15.1)—(15.3) are combinations of commutation rules of Fermi type (when s = s ) and of Bose type (when s s ). The appearance of commutation relations for Fermi operators when s = s means that the number of molecular excitations, i.e. the eigenvalue of the operator Pj-fp/, takes either the value 0 (the molecule in the ground state) or 1 (the molecule is excited). In turn, the Bose commutation rules for s A s are related to the fact that the operators with different indices s are acting on different arguments of the crystal wavefunction. 

As shown in Ch. 3, the crystal Hamiltonian, expressed in terms of the operators Pj-f and P/ (in the following the index / will be omitted) when the lattice vibrations are not taken into account, has the following form (see also eqn 3.39) 

When speaking of kinematic interaction, it should be noted that the problem of its separation in connection with the transition from Pauli operators to Bose operators is far from new. This problem arises, in particular, for the Heisenberg Hamiltonian, which corresponds, for example, to an isotropic ferromagnet with spin a = 1/2 when spin waves whose creation and annihilation operators obey Bose commutation relations are introduced. This problem was dealt with by many people, including Dyson (6), who obtained the low-temperature expansion for the magnetization. However, even before Dyson s paper, Van Kranendonk (7) proposed to take into account of the kinetic interaction by starting from a picture where one spin wave produces an obstacle for the passage of another spin wave, since two flipped spins cannot be located at the same site (for Frenkel excitons this means that two excitons cannot be localized simultaneously on one and the same molecule). 

In mathematical language, such an approach means adding to the initial Hamiltonian, in which the Pauli operators are replaced by Bose operators, a term that corresponds to the limiting strong repulsion of two bosons in one site. 

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