Hubbard Model for Polyallyl Chain

One of the simplest zr-electron models of a conjugated polymer with a macroscopic ground state spin is a polyallyl chain (Fig.2) described by the Hubbard Hamiltonian [20]  

According to Lieb s theorem [31], the ground state spin of this Hamiltonian 

Let us enumerate all the electrons of the chain in succession along the cells of the chain. Making use of the spin permutations, one can obtain the Hamiltonian (8) with (/ = oo in the form 

For one hole in the half-filled band, the exact energy spectrum of the chain with free ends formed by L unit cells is spin-degenerate, similar to the spectrum of the uniform Hubbard chain with U = oo. In the case of periodic boundary conditions an electron hopping between the first and the last unit cells of the chain leads to the additional term to the Hamiltonian (9). For one hole in the half-filled band this term has the following form  

In the case of periodic boundary conditions the chain Hamiltonian commutes with the operator that displaces all electrons by one unit cell cyclically. Therefore, its eigenfunctions must be characterized by the hole quasi- 

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