Singlets and Long-Range Spin-Pairing-Order

Singlets and long-range spin-pairing-order 

In order to reduce the dimension of the model space, several attempts have been made to find associated topological quantum numbers which would allow to separate the covalent VB space into non-mixing different subspaces (see Ref. 30 and references therein). For the linear chain, preliminary attempts were made early in 1979 [31], when a parameter measuring the spin-pairing long-range order was suggested, 

Recently, in Ref. 30, it has been shown that monoconfigurational covalent VB structures for ladder materials show a long-range spin-pairing order (LR-SPO). This LR-SPO can be easily generalized to our present strips. 

Let us take fn to be the right boundary of the cell n. Thus, fn unambiguously break up the strip in two regions a left region, Ln, and a right region, Rn (see Fig. 1). Furthermore, the fn-i and fn boundaries delimit the nth cell. 

We define P+ (P ) as the number of arrows penetrating the boundary fn with the arrowhead in the Rn (Ln) region (see Fig. 5). In is the number of arrows with both ends in the cell n. ln (rn) is the number of arrows starting in Rn (Ln-i) and with the arrowhead in Ln-1 (Rn), i-e. with no partner belonging to the cell nth. Finally, (w ) is the number of sites in the cell n belonging to the sublattice A (B). Then 

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