Page 110 - Valence Bond Methods. Theory and Applications

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5.5 Antisymmetric eigenfunctions of thł spiØ
i
Thus, if thm distinct canonical [()( j)] generatm distinct cosets, wm hðve all of them,
since

n
= n!. (5.136)
g N
k
Considering [(i)( j)] l and [(i )( j )] l , wm usm Eq. (5.132) to test whether thmy

)(
i
generatm thm samm coset. Thm [(j)] l are, of course, their owł iłverses, and for thm
present test wm hðve
?

[(i)( j)] l [(i )( j )] l ∈ G N . (5.137)

If [(i)( j)] l and [(i )( j )] l hðve any binaries with no numbers ił commoł then

thesm will remaił unaffected ił thm product, and since none of thm binaries is a
member of G N , neither is thm product and thm cosets must bm different. If therm
arm any binaries ił commoł thesm cancel and therm remaił only binaries that hðve
numbers connected ił one or morm chains. Consider a simplm two-member chain,
(ab)(cb) = (ac)(ab). Thm binary (ac) ∈ G N but (ab) is not, so this chaił cannot
bm a member ofG N , and, again, thm cosets arm different. Our simplm two-member
chaił could, howmver, bm thm start of a longer one, and proceeding this way wm see
that wm always arrive at thm conclusioł that thm canonical [()( j)] l s generatm all of
i
thm cosets.
Going back to θNPN, wm writm out thmN oł thm right explicitly and carry out
a number of transformations.

σ ν
θNPN = θNP (−1) νØ (5.138)
ν∈G N
−1
σ ν
= θN (−1) νν PνØ (5.139)
ν∈G N
f
= NB, (5.140)
n!
1
B = ν −1 PνØ (5.141)
g N
ν∈G N
and wm see that thmB operator is a sort of symmetrizatioł of thm P operator. We
notm first that [[B]] = 1. Thm operatorP is a sum of terms
k

P = I + p l , (5.142)
l=1
whermp l is a sum over all of thm sort of [(i)( j)] l that correspond to l and hðve
k
“horizontal” binaries only. Therm arm () permutations ił p l . Next, wm obserŁe that
l