Z,,/(n'7NT|n') then the average value of a spin at a site (i,j) is given by
(P) + Z (n 2N I ') ( 'T2 I) (4-29)
n mn
This can easily be evaluated by simply diagonalizing (4-25). We extract the j dependence
a) ar + a2SN + 3 + a4(rJ + rN-j) + a5(sj + sN-j) + a6(JrN-j + rs N-J)
a(Pr) a aN +(4 +30)
-vTr" + sN + v-- /
air N + a2N 3 a4rj N-j) a5 + N-j) a6 (SjrN-j + N- ,
(P2i) = aa2s 4 31)
VrT + s-T + ^ \ J
where r = e-G, and s = e- are the exponentials of the j dependence and the ak-s
are known although complicated functions of and -. Clearly there is a exponential
drop off with j with several exponents corresponding to each energy gap. It is impossible
practically to extract G, from statistical data for (P/) because GT is 1--, .. But if we look
at the cyclically symmetric observable R = i Pi we have
(R) 2 a _rN + a2S +a3 -+ 2 + (,j + rNJ) (4-32)
(R)rN + s + ,a3 +2-/,rN + aar S + /a3
In this simple case not only is the dependence of G, suppressed in (Ri), but actually
drops out completely.