9 where and Terms in (4.33) for different values of the summand com- mute. In the paired subspaces, the τ $^{α}$( k ) , α = x, y, z are indeed spin 1 / 2 operators. Thus, we can write V$_{2}$ ( ± ) as a product τ $^{z}$( k ) = ( F $^{†}$( k ) F ( k ) + F $^{†}$( − k ) F ( − k ) − 1) (4.34) τ $^{y}$( k ) = − i { F $^{†}$( − k ) F $^{†}$( k ) − F ( k ) F ( − k ) } (4.35) V$_{2}$ ( ± ) = ∏ 0 ≤ k<π exp [2 K$_{2}$ { τ $^{z}$( k ) cos k + τ $^{y}$( k ) sin k } ] (4.36) V$_{1}$ ( ± ) = ∏ 0 ≤ k<π exp [ − 2 K ∗ $_{1}$τ $^{z}$( k )] (4.37) Similarly, for the operator V$_{1}$ Thus, V $^{′}$( ± ) will also factorize: V $^{′}$( ± ) = ∏ 0 ≤ k<π V $^{′}$( k ) (4.38) where the k values are determined according to (4.32). The behavior at k = 0 and k = π must be specified carefully: V $^{′}$(0) = exp $^{[}$( K$_{2}$ − K ∗ $_{1}$) $^{(}$2 F $^{†}$(0) F (0) − 1 )] (4.39) V $^{′}$( π ) = exp $^{[}$− ( K$_{2}$ + K ∗ $_{1}$) $^{(}$2 F $^{†}$( π ) F ( π ) − 1 )] (4.40) and Then, the eigenstates of these operators are easily con- structed. For the remaining V $^{′}$( k ), note first that V $^{′}$( k ) F $^{†}$( ± k ) | 0 〉 = F $^{†}$( ± k ) | 0 〉 (4.41) In the subspace with vectors | 0 〉 and F $^{†}$( − k ) F $^{†}$( k ) | 0 〉 , the operators τ $^{y,z}$( k ) introduced in (4.34) and (4.35), along with the operator τ $^{x}$( k ) = F $^{†}$( − k ) F $^{†}$( k ) + F ( k ) F ( − k ) (4.42) behave like Pauli spin operators, as was observed in [15]. This shows that for k = 0 , π , V $^{′}$( k ) = cosh( γ ( k )) − sinh( γ ( k )) × (cosh( δ $^{′}$( k )) τ $^{z}$( k ) − sin( δ $^{′}$( k )) τ $^{y}$( k )) (4.43) cosh( γ ( k )) = cosh(2 K ∗ $_{1}$) cosh(2 K$_{2}$ ) − − sinh(2 K ∗ $_{1}$) sinh(2 K$_{2}$ ) cos( k ) (4.44) where the function γ ( k ) is determined by Equation (4.31) can be analyzed further by grouping equal and opposite k values together, giving where the sums over k are to be taken in the range Both periodic (for the minus sign) and antiperiodic (for the plus sign) wave numbers are required to reduce (4.27) to the form The next step is to introduce running wave vectors as follows: It should be noted that if a line of reversed K$_{2}$ bonds were to be inserted on every tier of the cylindrical lattice between site M and site 1, then the resultant partition function, which we denote Z a $_{f}$, is given by The operators P$_{±}$ are projectors onto the eigenstates of P$_{M}$ . Since P$_{M}$ | 0 〉 = | 0 〉 , it follows that so that, for instance, V ′ and P$_{M}$ can be diagonalized simultaneously. The operator P$_{M}$ has eigenvalues ± 1. Thus, we can write where and V ′ = P$_{+}$V $^{′}$(+) + P$_{−}$V $^{′}$( − ) (4.25) V $^{′}$( ± ) = V 1 / 2 1 V$_{2}$ ( ± ) V 1 / 2 1 (4.26) V$_{2}$ ( ± ) = exp$^{}$ $_{}$K$_{2}$$^{{}$$^{M}$$^{−}$$^{1}$ ∑ j =1 ( f † j − f$_{j}$ )( f † j +1 + f$_{j}$$_{+1}$ ) ∓ ( f † M − f$_{M}$ )( f † 1 + f$_{1}$ )$^{}]}$ (4.27) Z$_{f}$ = (2 sinh(2 K$_{1}$ )) M ( N − 1) / $^{2}$2 $^{M}$e − MK ∗ 1 〈 0 | ( V $^{′}$(+)) $^{N}$| 0 〉 (4.28) Z a f = (2 sinh(2 K$_{1}$ )) M ( N − 1) / $^{2}$2 $^{M}$e − MK ∗ 1 〈 0 | ( V $^{′}$( − )) $^{N}$| 0 〉 (4.29) F $^{†}$( k ) = M − 1 / 2 M ∑ j =1 e $^{ikj}$f † j (4.30) V$_{2}$ ( ± ) = exp$^{[}$ K$_{2}$ ∑ k ∈ Ω$_{M}$( ± ) e $^{ik}$( F $^{†}$( k ) − F ( − k )) ( F $^{†}$( − k ) + F ( k ))$^{]}$ (4.31) Ω$_{M}$( ± ) = { k : exp[ iMk ] = ∓ 1 , − π < k ≤ π } (4.32) V$_{2}$ ( ± ) = exp$^{}$ $_{}$2 K$_{2}$ ∑ 0 ≤ k<π { τ $^{z}$( k ) cos k + τ $^{y}$( k ) sin k }$^{}$  (4.33)