content_image/420933.jpg content_image/420934.jpg content_image/420935.jpg content_image/420937.jpg content_image/420938.jpg content_image/420939.jpg content_image/420931.jpg content_image/420932.jpg content_image/420936.jpg content_image/420902.jpg content_image/420903.jpg content_image/420904.jpg content_image/420910.jpg content_image/420911.jpg content_image/420919.jpg content_image/420905.jpg content_image/420908.jpg content_image/420914.jpg content_image/420916.jpg content_image/420918.jpg
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 

<img src='content_image/420933.jpg'>

The operators P$_{±}$ are projectors onto the eigenstates of P$_{M}$ . 

Since P$_{M}$ | 0 〉 = | 0 〉 , it follows that 

<img src='content_image/420934.jpg'>

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 

<img src='content_image/420935.jpg'>

The next step is to introduce running wave vectors as follows: 

Both periodic (for the minus sign) and antiperiodic (for the plus sign) wave numbers are required to reduce (4.27) to the form 

<img src='content_image/420937.jpg'>

where the sums over k are to be taken in the range 

<img src='content_image/420938.jpg'>

Equation (4.31) can be analyzed further by grouping equal and opposite k values together, giving 

<img src='content_image/420939.jpg'>

<img src='content_image/420931.jpg'>

<img src='content_image/420932.jpg'>

<img src='content_image/420936.jpg'>

where 

<img src='content_image/420902.jpg'>

and 

<img src='content_image/420903.jpg'>

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 

<img src='content_image/420904.jpg'>

Similarly, for the operator V$_{1}$ 

Thus, V $^{′}$( ± ) will also factorize: 

where the k values are determined according to (4.32). The behavior at k = 0 and k = π must be specified carefully: 

<img src='content_image/420910.jpg'>

and 

<img src='content_image/420911.jpg'>

Then, the eigenstates of these operators are easily con- structed. For the remaining V $^{′}$( k ), note first that 

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 

behave like Pauli spin operators, as was observed in [15]. This shows that for k ̸ = 0 , π , 

where the function γ ( k ) is determined by 

<img src='content_image/420919.jpg'>

<img src='content_image/420905.jpg'>

<img src='content_image/420908.jpg'>

<img src='content_image/420914.jpg'>

<img src='content_image/420916.jpg'>

<img src='content_image/420918.jpg'>

