8 Finally, it is straightforward to verify that for periodic (or toroidal) boundary conditions Z$_{t}$ = (2 sinh 2 K$_{1}$ ) MN/ $^{2}$Tr V N (4.19) Z$_{f}$ = (2 sinh 2 K$_{1}$ ) M ( N − 1) / $^{2}$2 M ×〈 0 | ( V$_{2}$V$_{1}$ ) N − $^{1}$V$_{2}$ | 0 〉 (4.11) Z$_{f}$ = (2 sinh 2 K$_{1}$ ) M ( N − 1) / $^{2}$2 $^{M}$e − MK ∗ 1 ×〈 0 | ( V $^{′}$) $^{N}$| 0 〉 (4.12) V ′ = V 1 / 2 1 V$_{2}$V 1 / 2 1 (4.13) where the operator V ′ is given by The final step is to insert V − 1 / 2 1 V 1 / 2 1 = 1 at the begin- ning and end of the product of operators in (4.11) and then use V 1 / 2 1 | 0 〉 = exp( − MK ∗ $_{1}$/ 2) | 0 〉 . This yields It is instructive to consider what can be deduced by du- ality. We can avoid having to consider the effect on eigen- vectors by considering (4.8) and (4.14) and examine the pair of operator products before taking the trace or the limit. A unitary transformation of the Fermi operators can be constructed [21] which interchanges ∑ M 1 ( − σ z $_{j}$) and σ M 1 σ x $_{j}$σ x j $_{+1}$; there is a technical problem involving the boundary, which has been overcome elsewhere [21]. This converts (4.8) into (4.15), but with K$_{0}$ replaced by K ∗ $_{0}$. The limit goes through and (4.17) is identical up to numerical factors, provided K ∗ 1 and K$_{2}$ are interchanged. Thus, if we have evaluated the right hand side of one or other of these two equations, the other follows directly. Duality in the usual form is discussed in the appendix. What we are using here is an operator version of it as an automorphism of the algebra of Fermi operators which appears to have been known to Onsager, although this work was not published, as far as we know. Furthermore, one can readily verify that as K$_{0}$ → 0, (2 sinh 2 K$_{0}$ ) exp(2 K ∗ $_{0}$) → 4. Cyclicity of the trace then reduces (4.8) to Given that the operator V ′ is self adjoint, a natural step is to make use of its eigenvalue decomposition as de- termined by Schultz, Mattis and Lieb [15]. Before doing this, we will show how the strip with “hard” walls ad- mits of a similar representation. We consider the same toroidal arrangement. This time, we alter the interac- tions between the spins in one of the M spin circles. That is, we replace the interaction strengths in one of the V$_{2}$ terms by K$_{0}$ where we now take K$_{0}$ to infinity. In addition, we add a term to the Boltzmann constant that keeps the partition function finite in the K$_{0}$ → ∞ limit. Then, (4.8) is replaced by In the LSM method Fermi operators f$_{m}$ , f † k are in- troduced by means of the Jordan-Wigner transformation [22] A. The LSM approach f$_{k}$ = P$_{j}$$_{−}$$_{1}$σ − j , 2 ≤ j ≤ M and f$_{1}$ = σ − 1 (4.20) P$_{j}$ = j ∏ l =1 ( − σ z $_{j}$) , j = 1 , . . . , M (4.21) where σ − j is the spin lowering operator, and P$_{j}$ is defined by It is easy to check that the f$_{j}$ and f † j are, indeed, Fermi operators, and that V$_{1}$ = exp$^{}$ $_{}$− K ∗ 1 M ∑ j =1 (2 f † $_{j}$f$_{j}$ − 1)$^{}$  (4.22) V$_{2}$ = exp$^{}$ $_{}$K$_{2}$$^{{}$$^{M}$$^{−}$$^{1}$ ∑ j =1 ( f † j − f$_{j}$ )( f † j +1 + f$_{j}$$_{+1}$ ) − P$_{M}$ ( f † M − f$_{M}$ )( f † 1 + f$_{1}$ )$^{}]}$ (4.23) The operator V$_{2}$ has a more subtle form: Given that the |±〉 are also eigenvectors of V$_{2}$ , we are led to another version of (4.14): where One can verify that [ V$_{1}$$_{,}$$_{2}$ , P$_{M}$ ] = 0 (4.24) V = V 1 / 2 2 V$_{1}$V 1 / 2 2 (4.18) Z$_{h}$ = (2 sinh 2 K$_{1}$ ) MN/ $^{2}$e − MK$_{0}$ $^{{}$〈 + | V $^{N}$| + 〉 + 〈−| V $^{N}$| −〉 } (4.17) σ x $_{m}$| ±〉 = ±|±〉 , m = 1 , . . . , M (4.16) where Given that Z$_{h}$ = (2 sinh 2 K$_{1}$ ) MN/ 2 lim K$_{0}$ $_{→∞}$Tr$^{(}$ ( V$_{1}$V$_{2}$ ) N − $^{1}$V$_{1}$ exp$^{[}$ K$_{0}$ M ∑ m =1 $^{(}$σ x $_{m}$σ x m +1 − 1 ) ] ) (4.14) lim K$_{0}$ $_{→∞}$exp$^{[}$ K$_{0}$ M ∑ m =1 $^{(}$σ x $_{m}$σ x m +1 − 1 ) ] = | + 〉〈 + | + |−〉〈−| (4.15)