Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 4 / b / Solution

Share each on-site field energy equally between its two neighboring bonds. In the ordered basis $\sigma =+1,-1$, the symmetric transfer matrix for a classical spin chain isThe product ${\prod }_j{T}_{{\sigma }_j{\sigma }_{j+1}}$ recovers each bond and each field term once. Summing the periodic indices closes a trace, giving $Z_N=\mathrm{Tr}{T}^N$. The two eigenvalues areThe square-root argument is positive. Moreover $\det T=e^{-2K}-e^{2K}<0$, so ${\lambda }_{-}<0$ even at nonzero field. Its sign must not be discarded merely because all matrix entries are positive. The Perron–Frobenius theorem ensures that the positive eigenvalue ${\lambda }_{+}$ has the larger modulus, and the infinite-chain free energy is $f_{\infty }=-{\beta }^{-1}\log {\lambda }_{+}$. For even $N$, both terms in the finite-size partition function are positive.