Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 35A (Statistical Physics)

Expanding the trace in the spin basis givesso the transfer matrix for the one-dimensional Ising model reproduces the partition function.

In the ordered basis $s=+1,-1$,Its eigenvalues areSince ${\lambda }_{+}>|{\lambda }_{-}|$ at every positive temperature, the thermodynamic limit gives the Helmholtz free energy per spinThis is an analytic function of $T$ and $B$ for $T>0$, so the one-dimensional short-range Ising model has no finite-temperature phase transition.

Insert $s_1{s}_i$ into the spin-basis expansion of the trace. Summing the spins along the first path from site $1$ to site $i$ produces $(M^{i-1}{)}_{st}$, while summing along the complementary path around the periodic chain produces $(M^{N-i+1}{)}_{ts}$. Therefore the two-point correlation function is

At $B=0$,with eigenvalues ${\lambda }_{+}=a+b=2\cosh (\beta J)$ and ${\lambda }_{-}=a-b=2\sinh (\beta J)$. Diagonalizing givesSubstitution in part (i), together with $Z={\lambda }_{+}^N+{\lambda }_{-}^N$, givesSince ${\lambda }_{-}/{\lambda }_{+}=\tanh (\beta J)$, the required One-dimensional Ising correlation function is