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

At $B=0$ a global reversal ${\sigma }_j\mapsto -{\sigma }_j$ leaves the energy and statistical weight unchanged but changes the sign of the one-spin observable. Pairing reversed configurations givesEquivalently, $⟨\pm |{\sigma }^z|\pm ⟩=0$ in the transfer eigenbasis, so $\mathrm{Tr}({\sigma }^z{T}^N)=0$. This holds for every finite even ring and all finite temperatures, and for the zero-field Gibbs limit. A symmetry-selected zero-temperature alternating state can have nonzero staggered magnetization; its uniform magnetization is still zero. The classical spin variable in this model has no separate quantum spin-operator dynamics despite the conventional matrix notation ${\sigma }^z$ used for the insertion.