Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2012 / Paper 1 / 1 / a

For the Transverse-field Ising model, use dimensionless spin one-half operators, so $S^x$ has eigenvalues $\pm 1/2$. At zero transverse field every aligned bond contributes $-J$. The two ground states areThey span an exactly two-dimensional ground space on the finite periodic ferromagnetic chain. Any linear combination is also a ground state; parity eigenstates are the symmetric and antisymmetric combinations. Choosing one ordered product state in the thermodynamic limit is spontaneous symmetry breaking.

For large positive field the leading term is $-2Jg{\sum }_n{S}_n^z$. Its unique minimizing state isThe exchange has zero first-order expectation in this state and produces virtual spin-pair fluctuations at finite g. Thus the polarized product is the limiting wavefunction, not an exact finite-field ground state. Bulk perturbation theory gives $E_0/N=-Jg-J/(4g)+O(J/g^3)$. The field-polarized phase has no twofold ordered-state degeneracy.