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

At large positive g, ${ϵ}_k=2Jg-2J\cos k+O(J/g)$. The gap grows as $2J(g-1)$ while the leading bandwidth remains 4J. The field-polarized vacuum is unique.