 Solution

ID: past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2012/paper-1/1/d/iii/solution

At g=1,

ϵ_{k} = 4 J ∣ sin (k /2) ∣ \sim 2 J ∣ k ∣

. The gap closes at zero momentum and

E_{0} / N = - 4 J / π

. With unit lattice spacing the critical velocity is

2 J /ℏ

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