Past exam of the field theories and collective phenomena in condensed matter course of the University of Oxford / 2025 / 3 / b

Solution

a = u c + v d^{†}, a^{†} = u c^{†} + v d, b = - v c^{†} + u d, b^{†} = - v c + u d^{†} .

c, d

, the nonzero same-mode anticommutators are

{a, a^{†}} = u^{2} {c, c^{†}} + v^{2} {d^{†}, d} = u^{2} + v^{2}, {b, b^{†}} = u^{2} + v^{2} .

The equal-type anticommutators satisfy

{a, a} = {b, b} = 0

, since every contributing equal-type or mixed-mode anticommutator of

c, d

vanishes. The potentially nontrivial cross term cancels:

{a, b} = - uv {c, c^{†}} + uv {d^{†}, d} = - uv + uv = 0.

Also

{a, b^{†}} = 0

, and the other cross relations follow by Hermitian conjugation. Therefore the necessary and sufficient condition is

u^{2} + v^{2} = 1.

The real transformation matrix is then an orthogonal matrix, with inverse

c = u a - v b^{†}, d = v a^{†} + u b .

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