My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
which describes a set of coupled harmonic oscillators; the coupling between the
oscillations of different k-vector is governed by the Hermitian matrix M.
To uncouple the oscillators, M must be diagonalised:
M kk ′ = ∑ i
U ki λ i U ∗ k ′ i. (7.30)
Here, U is the unitary matrix of eigenvectors and λ represents the eigenvalues of M.
From equation (7.25), it is clear that
M kk ′ = Mk ∗ ′ k = M(−k)(−k ∗ ′ ) = M (−k ′ )(−k). (7.31)
These symmetry relations have implications for U which will prove useful. The
eigenvalue equation is
∑
M kk ′U k ′ i = λ i U ki . (7.32)
k ′
Replacing k with −k, k ′ with −k ′ and taking the complex conjugate gives
∑
M(−k)(−k ∗ )U ′ (−k ∗ ′ )i = λ ∗ i U(−k)i. ∗ (7.33)
k ′
Equation (7.31) then implies that
∑
M kk ′U(−k ∗ ′ )i = λ i U(−k)i. ∗ (7.34)
k ′
Comparison with equation (7.32) shows that U(−k)i ∗ is also an eigenvector of M, with
the same eigenvalue λ i . If λ i is non-degenerate, the two eigenvectors are the same,
and
U(−k)i ∗ = U ki (7.35)
to within a phase factor, which is chosen to be zero. If λ i is degenerate, it is possible
to choose linear combinations of the degenerate eigenvectors to ensure that this
condition is met.
Using equation (7.25), and the fact that
∑
U ki Uk ∗ ′ i = ∑ i
i
U ki U † ik ′ = δ kk ′, (7.36)
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