My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 4.
ERRORS IN QMC SIMULATIONS
neously an eigenfunction of
Ĥ and of the set of all such translations:
ĤΨ = EΨ (4.2)
ˆT jR Ψ = T R Ψ for all R, j. (4.3)
The eigenvalue T R does not depend on j because of the antisymmetry of the wave
function.
Two consecutive translations correspond to another single translation; it follows
that
T R T R ′ = T (R+R ′ ). (4.4)
The eigenvalues therefore have the exponential form
T R = e ik·R , (4.5)
where at this stage the components of k are arbitrary complex numbers. 3 The
unitarity of the translation operator means that the eigenvalues must have modulus
1, which in turn implies that k is in fact real. The result is just Bloch’s theorem:
ˆT jR Ψ = e ik·R Ψ. (4.6)
Thus, when the Hamiltonian operator is periodic in real space, the wave function
must have the form
(
Ψ({r i }) = exp ik ·
N∑
r i
)U({r i }) (4.7)
where ˆT jR U = U. The wave vector k may be reduced into the first Brillouin zone
i=1
of the simulation cell: any remaining factors of
(
exp iK ·
N∑
r i
), (4.8)
where K is a simulation cell reciprocal lattice vector, are periodic and may be
included in U.
3 More detail can be found in the book by Ashcroft and Mermin [4]; this section follows their
proof of Bloch’s theorem.
i=1
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