My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
3.3.3 The ground-state wave function
In the previous section, a technique for simulating the development of a wave function
in imaginary time was established. To see why this is useful, consider the formal
solution of the Schrödinger equation in imaginary time:
|Ψ(τ)〉 = e −τĤ|Ψ(0)〉. (3.33)
In terms of the eigenfunctions of
Ĥ, the initial state may be written
Ψ(R, 0) =
∞∑
c i Φ i (R), (3.34)
i=0
which, on substitution into equation (3.33), gives
Ψ(R, τ) =
∞∑
c i e −τE i
Φ i (R). (3.35)
i=0
This shows that as τ increases, the eigenstate with the lowest energy provides the
dominant contribution to Ψ(R, τ). In the limit τ → ∞, only the ground-state solution
Φ 0 remains (as long as c 0 ≠ 0). Any wave function which has a non-vanishing
overlap with Φ 0 therefore evolves to the ground state in the limit of large imaginary
time. The value of the imaginary-time simulation described in the previous section
is that it allows the ground state wave function to be determined, thus reducing the
difficult quantum many-body problem of finding the ground-state wave function to
a much simpler problem in classical particle dynamics.
However, since each time step is required to be short, many steps are required
before the large-τ limit is reached. This is not the only problem with this method:
• the wave function has been tacitly assumed to be positive, like the particle
density in section 3.3.1, and there is no mechanism for the weight of a walker
to change sign;
• the walkers are free to diffuse around: this is at best inefficient, because time
is wasted sampling unimportant regions of configuration space, and at worst
disastrous, as in a finite system, where the walkers are effectively able to leave
the system;
40