CASINO manual - Theory of Condensed Matter

Noting that |Φ| ˆK ψ |Φ| = ∑ 〈
N
i=1 ∇ i ·[|Φ| 2 (q i A i −∇ i ψ)]/(2m i ), it is easy to show that |Φ| ∣ ˆK
∣ 〉 ∣∣
ψ |Φ| = 0.
Hence 〈
〉 〈
〉
∣
∣
Φ ∣Ĥ ∣ Φ = |Φ| ∣Ĥψ∣ |Φ| . (455)
So the ground-state eigenvalue of the fixed-phase Schrödinger equation Ĥψ|φ 0 | = E 0 |φ 0 | is equal to
the expectation value of the Hamiltonian Ĥ with respect to φ 0 = |φ 0 | exp(iψ), which is greater than
or equal to the Fermionic ground-state energy of Ĥ by the variational principle, becoming equal in
the limit that the fixed phase ψ is exactly equal to that of the Fermionic ground state [77].
37.4 Importance sampling
The fixed-phase imaginary-time Schrödinger equation is
(Ĥψ − E T
)
|Φ| = − ∂|Φ|
∂t , (456)
where E T is the reference energy. In the large-time limit the ground-state eigenfunction |φ 0 | of the
fixed-phase Hamiltonian is projected out.
Let the DMC wave function Φ have the same phase exp(iψ) as the trial wave function Ψ. Then
f ≡ Φ ∗ Ψ = |Φ||Ψ| is real. Substitute |Φ| = |Ψ| −1 f into Eq. (456) and rearrange to obtain the
importance-sampled fixed-phase imaginary-time Schrödinger equation,
N∑
i=1
1 [
−∇
2
2m i f + 2∇ i · (Re(V i )f) ] + [Re(E L ) − E T ]f = − ∂f
i ∂t , (457)
where V i = Ψ −1 ∇ i Ψ = |Ψ| −1 ∇ i |Ψ| + i∇ i ψ is the complex drift velocity. This is a straightforward
generalization of the usual fixed-node importance-sampled imaginary-time Schrödinger equation, with
the real part of the drift velocity appearing in the drift–diffusion term and the real part of the local
energy appearing in the branching term. After equilibration the algorithm produces configurations
distributed as φ ∗ 0Ψ = |φ 0 ||Ψ|. Noting that Re(E L ) = |Ψ| −1 Ĥ ψ |Ψ|, the mixed estimate of the energy
is equal to the pure estimate:
∫
φ
∗
0 ΨRe(E L ) dR
∫
φ
∗
0 Ψ dR
=
=
∫
|φ0 ||Ψ|Re(E L ) dR
∫
|φ0 ||Ψ| dR
〈
〉
∣
|φ 0 | ∣Ĥψ∣ |Ψ|
= E 0 =
〈|φ 0 | | |Ψ|〉
〈
〉
∣
|φ 0 | ∣Ĥψ∣ |φ 0 |
〈|φ 0 | | |φ 0 |〉
= 〈φ 0|Ĥ|φ 0〉
〈φ 0 |φ 0 〉 . (458)
For operators that do not commute with the Hamiltonian the mixed estimate is not equal to the pure
estimate. Extrapolated estimation can be used in the same fashion as for real wave functions.
37.5 Applying magnetic fields in CASINO
At present it is only possible to apply uniform magnetic fields in casino, although it would be
straightforward to generalize this. The vector potential is written as A(r) = A 0 + A 1 r, where A 1 is a
3 × 3 matrix. Clearly it is possible to satisfy the Coulomb gauge condition ∇ · A = 0 with this form.
To apply an external magnetic field, the complex wf keyword must be set to T in the input file and
the magnetic vector potential must be given in a UNIFORM MAGNETIC FIELD block in the
expot.data file. The format is:
START UNIFORM MAGNETIC FIELD
Vector A0
1.0 0.0 0.0
Matrix A1
0.0 0.0 0.0
1.0 0.0 0.0
0.0 0.0 0.0
END UNIFORM MAGNETIC FIELD
Please note that it is crucial to use a trial wave function with the correct phase behaviour corresponding
to the vector potential; otherwise the calculated energies will be nonsense.
206