CASINO manual - Theory of Condensed Matter

37 Magnetic fields and the fixed-phase approximation
37.1 Hamiltonian when an external magnetic field is present
Consider a many-particle system in the presence of an external magnetic field B(r) = ∇ × A(r). The
Hamiltonian is
N∑ 1
Ĥ = (ˆp i − q i A i ) 2 + V, (449)
2m i
i=1
where ˆp i = −i∇ i , the {m i } and the {q i } are the masses and charges of the particles, V (R) is the
usual electrostatic potential energy and A i ≡ A(r i ) is the magnetic vector potential for particle i.
The Hamiltonian does not have time-reversal symmetry in general, so that the eigenfunctions are
complex.
37.2 VMC in the presence of an external magnetic field
Let Ψ be a complex trial many-body wave function. The expectation value of Ĥ with respect to Ψ is
where the complex local energy is
E L = ĤΨ
Ψ
= N ∑
i=1
〈Ψ|Ĥ|Ψ〉
〈Ψ|Ψ〉
=
∫
|Ψ| 2 E L dR
∫
|Ψ|2 dR , (450)
(
1
− ∇2 i Ψ
2m i Ψ
+ q2 i |A i | 2 + 2iq i A i · ∇iΨ
)
Ψ + iq i∇ i · A i + V. (451)
Ĥ is Hermitian, so its expectation value is real. 30 Hence only the real part of the local energy should
be averaged when computing the VMC energy. On the other hand, the full complex local energy is
needed when evaluating the variance of the local energy.
σ 2 =
1
N C − 1
∑
R
|E L (R) − ¯ E L | 2 , (452)
where N C is the number of configurations sampled, because Ψ is eigenfunction of Ĥ if and only if the
complex local energy is constant (in which case the imaginary part of the local energy is zero). It is
possible for the real part of the local energy to be constant, but the imaginary part to vary, in which
case Ψ is not an eigenstate of the Hamiltonian. Therefore the variance of the real part of the local
energy is not a good objective function for wave-function optimization.
It is not currently possible to use magnetic fields or complex wave functions in conjunction with
linear-least-squares energy minimization in casino.
37.3 DMC in the presence of an external magnetic field
37.3.1 Fixed-phase Schrödinger equation
Let Φ = |Φ| exp(iψ) be a complex many-Fermion wave function. The phase ψ must change by π
whenever two particles are exchanged. Then
[ N
]
exp(−iψ)ĤΦ = ∑ 1 (
−∇
2
2m i + (q i A i − ∇ i ψ) 2) + V |Φ|
i=1 i
[ N
]
∑ 1
+ i (2[q i A i − ∇ i ψ] · ∇ i + ∇ i · [q i A i − ∇ i ψ]) |Φ| (453)
2m i
≡
i=1
[Ĥψ + i ˆK ψ
]
|Φ|. (454)
30 Averaging the complex local energy over a finite number of configurations distributed as |Ψ| 2 gives a small imaginary
component in the mean energy, which vanishes in the limit of a large number of samples.
205