5 Hirsch-Fye quantum Monte Carlo method for ... - komet 337

5.6 Nils Blümer
(i) Imaginary-time discretization (β = Λ∆τ)
(ii) Trotter decoupling (❀ systematic error O(∆τ 2 ))
(iii) Hubbard-Stratonovich transformation (exact)
Wick theorem:
G =
(iv) MC importance sampling over auxiliary Ising field {s} (❀ statistical error)
Fig. 2: High-level overview of the HF-QMC approach.
separated by use of the Trotter-Suzuki formula [28, 29] for operators  and ˆ B:
�
{s} M−1 det{M}
�
{s} det{M}
e −β(Â+ˆ B) = � e −∆τ Â e −∆τ ˆ B � Λ +O(∆τ), (10)
where ∆τ = β/Λ and Λ is the number of (imaginary) time slices. 6 Rewriting the action (7) in
discretized form
AΛ[ψ,ψ ∗ ,G,U] = (∆τ) 2�
σ
�Λ−1
−∆τU
l=0
�Λ−1
l,l ′ =0
ψ ∗ σl (G−1
σ )ll ′ψ σl ′
ψ ∗ ↑lψ↑l ψ∗ ↓lψ↓l , (11)
where the matrix Gσ consists of elements Gσll ′ ≡ Gσ(l∆τ − l ′ ∆τ) and ψσl ≡ ψσ(l∆τ), we
apply (10) and obtain to lowest order
exp(AΛ[ψ,ψ ∗ ,G,U]) =
Λ−1 �
l=0
� �
exp
(∆τ) 2�
�Λ−1
σ
l ′ =0
× exp � −∆τ U ψ ∗ ↑l ψ ↑l ψ∗ ↓l ψ ↓l
ψ ∗ σl(G −1
σ )ll ′ψσl ′
�
� �
. (12)
Shifting the chemical potential by U/2, the four-fermion term can be rewritten as a square
of the magnetization (in terms of operators: (ˆn↑ − ˆn↓) 2 = ˆn 2 ↑ + ˆn2 ↓ − 2ˆn↑ˆn↓ = ˆn↑ + ˆn↓ −
2ˆn↑ˆn↓) which makes it suitable for the following discrete Hubbard-Stratonovich transformation
[30] (the correctness of which is checked easily by inserting the four possible combinations
[(0,0),(0,1),(1,0),(1,1)] of eigen values of the associated density operators):
�
∆τU
exp
2 (ψ∗ ↑lψ↑l −ψ∗ ↓lψ↓l )2
�
= 1 �
exp
2
sl=±1
� λsl(ψ ∗ ↑lψ↑l −ψ∗ ↓lψ↓l )� (13)
6 Since β = 1/kBT is the inverse temperature, small ∆τ on each “time slice” corresponds to a higher temperature,
for which the operators effectively decouple. Thus, we may view the Trotter approach as a numerical
extension of a high-temperature expansion to lower temperatures.