Magnetic Field Induced Semimetal-to-Canted-Antiferromagnet ...

5 Outline of the Monte Carlo Technique
time Green function is a projector,
Gs(τ) 2 =
�
1 − U > �
s
= 1 − 2U > s
= 1 − U > s
U < s U > s
�
U < s U > s
� −1
U < s
�
U < s U > �−1 s U < s
� 2
� −1
U < s + U > s
�
U < s U > s
� −1
U < s U > s
� �� �
=1
�
U < s U > �−1 s U < s
= Gs(τ) . (5.16)
5.2.1 Efficient realization
An efficient implementation of the PQMC algorithm should result in a small warm-up
phase and low autocorrelation times. This assures that many valid measurements can be
taken which are both independent of each other and - to reflect the particle interaction
correctly - rely on a valid configuration of auxilliary fields. In the following we sketch the
essential steps leading to a fast and stable algorithm [5].
As mentioned before, the pivotal quantity is the equal time Green function Gs(τ),
Gs(τ) = 1 −
nτ �
n=1
= 1 − B > s
= 1 − U > s
e V(sn) e −∆τT P
�
B < s B > s
�
U < s U > s
� −1
B < s
� −1
U < s
�
P †
m�
n=1
e V(sn) e −∆τT P
� −1
P †
m�
n=nτ +1
e V(sn) e −∆τT
(5.17)
At this point it is appropriate to clarify on the use of the field variable s. Generally
spoken, there is an independent HS field for each lattice site i and time slice n. Those
individual fields take the values ±1 and are subject to the single spin flip decisions as
discussed above. That is every field fluctuates in Monte Carlo time between the values
±1. The matrix P is propagated along the imaginary time axis and at each time slice n
a sweep through the lattice is made resulting in a new configuration on that time slice.
This configuration is denoted by the index sn. The Green function Gs(τ) at any given
time τ being a non-local function relies on the entirety of fields i.e on the momentary
values of N × m field variables, which is indicated by the index s.
During the simulation a specified number of sweeps are done each of which consists of a
downward run from τ = 2θ to τ = ∆τ and a subsequent upward run from τ = ∆τ to
τ = 2θ. In the course of propagation the SDV stabilization scheme is applied periodically.
The total imaginary time interval 2θ = m∆τ is thus segmented into n sub-intervals of
length τ1 such that nτ1 = 2θ. This is shown schematically in Fig.5.1.
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