Any Monte Carlo expectation value is by definition an average of measurements over Monte Carlo configurations and hence inherently presents statistical errors. They can be reduced by generating longer Markov chains, i.e. by performing a larger number of measurement steps Nmand an exact result would be found in the limit of
infinite Monte Carlo measurement steps Nm → ∞. In addition, disordered sample-
to-sample fluctuations are another source of statistical errors. Adding more disorder realizations (bigger Ns) is the only way to reduce them. However, the computational
time at our disposition is finite meaning that an optimized strategy to obtain the smaller possible final uncertainties needs to be set in action. We already discussed how to choose the proper temperatures in an optimized fashion and this section will address the determination of the dominant source of uncertainties between Monte Carlo and disorder statistical errors and how to achieve the best possible precision in our final results.
2.3.1 Monte Carlo fluctuations vs disorder fluctuations
We try to determine a hierarchy between Monte Carlo and disorder statistical fluctuations to gain more insight on this problem, which means we need to quantify both of them. In order to do so, we take 10 representative disordered samples of a system of size N = 16 × 16 at a disorder strength of W = 4.6 inside the SF phase and perform the β-doubling scheme to reduce equilibration times, using Neq
steps. 10 independent bins of measurement using Nmmeasurement steps are carried
out for each sample, and we use three different values of Nm = 102, 103 and 104,
keeping a constant ratio Nm/Neq = 5. The results for the superfluid stiffness ρsf
and the BEC density ρ0 are plotted in figure IV.9. On one hand, for Nm = 100
(black circles) the estimates from a bin of measurement to another one (i.e. the Monte Carlo fluctuations) have a spread of the same order of magnitude as the spread of the estimates for different samples. On the other hand, for a number of measurement steps Nm ≥ 1000, the Monte Carlo fluctuations are smaller than the
disorder fluctuations.
We can partially conclude that at least 1000 measurement steps are needed to make the fluctuations due to the Monte Carlo sampling to be negligible as compared to disorder fluctuations. In order to check if 1000 indeed are enough we plot in fig- ure IV.10, for a system of size L = 16 and a disorder strength W = 5 inside the Bose glass phase, the full distribution of the logarithm of the superfluid response ln(ρsf)
using Ns ≈ 20000 independent disorder realizations together with the distributions
of the same quantity for 5 representative samples using Nm= 1000. For each one of
these samples around 300−400 independent Monte Carlo estimates with Nm= 1000
were carried out. The full distribution over disordered samples was multiplied by a factor of 20 for graphical reasons. The distribution for different Monte Carlo mea-
0 0.005 0.01 0.015 0.02 Nmc=10 , NEQ=20 NMC=103, NEQ=200 NMC=104, NEQ=2.103 0 0.005 0.01 0.015 0.02 0.025 0.03 sample 1 2 3 4 5 6 7 8 9 10
⇢
sf⇢
0 Nm= 10, Neq= 20 Nm= 103, Neq= 200 Nm= 104, Neq= 2000 L = 16, W = 4.6 MC bins 1· · ·10Figure IV.9: Superfluid and Bose condensed densities for a system of size N = 16 × 16 at disorder strength W = 4.6. The estimates for 10 independent disordered
samples are presented. For each one of them, the β-doubling procedure using Neq equilibration steps was used, followed by 10 independent bins of measurement, each with Nm measurement steps, performed only at the lowest temperature βt= 28. The 10 computed expectation values for a given sample and number of measurement steps is represented by a data point.
surements with 1000 steps inside a single disorder realization is clearly much narrower than the full distribution, specially so for the samples exhibiting big values of the stiffness. At most, the spread over Monte Carlo measurements is around one third of the spread of the disorder distribution (red distribution in figure IV.10). This corresponds to the samples with the lowest superfluid response, for which a larger number of measurement steps may be necessary since autocorrelation times for ρsf
are the longest precisely for these samples (cf. section 2.4).
As a conclusion, the fluctuations are vastly dominated by the disorder fluctuations and the best strategy to reduce the overall uncertainty of our results is to perform a large amount of disorder realizations using at least 1000 measurement steps, which usually are much larger than the autocorrelation times for the observables of interest, namely the superfluid stiffness ρsf and the BEC condensate ρ0. As a matter of
fact, there are some samples that present very large autocorrelation times for the superfluid stiffness ρsf, and 1000 measurement steps are not enough in this case.
-20 -15 -10 -5 0.1 1 10
Disorder
rea 1
rea 3
rea 4
rea 5
rea 7
ln(⇢
sf)
P
(ln(
⇢
sf))
L = 16, W = 5
sample 1 sample 2 sample 3 sample 4 sample 5 Full distribution⇥ 20Figure IV.10: Distribution of ln(ρsf) for a system of size N = 16 × 16 at disorder strength W = 5, inside the Bose glass phase. The black circles curve is the full dis- tribution over ≈20000 disordered samples, multiplied by a factor of 20 for graphical reasons. The colored lines are the histograms of ∼ 300 − 400 MC measurements made using Nm = 103 steps, shown for 5 representative disordered samples. The
spread of the MC measurements distributions is clearly much smaller than the spread stemming from disordered samples.
2.3.2 Obtaining disorder-converged estimates
Now that we have determined that disorder fluctuations are dominant in our problem, we have to tackle how to reduce them. As stated before, the only way to do so is by adding more disordered samples, keeping in mind that the total computation time grows with the number of disorder realizations (see eq. (IV.34)). In figure IV.10 it can be seen that the distributions of ln(ρsf) are very wide and need a very large
number of disorder realizations to be properly sampled so, a large enough number of samples is needed. In addition, Lin et al. found that, for the superfluid-insulator transition in the related model of quantum rotors, around 1000 samples do not give converged estimates [276]. So we may need to use the biggest possible number of samples as possible, without it being too large. To find a good compromise between these two constraints we plotted in figure IV.11 the cumulative averages of the superfluid stiffness ρsf and the BEC density ρ0 upon increasing the number of
included disorder realizations for two different system sizes L = 16 and L = 32 and several disorder strengths. The averages drift slowly towards their converged value and it becomes apparent that for the smaller size at least Ns ∼104 disorder samples
0 5000 10000 15000 20000 0.0029 0.003 0.0031 0.0032 0.0033 ρsf 0 5000 10000 15000 20000 0.005 0.00505 0.0051 0.00515 ρ0 0 2000 4000 6000 8000 10000 0.00042 0.00044 0.00046 0.00048 0.0005 0.00052 0.00054 ρsf 0 2000 4000 6000 8000 10000 0.0018 0.00185 0.0019 0.00195 0.002 ρ0 Ns Ns
¯⇢
sf¯⇢
0¯⇢
sf¯⇢
0 Ns Ns L = 16, W = 4.7 L = 16, W = 4.7 L = 32, W = 4.75 L = 32, W = 4.9Figure IV.11: Estimated averages of the superfluid stiffness ρsf and the Bose con- densed density ρ0 as a function of the number of disordered samples Ns for different
system sizes (N = 16 × 16, 32 × 32) and disorder strengths. A very large number of samples (Ns 1000) is needed to achieve disorder-converged averages, like in the
discussion for a related model [276].
are needed to achieve a converged estimate and at least Ns ≈ 5000 for the bigger
size. Since the distributions of ln(ρsf) are very wide (as will be discussed in details in
section 4) we choose an even bigger number of samples: for the smaller sizes L ≤ 22, Ns ∼20000 and for the bigger sizes L ≤ 24, Ns ∼10000.