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

Hirsch-Fye QMC 5.9
(a) (b)
1
0.8
0.6
0.4
0.2
0
-1 -0.5 0 0.5 1
x
f(x)
MC
I est
1.4
1.2
1
I est
1.27
1.26
0.8 exact
trapezoid
simple MC
0 0.01
1/N
0.6
0 20 40 60 80 100
2
Fig. 3: Illustration of simple MC: (a) instead of using deterministic quadrature schemes such
as the trapezoid rule, integrals over some function f(x) can be evaluated by averaging over
function values f(xi) of stocastically generated support points xi (and multiplying by the integration
volume). (b) The resulting statistical error decreases as 1/ √ N with the number of
function evaluations (crosses and error bars). The deterministic trapezoid discretization scheme
(circles) converges with an error proportional to N −2/d , i.e., is superior in dimensions d < 4;
furthermore, an extrapolation of the systematic error is straightforward (inset).
where we may regard pl as a (unnormalized) probability distribution for the indices and ol as
a remaining observable. If both the normalization c is known (i.e, the sum over the weights pl
can be performed exactly) and the corresponding probability distribution can be realized (by
drawing indicesl with probabilityP(l) = pl/c), we obtain
X MC
imp = c
N
N�
j=1
olj and ∆X MC
�
vo
imp = c . (26)
N
Thus, the error can be reduced (vo < c 2 vx), when the problem is partially solvable, i.e., the
sum over pl with pl ≈ xl can be computed. 10 Since this is not possible in general, one usually
has to treat the normalization c as an unknown and realize the probability distributionP(lj) =
plj /c in a stochastic Markov process: Starting with some initial configuration l1, a chain of
configurations is built up where in each step only a small subset of configurationsl ′ is accessible
in a “transition” from configuration l. Provided that the transition rules satisfy the detailed
balance principle,
plP(l → l ′ ) = pl ′P(l′ → l), (27)
and the process is ergodic (i.e., all configurations can be reached from some starting configuration),
the distribution of configurations of the chain approaches the target distribution in the
limit of infinite chain length, as illustrated in Fig. 4. Since the normalization remains unknown,
importance sampling by a Markov process can only yield ratios of different observables evaluated
on the same chain of configurations. Another consequence of using a Markov process is
that initial configurations have to be excluded from averages since the true associated probabil-
10 The possible reduction of the variance is limited when xl is of varying sign. This “minus-sign problem”
seriously restricts the applicability of Monte Carlo methods for finite-dimensional fermion problems.
N