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

5.8 Nils Blümer
The ratio of the determinants of new and old matrix can be easily determined using the inverse
of the old matrix: 9
R σm := det(Mσ ′ )
det(Mσ) = det� 1+∆ σm (Mσ) −1�
= 1+2∆τ λσsm(Mσ) −1
mm
The inversion ofMis also elementary, one obtains:
(Mσ ′ ) −1 = (Mσ) −1 + 1
R σm (Mσ) −1 ∆ σm (Mσ) −1
This reduces the effort for the recalculation of a term of (16) after a spin flip toO(Λ 2 ).
Only for Λ � 30 can all terms be summed up exactly. Computations at larger Λ are made
possible by Monte Carlo importance sampling which reduces the number of terms that have to
be calculated explicitly from2 Λ to orderO(Λ).
2.2 Monte Carlo importance sampling
Monte Carlo (MC) procedures in general are stochastic methods for estimating large sums (or
high-dimensional integrals) by picking out a comparatively small number of terms (or evaluating
the integrand only for a relatively small number of points). Let us assume we want to
compute the averageX := 1
M
and x some observable with the (true) variance vx = 1
M
(21)
(22)
�M l=1xl, wherel is an index (e.g., an Ising configurationl ≡ {s})
� M
l=1 (xl − X) 2 . In a simple MC ap-
proach, one may select a subset of N ≪ M indices independently with a uniform random
distributionP(lj) = const. (for1 ≤ j ≤ N),
XMC = 1
N
N�
j=1
xlj
∆XMC := 〈(XMC −X) 2 〉 = vx
N ≈
1
N(N −1)
N�
j=1
(23)
(xlj −XMC) 2 . (24)
Here, the averages are taken over all realizations of the random experiment (each consisting of
a selection of N indices). In the limit of N → ∞, the distribution of XMC becomes Gaussian
according to the central limit theorem. Only in this limit is the estimate of vx from the QMC
data reliable. An application for a continuous set is illustrated in Fig. 3.
Smaller errors and faster convergence to a Gaussian distribution for the estimate may be obtained
by importance sampling. Here, the functionxl is split up,
xl = plol; pl ≥ 0;
M�
pl = c, (25)
9 Since ∆ σm has only one non-zero element, it is clear that only the row m of ∆ σm (Mσ) −1 will contain
non-zero elements. The determinant of1+∆ σm (Mσ) −1 is then equal to the product of its diagonal elements.
l=1