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

Hirsch-Fye QMC 5.17
be reduced by about an order of magnitude and a smooth curve can be obtained by choosing an
appropriate boundary condition (short-dashed line) which has here been obtained by a numerical
minimization procedure for the variation of the second derivatives (see [38]). It is possible
to avoid the minimization by computing the correct second derivativeG (2) (0) analytically (for
arbitrary interactionU):
d2G(τ) dτ2 �
�
� = −
τ=0+
1
�
1+
2
U2
�
; (37)
4
in any case, however, a cubic spline interpolation of the full Green function suffers from the
problem that derivatives of fourth and higher order vanish on segments of the splines while they
can be large for the true Green function, in particular at τ = 0. 15
Splining the difference Green function using a model self-energy
A related method for improving on the natural spline scheme is to interpolate not {G(τl)} Λ l=0 ,
but only the difference with respect to some reference Green function obtained from, e.g., plain
second order perturbation theory or IPT. This approach, discussed in a general context by Deisz
[39], is implemented in Jarrell’s code [5]. In addition, the high-frequency part may be directly
stabilized by supplementing the QMC estimates with IPT (using low-pass/high-pass filters).
In the following, we will describe a stable and accurate method which achieves excellent highfrequency
behavior without requiring any filters [38, 40, 41]. It is based on the following exact
high-frequency expansion for the self-energy [42]: 16
Σσ(ω) = U (〈ˆn−σ〉− 1 〈ˆn−σ〉(1−〈ˆn−σ〉)
)+U2 +O(ω
2 ω
−2 ). (38)
One possible choice of a model self-energy with this asymptotic behavior (including the ω −1
term) which is nonsingular (and purely imaginary) on the imaginary axis is given by the following
two-pole approximation [38, 40, 41].
Σmodel,σ(ω) = U (〈ˆn−σ〉− 1 1
)+
2 2 U2 〈ˆn−σ〉 � 1−〈ˆn−σ〉 �� 1
+
ω +ω0
1 �
. (39)
ω −ω0
While the quality of the low-frequency part of this fit could be tuned by adjusting the parameter
ω0, it does not depend very sensitively on it as long as it is not much larger than the bandwidth
or the Nyquist frequency. By evaluatingΣmodel on the imaginary axis, the corresponding Green
function Gmodel can be computed for an arbitrary number of Matsubara frequencies. Consequently,
the Fourier transformation to imaginary time is unproblematic (when the “free” term
1/(iωn) is taken care of analytically). Thus, the difference of the Green functions can be accurately
evaluated at all time slices. Since the second derivative of the transformed model Green
function at τ = 0 exactly reproduces that of the true Green function, the difference is well
represented by a natural spline. The Green function is obtained as a function of Matsubara frequencies
by Fourier transforming the oversampled spline and adding the Matsubara-frequency
model Green function to the result.
15 This also implies that the boundary value (37) may not be the best choice for the whole intervalτ ∈ [0,∆τ].
16 The multi-band generalization [40] contains additional pair occupancy terms which have to be computed
numerically even at fixed band filling.