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Momentum Distribution
In the path integral formalism, the momentum distribution can be
derived by projecting out a many particle state with momentum
. The projection operator is given by,
Applying it to the thermal density matrix using
Eq. 2.7 leads to

(213) 
To find the single particle momentum distribution, one averages over
the momentum of all particles except the first, which is equivalent to
performing the integrals
. Including an
extra normalization factor of , this leads to the single
particle momentum distribution,

(214) 
where
is the oneparticle reduced density
matrix defined in Eq. 5.1. The normalizations are given by
and
. For isentropic homogeneous systems, the oneparticle
reduced density matrix is only a function of
, which
allows one to introduce the function
with . It represents
the distribution function of the separation of open ends in a PIMC
simulation with one open path.
Classical particles exhibit the Maxwellian momentum
distribution. Therefore, the single particle density matrix is a
Gaussian,
For an ideal Fermi gas at , the momentum distribution is a Fermi function,
The single particle density matrix is proportional to the
spherical Bessel function , which oscillates around zero. In
the PIMC simulations, can be obtained in form of a histogram,
in which the separations of the open ends weighted with the sign of the
permutation are entered. At separations where it is negative, odd
permutations leading negative contributions must outweigh even
permutations making positive contributions. decays slowly
as , which requires macroscopic exchange cycles to
occur. Those are solely a consequence of the discontinuity of
at .
Before we discuss results from PIMC simulation of interacting systems
we will examine the scaling behavior of the offdiagonal sampling
method. In order to study a certain number of oscillations in ,
one can make a simple estimation of how many particles are
required. For this purpose, we neglect the fact that Eq. 5.11
was derived in thermodynamic limit, . In a simulation
using a 3D box of size , one can directly measure up
to and indirectly up to
. This leads
to the estimates for the required number of particles given in
Tab. 5.1.
Figure 5.2:
for systems of 33 spinparallel electrons at
and (
)
(solid line). At this temperature, fermionic effects are only
a small correction to the classical behavior. Therefore
negative contributions (dashdotted line) are negligible and
the sum of the positive contributions (dotted line) are
almost identical to the full . We also found that the
repulsive interactions do not lead to significant
modification to the noninteracting case given by the
Gaussian function in Eq. 5.9.

Figure 5.3:
from the system shown in Fig. 5.2 but here for a
significantly lower temperature of
,
where fermionic effects dominate (
). This
leads to negative regions in , as expected from the
zero temperature limit given by Eq. 5.11. In these
regions, odd permutations from exchange cycles with an even
number of particles dominate. The functions decrease rapidly
at as a result of finite size effects because
the minimum image method is applied.

Figure 5.4:
from Fig. 5.2 using the same line styles.
The extra factor emphasizes the small fermionic effects
that lead to some negative contributions.

Figure 5.5:
from Fig. 5.3 (thick lines). It shows the
oscillating behavior of n= in fermionic systems
(Eq. 5.11). One finds 3 zeros as expected for free
particles (see table 5.1). The thin lines show
finite size corrections for .

Figure 5.6:
Momentum distribution () for a finite system of 33
spinparallel electrons at
and
(
) also studied in
Fig. 5.2. The symbols show the momentum
distribution for a finite system of noninteracting fermions
under these conditions and the display the
MaxwellBoltzmann distribution.

Figure 5.7:
Momentum distribution as in Fig. 5.6 but
for a lower temperature of
. This leads to
a degenerate Fermi gas described by a FermiDirac
distribution, which is also shown for a system of 33
noninteracting fermions at this () and at zero
temperature (solid line). The difference to the
MaxwellBoltzmann distribution () is
substantial.

Table 5.1:
Minimal number of particles required to observe the th zero in
the single particle density matrix




1 
4.493 
12.2 
2.4 
2 
7.725 
62.3 
12.0 
3 
10.904 
175.1 
33.7 
4 
14.067 
376.0 
72.4 
From this table, one can quickly realize that the computational
demand grows rapidly with the number of zeros because and CPU time
. Furthermore, it should be
noted that one needs to go to sufficiently low temperatures to observe
the fermionic effects and that the CPU time scales linearly with the
number of time slices. Additionally, positive and negative
contributions cancel, which leads to fluctuations in the
observables. The fluctuation do not increase as rapidly as for the
direct fermion method (see section
2.6.5) since we still
use a nodal restriction but one needs converged results from all cycle
lengths, which becomes increasingly difficult at low temperature. A
detailed analysis of the scaling behavior with temperature remains to
be done.
As a first application of the offdiagonal density matrix sampling
method, we chose to study the electron gas because of its
simplicity. The method can be easily extended to hydrogen or
spinpolarized hydrogen. We looked at a system of 33 closed shell
spinparallel electrons at a density of (
)
and selected the two temperatures and
that represent the classical case at high temperature and,
respectively, the degenerate electron gas at low temperature. For
, the observed reduced density matrix in
Fig. 5.2 is in good approximation the classical Gaussian
function. Due to the high temperature, permutations are relatively
rare. Their contribution can be seen best in Fig. 5.4 where
is shown. Since we multiplied by the volume element , the graph can be interpreted as the probability of finding the
two ends of the open path separated by . The corresponding momentum
distribution in Fig. 5.6 was calculated directly from
MC average,

(219) 
rather then using a Fourier transform of , which would have
required an extrapolation for large or to store on a 3D
grid because the spherical symmetry is broken by the cubic simulation
box. The observed momentum distribution lies between the
MaxwellBoltzmann distribution and the Fermi distribution for the
corresponding noninteracting finite system. All three curves are
rather close together because the simulation is performed in a
classical regime. The deviations of the PIMC result from the free
Fermi distribution show the effect of the repulsive interactions
between the electrons, which leads to a depletion of the occupation
probability at small values.
This effect is also present in the low temperature results at
where one finds a degenerate electrons gas. The
momentum distribution in Fig. 5.7 is a Fermi function rather
than a MaxwellBoltzmann distribution, which can reach arbitrarily
higher occupation for because it is not limited by the Pauli
exclusion principle. The solid line denotes the ideal Fermi gas at
given by Eq. 5.10. Thermal excitations as well as the
Coulomb interaction lead to the population of momentum states above
the ideal Fermi momentum. For interacting systems at , a
discontinuity in the momentum distribution is still present but some
states are pushed to higher values (Ortiz and Ballone, 1994). The comparison
with the ideal Fermi gas at gives an estimate for the thermal
excitations at this temperature. The degree of degeneracy is rather
high, which has a significant consequence for the reduced density
matrix shown in Figs. 5.3 and 5.5. The latter graph
shows how positive contributions dominate at small separations
. Then the function goes through zero and even permutations
dominate. After that, it becomes positive again and finally approaches
zero near
, which is in good agreement with the
estimate given in Tab. 5.1. We corrected for the finite
size effects for by dividing out the reduction in the volume
element. This correction could also have been done by using the
Fourier transform of the sampled but it would have required
a higher number of kvectors than we kept. The figure also
shows that the magnitude of the positive and the negative
contributions still grows for
but their difference is
smaller, which leads to the expected decay of . The reason why
the magnitude of the positive and negative contributions still
increases can be understood from Fig. 5.8 where the
contributions from the individual cycle lengths are shown. Generally,
one finds that cycles of an odd (even) number of particles lead mainly
to positive (negative) contributions despite the possibility that
permutations of nearby closed paths could change the sign since it is
the sign of the total permutation that enters in the average. At small
separations the positive contributions from open 1cycles
dominate. Two particle permutations give rise to the biggest fraction
of negative contributions for
. For
, the
contributions from and longer cycles still increase because
the average separation of an open cycle of length is given by
. The cancellation
between odd and even cycles makes the function decay faster
than its positive and negative summands.
Figure 5.8:
Distribution from Fig. 5.5 split into the different
permutation cycles, shown for 1 to 6 particles. Cycles with
an odd number of particles mainly lead to positive
contributions and those even numbers to negative
contributions. The thins lines show the finite size
corrections for . The figure shows how the
distribution of the different cycles in the restricted path
integral method lead to the oscillations in the
function at sufficiently low .

Next: Natural orbitals
Up: OffDiagonal Density Matrix Elements
Previous: Nodal Restriction for Open
Contents
Burkhard Militzer
20030115