Odd-particle systems in the shell model Monte Carlo method: circumventing a sign problem.

Abstract

The shell model Monte Carlo method is a powerful technique to calculate thermal and ground-state properties of strongly correlated finite-size systems. However, its application to odd-particle-number systems has been hampered by the sign problem that originates from the projection on an odd number of particles. We circumvent this sign problem for the ground-state energy by extracting the ground-state energy of the odd-particle-number system from the asymptotic behavior of the imaginary-time single-particle Green's function of the even-particle-number system. We apply this method to calculate pairing gaps of nuclei in the iron region. Our results are in good agreement with experimental pairing gaps.

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arXiv:1201.3341v1 [nucl-th] 16 Jan 2012
Odd-particle systems in the shell model Monte Carlo: circumventing a sign problem
Abhishek Mukherjee and Y. Alhassid
Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, CT 06520
(Dated: January 17, 2012)
We introduce a novel method within the shell model Monte Carlo approach to calculate the
ground-state energy of a finite-size system with an odd number of particles by using the asymptotic
behavior of the imaginary-time single-particle Green’s functions. The method circumvents the sign
problem that originates from the projection on an odd number of particles and has hampered direct
application of the shell model Monte Carlo method to odd-particle systems. We apply this method
to calculate pairing gaps of nuclei in the iron region.
experimental pairing gaps.
Our results are in good agreement with
PACS numbers: 21.60.Ka, 21.60.Cs, 21.60De, 21.10.Dr, 27.40.+z, 27.40.+e, 26.50.+x
Introduction. The shell model Monte Carlo (SMMC)
approach [1–4] has been used successfully to calculate sta-
tistical properties of nuclei [5–7] within the framework of
the configuration-interaction shell model. Recently, this
method has also been applied to trapped cold atom sys-
tems [8, 9]. The SMMC method enables calculations in
model spaces that are many orders of magnitude larger
than those that can be treated by conventional diagonal-
ization methods.
For typical effective nuclear interactions, the SMMC
method breaks down at low temperatures because of the
so-called fermionic sign problem, leading to large statis-
tical errors. In the grand-canonical ensemble the sign
problem can be avoided by constructing good-sign inter-
actions that include the dominant collective components
of effective nuclear interactions [10]. The remaining part
of the effective interaction can be accounted for by using
the method of Ref. 2.
In finite-size systems, such as nuclei, it is necessary
to use the canonical ensemble, in which the number of
particles is fixed. This particle-number projection gives
rise to an additional sign problem when the number of
particles is odd, leading to a rapid growth of statistical
errors at low temperatures even for good-sign interac-
tions. Consequently, it has been a major challenge to
make accurate estimates for the ground-state energy of
odd-particle systems in SMMC. Accurate ground-state
energies are necessary for the calculation of level densi-
ties and pairing gaps (i.e., odd-even staggering of binding
energies).
Here we develop a method based on the asymptotic
behavior of the imaginary-time single-particle Green’s
functions of an even-particle system to calculate ground-
state energies of neighboring odd-particle systems. This
method is somewhat similar in spirit to a technique used
in lattice quantum chromodynamics to extract hadron
masses (see, e.g., in Ref. 11). We apply our Green’s func-
tion method to calculate pairing gaps of nuclei in the iron
region using the complete fp + g9/2shell model space.
Green’s functions in SMMC. The SMMC method is
based on the Hubbard-Stratonovich representation of the
imaginary-time propagator, e−βH=
where β is the inverse temperature, H is the Hamilto-
nian, D[σ] is the integration measure, G(σ) is a Gaussian
weight, and Uσ(β) is the propagator of non-interacting
nucleons moving in external auxiliary fields σ that de-
pend on the imaginary time τ (0 ≤ τ ≤ β). The canonical
thermal expectation value of an observableˆO is given by
?ˆO? =
?D[σ]G(σ)TrA[ˆOUσ(β)]/?D[σ]G(σ)TrAUσ(β),
where TrA denotes a trace over the subspace of a fixed
number of particles A. In actual calculations we project
on both proton number Z and neutron number N, and
in the following A will denote (Z,N).
For a quantity Xσthat depends on the auxiliary fields
σ, we define
?D[σ]G(σ)Uσ(β),
Xσ≡
?D[σ]|W(σ)|XσΦσ
?D[σ]|W(σ)|Φσ
,(1)
where W(σ) = G(σ)TrAUσ and Φσ = W(σ)/|W(σ)| is
the sign. With this definition, the above thermal expec-
tation of an observableˆO can be written as ?ˆO? = ?ˆO?σ,
where ?ˆO?σ = TrA[ˆOUσ(β)]/TrAUσ(β). In SMMC we
choose M samples σk according to the weight function
|W(σ)|, and estimate the average quantity in (1) by
Xσ≈?
For an even number of particles with a good-sign in-
teraction, the average value of the sign Φσremains close
to 1. However, when the number of particles is odd, the
average sign decays towards zero as the temperature is
lowered. This leads to rapidly growing errors, hampering
the direct application of SMMC at low temperatures for
odd-particle systems.
For a rotationally invariant and time-independent
Hamiltonian, we define the following scalar imaginary-
time Green’s functions [12]
kXσkΦσk/?
kΦσk.
Gν(τ) =TrA
?e−βHT?
maνm(τ)a†
TrAe−βH
νm(0)?
, (2)
where ν ≡ (nlj) labels the nucleon single-particle orbital
with radial quantum number n, orbital angular momen-
tum l and total spin j. Here T denotes time ordering

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and aνm(τ) ≡ eτHaνme−τHis an annihilation operator
of a nucleon at imaginary time τ (−β ≤ τ ≤ β) in a
single-particle state with orbital ν and magnetic quan-
tum number m (−j ≤ m ≤ j).
Using the Hubbard-Stratonovich transformation, the
Green’s functions defined in (2) can be written in a form
suitable for SMMC calculations
Gν(τ) =











?
m
[Uσ(τ)(I − ?ˆ ρ?σ]νm,νm
for τ > 0
?
m
??ˆ ρ?σU−1
σ(|τ|)?
νm,νm
for τ ≤ 0
,
(3)
where we have used the notation in Eq. (1). Here Uσ(τ)
and I are matrices in the single-particle space repre-
senting the propagator Uσ(τ) and the identity, respec-
tively. ?ˆ ρ?σis a matrix in the single-particle space whose
νm,ν′m′matrix element ?ˆ ρνm,ν′m′?σis defined in terms
of the one-body density operator ˆ ρνm,ν′m′ = a†
ν′m′aνm.
Assuming A is an even-even nucleus, A±≡ (Z,N ±1)
are neighboring odd-even nuclei with odd number of neu-
trons. We denote by Jn (n = 0,1,2,...) the n-th excited
state with total spin J and define the energy differences
∆EJ(A±) = EJ0(A±) − E00(A), where EJn(A) is the
energy of the state Jn with particle number A. Assum-
ing that the ground state of the even-even nucleus has
spin zero, ∆EJ(A±) is the energy difference between the
lowest state of a given spin J in the odd-even nucleus
A± and the ground state of the even-even nucleus A.
Assuming that the ground state of the odd-even nucleus
A± is J = j, where j is one of the single-particle or-
bital spin values, its corresponding energy is given by
Egs(A±) = E00(A) + ∆Emin(A±), where ∆Emin is the
minimum of ∆Ej(A±) over the possible values of j.
The neutron Green’s function Gν(τ) that corresponds
to an orbital with angular momentum j can be written
as
Gν(τ) = C(β)e−∆Ej(A±)|τ|


1 +
?
Jn?=00
J′n′?=j0
RJ′n′
Jn(A±,ν)e−|τ|[EJ′n′(A±)−Ej0(A±)]e−(β−|τ|)[EJn(A)−E00(A)]



(4)
where the + (−) subscript should be used for τ >
0 (τ≤0) and C(β) is a τ-independent con-
stant.RJ′n′
Jn(A±,ν) are scaled weights defined by
RJ′n′
Jn(A+,ν)=|(J′n′||a†
RJ′n′
(J′n′||a†
ments of a†
νand aν between the state Jn in A and the
states J′n′in A+and A−, respectively.
When all terms in the summation on the r.h.s. of Eq.
(4) are small, the Green’s function can be well approxi-
mated by a single exponential, Gν(τ) ∼ e−∆Ej(A±)|τ|. In
this asymptotic regime for τ, we can calculate ∆Ej(A±),
and hence Egs(A±) from the slope of lnGν(τ). This is
the method we use here to calculate the ground-state en-
ergy of odd-A nuclei with odd number of neutrons. The
ground-state energy of odd-A nuclei with odd number
of protons can be similarly calculated using the proton
Green’s functions.
In principle, the asymptotic regime is accessed in the
limit β → ∞. However, in a shell-model Hamiltonian
with discrete, well separated energy levels, only a few
transitions give significant contributions. If the relative
contribution from the sum in Eq. (4) is less than a few
percent, then (assuming that |τ| ∼ 1 MeV) the sensitiv-
ity of the slope of lnGν(τ) to this contribution is about
ν||Jn)|2/|(j0||a†
ν||00)|2
and
Jn(A−;ν) = |(J′n′||aν||Jn)|2/|(j0||aν||00)|2, where
ν||Jn) and (J′n′||aν||Jn) are reduced matrix ele-
a few tens of keV, which is comparable to our target ac-
curacy. For low- and medium-mass nuclei, we expect the
energy differences to be ? 1 MeV and the scaled weights
to be much smaller than one. Thus, calculations with β
of a few MeV−1and with an asymptotic regime of τ ∼ 1
MeV should be sufficient. This can be validated explic-
itly in sd-shell nuclei (see below), whose Hamiltonian can
be diagonalized numerically. For larger model spaces, it
is not possible to calculate explicitly the corrections in
the sum of Eq. (4), and the asymptotic region has to be
determined by the goodness of the linear fits to lnGν(τ).
Results. We first tested the Green’s function method
in sd-shell nuclei and then applied it to medium-mass
nuclei in the complete (pf + g9/2) shell. In these nuclei,
we carried out calculations for several values of β in the
range 3 MeV−1≤ β ≤ 4 MeV−1. For each β, we calcu-
lated Gν(τ) for a range of values of τ in steps of 1/32
MeV−1. We chose the asymptotic region in τ such the
linear fits to lnGν(τ) have a χ2per degree of freedom
∼ 1 or less in all cases considered. We find that a good
asymptotic region is 0.5 MeV−1≤ τ ≤ 2 MeV−1.
Within the asymptotic region, we fit a straight line to
lnGν for each possible subset of points in τ for which
Gν(τ) has been calculated. The mean and standard de-
viation of the slopes so obtained are used to estimate

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0123
|τ| (MeV-1)
0
10
20
30
|ln G1p3/2|
12
|τ| (MeV-1)
-0.2
0
0.2
12
|τ| (MeV-1)
-0.1
0
0.1
56Fe
58Fe
FIG. 1: The absolute value of logarithm of the Green’s func-
tion (2) for the neutron orbital ν = 1p3/2in56Fe (lower curve,
τ > 0) and58Fe (upper curve, τ ≤ 0) at β = 4 MeV−1. The
solid blue lines are linear fits for 0.5 MeV−1≤ |τ| ≤ 2 MeV−1.
The insets show the deviations from these linear fits.
∆Emin(A±) and its statistical error, respectively, at each
β. A weighted average of the results at different values
of β is then taken.
In a few selected cases, we also performed calculations
for larger values of β (i.e., β > 4 MeV−1), and found the
corresponding values of ∆Emin(A±) to be consistent with
those obtained in the region 3 MeV−1≤ β ≤ 4 MeV−1.
This indicates that for the model spaces and particle
numbers considered, the above chosen values of β are
sufficiently large to isolate the ground state of the corre-
sponding even-even nucleus.
For a given odd system (an odd-even nucleus) there are
two neighboring even systems (even-even nuclei), and our
method can be used by starting from either of the even
systems. Unless noted otherwise, the results we report
here are the average of both of these calculations.
To test the validity and accuracy of our method, we
performed calculations in the sd shell using a schematic
good-sign Hamiltonian. In all cases, our results devi-
ated no more than 0.1% from the exact ground-state en-
ergies, obtained by diagonalizing the Hamiltonian with
the OXBASH code [13]. For example for29Si we found a
ground-state energy of −133.98 ± 0.04 MeV compared
with the exact result of −133.95 MeV. Our method also
reproduced correctly the ground-state spin in all cases.
We applied our method to nuclei in the (pf+g9/2) shell,
using the isospin-conserving Hamiltonian of Ref. [5].
Typical results are demonstrated in Fig. 1, in which the
absolute value of the logarithm of the Green’s functions
for the neutron orbital ν = 1p3/2in56Fe (τ > 0) and
in58Fe (τ ≤ 0) are plotted versus |τ| for β = 4 MeV−1.
The linear fits (solid lines) were used in the calculation
of the ground-state energy of57Fe. The deviations from
the linear fits are shown in the insets of Fig. 1.
A direct application of the SMMC method to the odd-
33.23.4
β (MeV-1)
3.63.84
-206
-204
-202
-200
E (MeV)
3.2 3.64
β (MeV-1)
0.01
0.1
1
σE (MeV)
FIG. 2: The energy of the57Fe nucleus calculated from the
present method and direct SMMC are shown by solid and
open squares, respectively. The error bars describe the statis-
tical errors. Inset: the statistical errors for the energy of57Fe
in the present method (solid squares) and in direct SMMC
calculations (open squares) are shown on a logarithmic scale.
The statistical errors for the energy of
Hamiltonian are shown by open circles.
56Fe using the same
particle systems suffers from a sign problem which leads
to very large statistical errors at low temperatures. In
contrast, the method presented here does not have such
problem.This is illustrated in Fig. 2 where we com-
pare the energy and its statistical error for the57Fe nu-
cleus in the present method (using the neutron Green’s
functions of56Fe) with the results obtained from the di-
rect method. The errors in the present method remain
roughly constant with β. At β = 3 MeV−1the statistical
error in the direct method is about 5 times larger than
the present method while at β = 4 MeV−1it is about 20
times larger. The inset shows the statistical errors on a
logarithmic scale. For comparison we have also included
the statistical error in the energy of the even-even nucleus
56Fe using the same Hamiltonian.
We applied our Green’s function method for fami-
lies of odd-neutron isotopes:
59−65Ni,63−67Zn and71−73Ge. The ground-state spins
we determine are in agreement with experimental values
in all cases except for47Ti,57Fe and63Ni. The anomalous
ground-state spin of57Fe from the shell model perspec-
tive is well documented in the literature [14].
In our method we extract directly the odd-even ground
state energy differences, and therefore this method is par-
ticularly suitable for accurate calculations of pairing gaps
(i.e., odd-even staggering of masses).
When extracting an odd-even ground-state energy dif-
ference such as ∆Emin(A+) we use the Hamiltonian of the
A+nucleus for both the A+and A nuclei. Since the fp+
g9/2-shell Hamiltonian we use is nucleus-dependent [5], it
is necessary to correct the ground-state energy of the A
nucleus. As the latter is an even-even nucleus, this cor-
47−49Ti,51−57Cr,53−61Fe,

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48505254
0.5
1
1.5
2
∆n (MeV)
Ti
Cr
54565860
Fe
Exp.
Theory
606264
0.5
1
1.5
2
Ni
646668
70
ZnGe
A
FIG. 3: Neutron pairing gaps ∆n as a function of mass num-
ber A in fp + g9/2-shell nuclei. The gaps calculated with the
present Green’s function method (solid circles connected by
solid lines) are compared with the experimental gaps (open
circles connected by dashed lines). The theoretical statistical
errors are smaller than the size of the symbols.
rection can be found in direct SMMC calculations for
the A nucleus. However, this correction can also be esti-
mated as follows. The dependence of the interaction on
the nucleus is rather weak; the strengths of the multipole-
multipole interactions depend weakly on the mass num-
ber A (∝ A−1/3) and the monopole pairing strength is
constant through the shell. The largest variation among
neighboring nuclei is that of the single-particle energies
εµ(A) of the orbitals µ. Correcting for this variation, the
neutron separation energy for the A+nucleus is given by
Sn(A+) = −∆Emin(A+) +
?
µ
[εµ(A) − εµ(A+)]?nµ?A,
(5)
where ?nµ?Aare the average occupation numbers for the
A nucleus using the Hamiltonian for the A+nucleus. The
second term on the r.h.s. of (5) approximates the differ-
ence between the ground-state energies of the A nucleus
when calculated using the respective Hamiltonians for
the A and A+nuclei. We verified (in sd-shell nuclei) that
this approximation is highly accurate and well within a
typical statistical error. In our calculations we used (5)
since the resulting statistical error is much smaller than
the statistical error of direct SMMC calculations.
The neutron separation energy for the A nucleus is
given by a similar expression. The neutron pairing gaps
can then be calculated from the differences of separation
energies ∆n(A) = (−)N[Sn(A+) − Sn(A)]/2, where A
can now be either an even-even or an odd-even nucleus.
Our calculated pairing gaps are shown in Fig. 3 (solid
circles), where they are compared with the experimental
pairing gaps (open circles) as determined from odd-even
staggering of binding energies. Our results agree quite
well with the experimental values; in most cases the de-
viation of the theoretical pairing gaps from their experi-
mental counterparts is less than 15%. Systematic devia-
tions are observed for the iron isotopes above A = 59 and
for the germanium isotopes. For the germanium isotopes
the size of the model space might be insufficient, while
the deviation for the iron isotopes indicates the necessity
to refine our isospin-conserving Hamiltonian.
Conclusion.
that circumvents a sign problem for calculating the
ground-state energy of odd-particle systems in the shell
model Monte Carlo approach. We have demonstrated
the usefulness of the method by calculating pairing gaps
of nuclei in thefp + g9/2shell. This method can also
be applied to other finite-size many body systems such
as trapped cold atoms. In principle this method can be
used more generally to calculate the lowest energy state
for a given spin. However, when such a state is an ex-
cited state, the statistical errors are larger and it is more
difficult to identify the asymptotic regime.
We have described a practical method
Acknowledgements. This work was supported in part
by the U.S. Department of Energy Grant No. DE-FG02-
91ER40608. Computational cycles were provided by the
High Performance Computing Center at Yale University.
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