218
5 Formulation and Implementation of the Gamow Shell Model
when considering the Berggren basis. Identically, it is sufficient to consider the 2p2h part of the ˆ
H matrix, whereby SD f | ˆ
H |SD i = ± ±α f β f | ˆ
V |α i β i , as they form
the vast majority of nonzero matrix elements in the ˆ
H matrix. In this approximate
picture, the dimension of the ˆ
H matrix in the Gamow shell model approach is
proportional to N s
N v , so that the probability to have a nonzero matrix element is
proportional to (1/N s ) N v −2 , as all states in |SD i and |SD f must be equal, except
for |α i β i = |α f β f . Hence, the total number of nonzeros matrix elements is of the
order of d 1+2/N v , which corresponds to d 1.67 and d 1.5 when one truncated the model
space so that 3 or 4 valence nucleons are allowed in the continuum, respectively. In
particular, for a dimension d ∼ 10 9 , one has typically d 1.4 nonzero matrix elements
in standard shell model [29], so that the number of nonzero matrix elements in
the Gamow shell model is typically one to two orders of magnitude larger that
of the standard shell model for that same dimension. Consequently, computation
is more expensive in Gamow shell model than in standard shell model, so that
message passing interface communications are expected to play less important a
role in Gamow shell model than in standard shell model.
Let us consider two many-body Gamow shell model vectors, i.e., the initial |Ψ i
and final |Ψ f many-body states, verifying ˆ
H |Ψ i = |Ψ f . One can distribute the
basis states of the model space over all processors, so that the parts of the basis
model space on each processor are much smaller than the full space. In particular,
the parts of the basis model space to which |Ψ i and |Ψ f belong are deemed as
the initial and final spaces, respectively. They correspond to the number of rows
and columns, respectively, similarly to the code of many-body fermion dynamics
applied to nuclear structure [34]. Hence, one stores matrix elements of the form:
SD int |a α |SD i and SD f |a †
α |SD int SD int |a α a β |SD i , and SD f |a †
α a
†
β |SD int
where |SD int is an intermediate Slater determinant, chosen so that the additional
data to store are minimal. Any one-body or two-body observable can be calculated
with this scheme. One has indeed:
SD f |a
†
α a β |SD i = =SD f |a
†
α |SD int SD int |a β |SD i
(5.68)
SD f |a
†
α a
†
β a δ a γ |SD i = =SD f |a
†
α a
†
β |SD int SD int |a δ a γ |SD i . (5.69)
The phase matrix, containing by definition these matrix elements, is stored as a
sparse matrix. For this, one fixes |SD i , so that all the basis states |SD f which vary
by one or two states are generated. Hence, the relative phase between |SD i and
|SD f , as well the indices of |SD f and of associated one-body states, denoted as α,
β, γ , and δ in Eq. (5.69), must be stored.
It is most efficient to firstly loop over configurations, so that only the shell indices
associated to α, β, γ , and δ and the configuration index associated to |SD f must be
stored. The loop over Slater determinants is done afterward for fixed configurations,
so that only m-dependent values are stored at this level. The phase matrix can then
be recovered from this information. The number of phases associated to |SD int
is proportional to 2N v for one-body phases and N v (N v − 1) for two-body phases,
where N v is the number of valence nucleons. The factor 2 comes the fact that one
5 Formulation and Implementation of the Gamow Shell Model
when considering the Berggren basis. Identically, it is sufficient to consider the 2p2h part of the ˆ
H matrix, whereby SD f | ˆ
H |SD i = ± ±α f β f | ˆ
V |α i β i , as they form
the vast majority of nonzero matrix elements in the ˆ
H matrix. In this approximate
picture, the dimension of the ˆ
H matrix in the Gamow shell model approach is
proportional to N s
N v , so that the probability to have a nonzero matrix element is
proportional to (1/N s ) N v −2 , as all states in |SD i and |SD f must be equal, except
for |α i β i = |α f β f . Hence, the total number of nonzeros matrix elements is of the
order of d 1+2/N v , which corresponds to d 1.67 and d 1.5 when one truncated the model
space so that 3 or 4 valence nucleons are allowed in the continuum, respectively. In
particular, for a dimension d ∼ 10 9 , one has typically d 1.4 nonzero matrix elements
in standard shell model [29], so that the number of nonzero matrix elements in
the Gamow shell model is typically one to two orders of magnitude larger that
of the standard shell model for that same dimension. Consequently, computation
is more expensive in Gamow shell model than in standard shell model, so that
message passing interface communications are expected to play less important a
role in Gamow shell model than in standard shell model.
Let us consider two many-body Gamow shell model vectors, i.e., the initial |Ψ i
and final |Ψ f many-body states, verifying ˆ
H |Ψ i = |Ψ f . One can distribute the
basis states of the model space over all processors, so that the parts of the basis
model space on each processor are much smaller than the full space. In particular,
the parts of the basis model space to which |Ψ i and |Ψ f belong are deemed as
the initial and final spaces, respectively. They correspond to the number of rows
and columns, respectively, similarly to the code of many-body fermion dynamics
applied to nuclear structure [34]. Hence, one stores matrix elements of the form:
SD int |a α |SD i and SD f |a †
α |SD int SD int |a α a β |SD i , and SD f |a †
α a
†
β |SD int
where |SD int is an intermediate Slater determinant, chosen so that the additional
data to store are minimal. Any one-body or two-body observable can be calculated
with this scheme. One has indeed:
SD f |a
†
α a β |SD i = =SD f |a
†
α |SD int SD int |a β |SD i
(5.68)
SD f |a
†
α a
†
β a δ a γ |SD i = =SD f |a
†
α a
†
β |SD int SD int |a δ a γ |SD i . (5.69)
The phase matrix, containing by definition these matrix elements, is stored as a
sparse matrix. For this, one fixes |SD i , so that all the basis states |SD f which vary
by one or two states are generated. Hence, the relative phase between |SD i and
|SD f , as well the indices of |SD f and of associated one-body states, denoted as α,
β, γ , and δ in Eq. (5.69), must be stored.
It is most efficient to firstly loop over configurations, so that only the shell indices
associated to α, β, γ , and δ and the configuration index associated to |SD f must be
stored. The loop over Slater determinants is done afterward for fixed configurations,
so that only m-dependent values are stored at this level. The phase matrix can then
be recovered from this information. The number of phases associated to |SD int
is proportional to 2N v for one-body phases and N v (N v − 1) for two-body phases,
where N v is the number of valence nucleons. The factor 2 comes the fact that one