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arXiv:1109.4292v1 [nucl-th] 20 Sep 2011
Exactly solvable pairing Hamiltonian for heavy nuclei
J. Dukelsky,1S. Lerma H.,2L. M. Robledo,3R. Rodriguez-Guzman,1and S. M. A. Rombouts1,4
1Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain
2Departamento de F´ ısica, Universidad Veracruzana, Xalapa, 91000, Veracruz, Mexico
3Departamento de F´ ısica Te´ orica, M´ odulo 15, Universidad Aut´ onoma de Madrid, E-28049 Madrid, Spain
4Departamento de F´ ısica Aplicada, Universidad de Huelva, 21071 Huelva, Spain
We present a new exactly solvable Hamiltonian with a separable pairing interaction and non-
degenerate single-particle energies. It is derived from the hyperbolic family of Richardson-Gaudin
models and possesses two free parameters, one related to an interaction cutoff and the other to the
pairing strength. These two parameters can be adjusted to give an excellent reproduction of Gogny
self-consistent mean-field calculations in the canonical basis.
PACS numbers: 02.30.Ik, 21.60.Fw, 21.60.Jz
Pairing is one of the most important ingredients of
the effective nuclear interaction in atomic nuclei as it
was early recognized by Bohr, Mottelson, and Pines [1]
in an attempt to explain the large gaps observed in
even-even nuclei. They suggested that the recently pro-
posed Bardeen-Cooper-Schriefer (BCS) [2] theory of su-
perconductivity could be a useful tool in nuclear struc-
ture although care should be taken with the violation
of particle number in finite nuclei. Since then, BCS or
the more general Hartree-Fock-Bogoliubov (HFB) the-
ory combined with effective or phenomenological nuclear
forces has been the standard tool to describe the low en-
ergy properties of heavy nuclei. Improvements over BCS
or HFB came through the restoration of broken symme-
tries, specially particle number projection which is still
a problem not satisfactory solved with density depen-
dent forces [3]. From a different perspective, Richardson
found an exact solution of the constant pairing problem
with non-degenerate single particle energies as early as in
1963 [4]. Though highly schematic, the constant pairing
force has been used for decades in nuclear structure with
several approximations (BCS, RPA, PBCS, etc.)
scarcely resorting to the exact solution. Almost forgot-
ten, the exact Richardson solution was recovered within
the framework of ultrasmall superconducting grains [5],
in which not only number projection but also pairing fluc-
tuations were essential to describe the disappearance of
superconductivity as a function of the grain size.
but
By combining the Richardson exact solution with the
integrable model proposed by Gaudin [6] for quantum
spin systems, it was possible to derive three families of
integrable models called Richardson-Gaudin (RG) mod-
els [7]. The rational family, extensively used since then,
contains the Richardson model as a particular exactly
solvable Hamiltonian as well as many other exactly solv-
able Hamiltonians of relevance in quantum optics, cold
atom physics, quantum dots, etc. [8]. However, the other
families did not find a physical realization up to very
recently when it was shown that the hyperbolic family
could model a p-wave pairing Hamiltonian in a 2 dimen-
sional lattice [9], such that it was possible to study with
the exact solution an exotic phase diagram having a non-
trivial topological phase and a third order quantum phase
transition [10]. In this letter we will show that the hyper-
bolic family give rise to a separable pairing Hamiltonian
with 2 free parameters that can be adjusted to reproduce
the properties of heavy nuclei as described by a Gogny
HFB treatment.
Let us start our derivation with the integrals of motion
of the hyperbolic RG model [7], which can be written in
a compact form [11] as
Ri
= Sz
i−
?
(1)
?
2γ
j?=i
? √ηiηj
ηi− ηj
?S+
iS−
j+ S−
iS+
j
?+ηi+ ηj
ηi− ηjSz
iSz
j
,
where Sz
algebra of copy i with spin representation si such that
?S2
i = 1,...,L. The L operators Ricontain L free param-
eters ηiplus the strength of the quadratic term γ. The
integrals of motion (1) commute among themselves and
with the z component of the total spin Sz=?L
are parametrized by the ansatz
i, S±
i, are the three generators of the SU(2)i
i? = si(si+ 1). We assume L SU(2)-algebra copies,
i=1Sz
i.
Therefore, they have a common basis of eigenstates which
|ΨM? =
M
?
β=1
S+
β|ν?, S+
β=
?
i
√ηi
ηi− EβS+
i, (2)
where |ν? is the vacuum of the lowering operators
S−
i|ν? = 0 and the Eβ (β = 1,··· ,M) are the pair en-
ergies or pairons which are determined by the condition
that the ansatz (2) must satisfy the eigenvalue equations
Ri|ΨM? = ri|ΨM? for every i.
In the pair representation of the SU(2) algebra, the
generators are expressed in terms of fermion creation and
annihilation operators S+
c†
ici− 1)/2. Each SU(2) copy is associated with a single
particle level i, with i the time reversed partner, and M
is the number of active pairs. The vacuum |ν? is defined
by a set of seniorities, |ν? = |ν1,ν2,...,νl?, where the
i= c†
ic†
i= (S−)†, Sz
i= (c†
ici+
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2
seniority νi= 0,1 is the number of unpaired particles in
level i, which determines the spin associated to the level
as si= (1 − νi)/2. The blocking effect of the unpaired
particles reduces the number of active levels to Lc =
L −?
generates an exactly solvable Hamiltonian, we will re-
strict ourselves in this presentation to the simple lin-
ear combination H=λ?
nipulations the Hamiltonian reduces to
iνi.
Although any function of the integrals of motion
iηiRi. Defining λ=
(1 + 2γ(1 − M) + γLc)−1, and after some algebraic ma-
H =
?
i
ηiSz
i− G
?
i,i′
√ηiηi′S+
iS−
i′, (3)
where G = 2λγ is a free parameter.
This Hamiltonian, expressed in a 2 dimensional mo-
mentum space basis gave rise to the celebrated px+ ipy
model of p-wave pairing [9, 10]. However, if we interpret
the parameters ηias single particle energies correspond-
ing to a nuclear mean-field potential, the pairing interac-
tion has the unphysical behavior of increasing in strength
with energy. In order to reverse this unwanted effect we
define ηi= 2(εi− α), where the free parameter α plays
the role of an energy cutoff and εiis the single particle
energy of the mean-field level i. Making use of the pair
representation of the SU(2), the exactly solvable pairing
Hamiltonian (3) takes the form
H =
?
−2G
i
εi
?
c†
ici+ c†
ici
?
(4)
?
ii′
?
(α − εi)(α − εi′)c†
ic†
ici′ci′,
with eigenvectors given by (2) and eigenvalues
E = 2αM +
?
i
εiνi+
?
β
Eβ. (5)
Here the pairons Eβ correspond to a solution of the set
of non-linear Richardson equations
?
i
si
ηi− Eβ
−
?
β′(?=β)
1
Eβ′ − Eβ
=
Q
Eβ,(6)
where Q =
Eq. (6) defines a unique eigenstate (2).
In order to get an insight into the solutions of (6) we
show in figure 1 the ground-state pairon dependence on
the pairing strength G for a schematic system of M = 10
pairs moving in a set of L = 24 equally spaced single-
particle levels (εi= i) and a cutoff α = 24. For G → 0
the pairons are all real and stay close to a set of M pa-
rameters ηi(the M lowest η′s for the G.S. configuration)
in order to cancel the divergence in the r.h.s. of (6). As G
increases the pairons move down in energy till they reach
1
2G−Lc
2+M −1. Each particular solution of
?
?
?
?
?
?
?
?
Re
?Eβ
?
Im
?Eβ
?
G
Im[Eβ]
Re [Eβ]
G=0.020 G=0.035
FIG. 1: Real and imaginary parts of the ground-state pairons
as a function of pairing strength, for a set of 24 equally spaced
single-particle levels (εi = i), a cutoff α = 24 and M = 10
pairs. The inset shows the pairon distribution in the complex
plane for two different pairing strengths.
a critical value of G ≈ 0.012 for which the two pairons
closest to the Fermi level collapse to η = −30. Immedi-
ately after they acquire an imaginary part and expand
in the complex plane as a complex-conjugate pair. The
same phenomenon happens to the other pairons as G is
further increased forming an arc in the complex plane as
can be seen in the inset of Fig. 1. Even though the be-
havior of the pairons resembles that of the rational model
[8], there are qualitative differences associated to the non-
constant form of the pairing interaction that will turn out
to be essential for the description of heavy nuclei.
In what follows we will derive the two free parame-
ters G and α of the integrable Hamiltonian (4) by fit-
ting its BCS wavefuntion to a Gogny HFB calculation
in the canonical basis. The HFB calculations with the
Gogny force have been carried out with the standard D1S
parametrization [12], and the canonical basis obtained
by diagonalizing the Hartree-Fock (HF) field.The pairing
tensor is not exactly diagonal, but we have checked that
the off diagonal contributions are much smaller than the
diagonal ones. In this approximation HFB in the canon-
ical basis is equivalent to BCS.
Due to the separable character of the integrable pair-
ing interaction the state dependent gaps and the pairing
tensor in the BCS approximation are
∆i= 2G√α − εi
?
i′
√α − εi′ui′vi′ = ∆√α − εi, (7)
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3
?
?
?
?
?
?
?
?
?
?
∆i
∆i
uivi
uivi
εi
FIG. 2: State dependent gaps ∆i, and pairing tensor uivi
for protons in238U and154Sm. Open circles are Gogny HFB
calculations in the canonical basis while the continuous lines
are the BCS results of the integrable Hamiltonian.
uivi=
∆√α − εi
2?(εi− µ)2+ (α − εi)∆2. (8)
Note that the gaps ∆iand the pairing tensor uivide-
pend on a single gap parameter ∆ and have a square
root dependence on the single particle energy. Hence,
the model has a highly restricted form for both magni-
tudes that we will test against the Gogny gaps ∆G
?
ical basis and (uGvG) is the HFB eigenvector. We take
the single particle energies εiof the integrable Hamilto-
nian from the HF energies of the Gogny HFB calculations
and we set up an energy cutoff of 30 MeV on top of the
Fermi energy. Occupation probabilities above this cutoff
are lower than 10−3and oscillate randomly. In order to
fit the two parameters of the model α and G and to fulfill
the BCS equations for the chemical potential µ and the
gap ∆, we solve the following three coupled equations for
the chemical potential µ, the gap ∆ and the parameter
α:
i
=
i′Vii,i′i′uG
are the matrix elements of the Gogny force in the canon-
i′vG
i′ and pairing tensor uG
ivG
i, where Vii,i′i′
2M − L +
?
i
ξi
Ei
= 0, (9)
MLG∆EG
Corr EBCS
Corr EExact
Corr
154Sm 31 91 2.24×10−30.1577 1.3254 1.0164 2.9247
238U 46 148 1.99×10−30.1594 0.8613 0.5031 2.6511
TABLE I: Parameter values and correlation energies for pro-
tons in154Sm and238U
iF+n+1
?
i=iF−n
?
uG
ivG
i−∆
2
ti
Ei
?tiξ2
i
E3
i
= 0,(10)
?
i
?∆G
i− ∆√α − εi
√α − εi
?
= 0,(11)
where ti =√α − εi, ξi = (εi− µ), and the quasipar-
ticle energy Ei =
?
Eq. (10) is a fitting of the Gogny pairing tensor uG
with respect to the gap parameters ∆, i.e. we minimize
?iF+n+1
to enhance the quality of the fit for the most correlated
levels. We typically choose n ∼ 10. Finally, Eq. (11)
fixes the interaction cutoff α by minimizing the differ-
ences?
µ, α and ∆ are fixed, the pairing strength is determined
from Eq. (7,8)
ξi2+ ∆i2. Eq.(9) is the BCS
number equation that fixes the chemical potential µ.
ivG
i
i=iF−n
lect n levels above and below the Fermi energy in order
?uG
ivG
i− uivi
?2with respect to ∆. Here we se-
i
?∆G
i− ∆√α − εi
?2between the state depen-
dent Gogny gaps ∆G
iand ∆i, with respect to α. Once
1
G=
?
i
(α − εi)
i+ (α − εi)∆2.
?ξ2
As a first step in ascertain the quality of the hyperbolic
Hamiltonian (4) to reproduce the superfluid features of
heavy nuclei, we show, in Fig. 2, the state dependent
gaps ∆i and the pairing tensor uivi for protons corre-
sponding to two heavy nuclei,154Sm and238U. Following
the fitting procedure we consider all levels below 30MeV
above the Fermi energy and solve selfconsistently equa-
tions (9-11) for the chemical potential µ, the gap param-
eter ∆ and the interaction cutoff α. Fig. 2 shows a re-
markable agreement between the Gogny force and the hy-
perbolic Hamiltonian for the pairing tensor. The Gogny
state dependent gaps exhibit large fluctuations due to the
details of the two-body Gogny force. However, the gen-
eral trend of the gaps is very well described by the square
root
?(α − εi) of the hyperbolic model. Although238U
the mapping is excellent for both nuclei. It is interesting
to note that the rational model, leading to the constant
pairing exactly solvable Richardson Hamiltonian, has a
constant gap (a horizontal line) failing completely to de-
scribe the Gogny gaps. Table I shows the number of pairs
has 50% more proton pairs than154Sm the quality of
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4
?
?
?
?
?
Re
?Eβ
?
Re
?Eβ
?
Im
?Eβ
?
FIG. 3: Pair energies (grey circles) of the exact ground state
solution for protons in238U and154Sm. The horizontal seg-
ments in the real axis represent the parameters ηi = 2(εi−α).
M, the number of active levels L within the energy cut-
off, the pairing strength G, the gap parameter ∆ and the
correlations energies for both nuclei.
Once we have set up the procedure to define the pa-
rameters of the hyperbolic Hamiltonian in the BCS ap-
proximation, we are ready to explore the exact solution.
For a general pairing Hamiltonian the dimension of the
Hilbert space is given by the Binomial B(L,M). Using
the information of Table I, the dimensions are 1.98 ×
1024for154Sm and 4.83 × 1038for238U, well beyond
the limits of a large scale diagonalization. However, the
integrability of the hyperbolic Hamiltonian allows us to
obtain the exact solution by solving the set of M non-
linear coupled Richardson equations (6). The exact cor-
relation energy shown in Table I are in both nuclei consid-
erable greater than the mean-field results, reflecting the
importance of beyond mean-field quantum correlations
and number fluctuations. The exact ground state wave-
function is completely determined by the position of the
M pairons in the complex plane. Fig. 3 shows the exact
ground states for both nuclei. Considering the structure
of the pair wavefunctions (2) we may argue that238U
has 4 correlated Cooper pairs, while154Sm has only 2.
Further analysis of the Cooper pair wavefunction from
the exact solutions as was carried out in [13] for cold
atoms and in [14] for nuclei within the rational model is
straightforward but beyond the scope of this letter.
In summary, we have presented a new exactly solvable
Hamiltonian with separable pairing interaction and non-
degenerate single particle energies (4), which arises as a
particular linear combination of the hyperbolic integrals
of motion (1). The separable form of the pairing matrix
elements could be derived from a novel Thomas-Fermi
approximation for a contact interaction in a square well
potential [15]. We have shown that the separable Hamil-
tonian (4) with 2 free parameters is able to reproduce
qualitatively the general trend of the state dependent
gaps as described by the Gogny force in the canonical
basis. At the same time, it reproduces accurately the
HFB wavefunction represented by the pairing tensor. As
such, our exactly solvable Hamiltonian is an excellent
benchmark for testing approximations beyond HFB in
realistic situations for even and odd nuclei. Moreover, a
self-consistent HF plus exact pairing approach could be
set up along the lines of Ref. [16] for well bound nu-
clei. The inclusion of exact T=1 proton-neutron pairing
within this self-consistent approach is also possible [17].
We acknowledge support from a Marie Curie Action
of the European Community Project No. 220335, the
Spanish Ministry for Science and Innovation Project
No. FIS2009-07277, and FPA2009-08958, the Mexi-
can Secretariat of Public Education Project PROMEP
103.5/09/4482.
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