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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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?

?

?

?

?

?

?

?

?

?

∆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

Page 4

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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