On the formulation of a functional theory for pairing with particle number restoration

Guillaume HupinGANIL, Caen FRANCE

Collaborators : • M. Bender (CENBG) • D. Lacroix (GANIL)• D. Gambacurta (GANIL)

2

Brief summary of the SC-EDF functional

The SC-EDF functional

• SC-EDF• MR-EDFnon-regularized

No regularizationpossible

SC-EDF is

SR-EDF and MR-EDF

3

Practice

Expressing the 2-body densities a function of xi

with

withBCS

occupation probabilities

Or recurrence relation using

4

Variation After Projection in EDF (VAP)E

(MeV

)

SR-EDF broken symmetry minimum

SC-EDF minimum

PAV

VAPBCS

Exact

Coupling strength

Cor

rela

tion

ener

gy

Pairing Hamiltonian

VAP

Threshold effects

Sin

gle

parti

cle

ener

gy

Δε

Motivations

Better reproduction of the energy.

Correct finite size effects (no threshold).

Optimization of the auxiliary state .

Flexibility of EDF.

5

Variational principle

With the SC-EDF functional

Applied using the parameters of auxiliary state

K. Dietrich et al. Phys. Rev. 135 (1964)MV Stoitsov et al. PRC (2007)…

6

Numerical methods

76Kr

Preliminary : simplification of F(vk) to reduce the numeric to a minimum search.

1 - Imaginary time step method to diagonalized MF Hamiltonian.

2 - Gradient method to solve the secular equations with respect to vi.

Solve minimization with gradient method

New set of vi and MF potentials

Evolve imaginary time

New set of φi

ConvergenceVAP

BCS BCSVAP

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Functional Theory : flexibility of SC-EDF

Solves the BCS threshold problem.

Avoids complex numeric.

Easy to implement in existing Mean-Field codes.

→ 1 Mean-Field.

Functional theory allows to modify the expression of the energy.

VAP

BCS

VAP

BCS

Original VAP VAP with ni nj

8

Achievements of this work

1. MR-EDF when regularized can be viewed as a DFT.

2. A modification of the regularization makes possible to associate MR-EDF with a correlated auxiliary state.

3. SC-EDF formalism allows to use density dependent interaction.

DFT

Here

Question : Is it possible to express the energy as a function of ρ1 [N] ?

It is already the case of the BCS theory :

9

Density Matrix Functional Theory (DMFT)- alternative path

Exact

DMFT

Focus on one body observables

DFT

Describes at the minimum of the functional energy

T.L. Gilbert PRB 12 (1975)

Cor

rela

tion

ener

gy

Full one body observables

N. N. Lathiotakis et al. PRB 75 (2007)

Recently applied in electronic systems

Example : Homogenous Electron Gas (HEG)

10

Sin

gle

parti

cle

ener

gy

DMFT from a projected BCS state

BCS PBCS

?

with

Δε

11

Applications and benchmarks of the new functional

?A new systematic 1/N expansion beyond BCS:

BCS

Lacroix and Hupin, PRB 82 (2011)

EH

F - E

Coupling

Objective : invert into

Finite size effectsOK when all terms

are included

Exact

Exact

BCS

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Resummation into a compact functional

All contributions can be approximately summed to give:

with BCS

EH

F - E

Coupling

EH

F - E

EH

F - E

Coupling Coupling

4 particles 16 particles 44 particles

BCS

Exact

New func.

13

Applications : more insights

Hupin et al. PRC83 (2011)

What is required for realistic situations in Condensed Matter and Nuclear physics ?

1. Can be applied to odd system.

2. Functional applicable to small and large systems while reproducing the desired physics (here the finiteness of systems).

3. The single particle spectra upon which is applied the functional should not be constrained.

Richardson model Any spectra

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Applications : odd systems

Great improvement over BCS

+Energy of odd systems is better

reproduced

Odd systems have been described in terms of a blocked state – the last occupied state (i) of the Fermi sea.

PBCS

Richardson

Particle number

We define the mean gap (BCS gap in thermo. limit)

BCS Functional

Exact

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Applications : thermodynamic limit

Lacks some correlations at small

number of particles

+The functional does as good as

the PBCS ansatz

G. Sierra et al. PRB 61 (2000)

Cor

rela

tion

ener

gy

~1/A

Sin

gle

parti

cle

ener

gy

Δε=dFiniteness of physical systems is also of interest in condensed matter

SuperconductingNanoscale grains

Parameterization of the SP energy splitting and particle number (A) :

BCSFunctional

Exact

Dot = odd systems

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Applications : random single particle spectra

This functional is efficient with

any SP spectra can be used

with self consistent methods

?Generate SP energy levels

Normalized to unity the average SP splitting

Solve minimization with gradient method

New set of vi and MF potentials

Evolve imaginary time

New set of φi

For instance

In a SC scheme

BCS

Functional

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Extension : functional for finite temperature

DMFT : information reduced to one body observables

Functional build from Hamiltonian finite temperature

Entropy reduced to a set of one body observables

Balian, Amer. J. Phys. (1999)

D. Gambacurta (GANIL)

Esebbag, NPA 552 (1993)

Gibbs free energy

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Conclusions and Perspectives

Restoration of particle number in MR-EDF

Reanalyzed the MR-EDF method with its regularization.

Proposed and alternative method that is a functional of the projected state SC-EDF.

• PAV : direct use of the SC-EDF functional

• VAP : variation of the functional

E (M

eV)

PAV

VAPSC-EDF

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Conclusions and Perspectives

SC-EDF (↔ MR-EDF regularized) is a framework for the restoration of particle number in functional theory.

However, the SC-EDF restores the functional flexibility (ρα ).

Refitting of the pairing functional.

Application to others symmetries ?

Neutron / Proton pairing with finite size correction.

Pai

ring

ener

gy

N/Z

ExpBCS/HFB

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Conclusions and Perspectives

New DMFT functional for finite size systems with pairing

Proposition.

Benchmark with exact solution of Richardson model.

Check the applicability in realistic cases.

Thermodynamics and dynamics of finite systems.

Quantum phase transition exploration.

• Large N• Odd even• Random spectra