NUCLEAR DFT: SPECIAL PROBLEMS, SOME SOLUTIONS

1. NUCLEAR DFT: SPECIAL PROBLEMS, SOME SOLUTIONS Back to basics State potatoes and slices Hamiltonian Translation invariance, c.m. trap Rotation invariance, radial DFT Intrinsic frame, projectordependence Particlenumber, concavity Constructive DFT, polynomials B. G. Giraud, IPhT Saclay, oct.'11

2. Back to basics • In practice, the nuclear DF and/or EDF calculationswhich have been published look verymuchlikeHartree-Fock or Hartree-Bogoliubovcalculations, where one has to guess a good form for the functional, with efficient parameters for the variousenergyterms, hence an efficient Kohn-Shampotential. • This approach has hadquite an amount of success for the reproduction of experimental data, but thissuccesspushedseveral basic problems to bepartlyforgotten. • Present DF solutions are alwayslocalized, oftendeformed and often do not have a welldefinedparticlenumber. The discussion to-dayreturns to the basic factthat a DFT or EDFT resultsfrom an energyminimizationunderconstraint. This gives a new light to the restoration of (E)DF brokennuclearsymmetries. B. G. Giraud, IPhT Saclay, oct.'11

3. Potato of trial states, slices of densityconstraint, locus of energy minima Minima or Infima? Degeneracies? Analyticity for locus? Verifyconcavity of F[\rho] B. G. Giraud, IPhT Saclay, oct.'11

4. Slices, constraintcalculus • A proper (E)DFT can consider nuclear states that are pure states, with a many-body density operator | \psi >< \psi |. But sets of mixed states, such as \sumn wn | \psin >< \psin |, can be most useful to minimize the energy. • Many states have the same one-body density \rho. Such states with the same \rho make a “slice”, labeled by \rho. • Energy shall be minimized within each slice constrained by its label \rho. The obtained minimum (or infimum) is a function of the slice, hence a functional, F[\rho], of the density. • This functional is necessarily, mathematically, CONCAVE. Typically, F[(\rho+\rho’)/2] < (F[\rho]+F[\rho’])/2. This is a major property of minimizations under constraint(s). B. G. Giraud, IPhT Saclay, oct.'11

5. Hamiltonian and reduced information • As willbeexplainedstep by step, weadvocatecompleting the usualnuclearHamiltonian by: • a center-of-massharmonictrap: k R2, and • a concavityterm, c (N2+Z2). • Wealsoadvocate a theorywith few radial positions for measuring the density. The whole profile is not necessary. B. G. Giraud, IPhT Saclay, oct.'11

6. LocalizorHamiltonian The usual, Galilean invariant Hamiltonian, makesimpractical the definition of a density: the center of mass (CM) iseverywhere. B. G. Giraud, IPhT Saclay, oct.'11

7. Internaldensity vs Labdensity Now, because of the FACTORIZATION of the wavefunctionwith the trapped CM, bothdensitiesturn out to berelated by a convolution. The CM trapdoes not change the internalphysics. It isthuseasy to use a DFT in the Lab system and recover the internaldensity by a deconvolution. Alternatetheory: seeMessudet al, Phys. Rev. C 80 (2009) 054314 B. G. Giraud, IPhT Saclay, oct.'11

8. Consequence of rotational invariance of H Let Z,N,A ≡ Z + N be the proton, neutron, and mass numbers, respectively. The nuclearHamiltonianH is invariant under rotations. Therefore, besidesZ and N, nuclearg.s. carry good quantum numbers, J and M, for the total angularmomentum and itsz-component. Two cases occur: (i) either J = 0, hence a nondegenerateg.s., the density of whichis ISOTROPIC, or (ii) J > 0, hence a trivial degeneracy for a magnetic multiplet of g.s., the densities of which, nonisotropic, containseveralmultipoles, with the same monopole part of the density for all members of the multiplet. In both cases, the many-bodydensityoperator for the (set of) ground state(s) of a given nucleus canread, D INVARIANT under rotation, whether the nucleus isodd or even, deformed or spherical. The correspondingone-bodydensitymakes an ISOTROPIC profile. B. G. Giraud, IPhT Saclay, oct.'11

9. The radial DFT (RDFT) • For anydensitymatrix D in many-bodyspace, the energyreads, E=Tr (H D), where Tr means the trace in many-bodyspace of such a product of operators. • The previousslideshowedthat, for any nucleus, itsg.s. energy, degenerate or not, resultsfrom a rotation invariant D. • The density of such a D istherefore a scalar. Weneedjust to studyfunctionals of radial profiles. • Notice: adding to H a term k J2, k small and variable, tells the spin. • Notice that the methodis compatible with the CM trap, whichis invariant under rotation. • This reduces (E)DFT to one-dimensionalcalculations. Partial filling of sphericalorbitalswilldefinesphericalKohn-Shampotentials. B. G. Giraud, IPhT Saclay, oct.'11

10. About deformed solutions of the usual (E)DFT • Advantage: they do signal deformations and, whenangularmomentum projection isavailable, theygive a whole band. • Defect: 3-dim or 2-dim calculations, while the RDFT means 1-dim. • Defect: angularmomentum projection costly; furthermore, variation after projection costlier. • Advantage: density \tau in the intrinsic frame. • Can one definerigorously a DFT in an intrinsic frame? Yes! • Let D be a many-bodydensityoperator, not rotation invariant, and PJbe an angularmomentumprojector (magnetic label understood). Define a slice of D’shaving the sameone-bodydensity \tau, deformed. Then, within the slice, minimize the ratio, ( Tr H PJ D ) / (Tr PJ D). • This minimum, a function of the slice, defines a functional FJ(\tau). • Defect: J dependence! B. G. Giraud, IPhT Saclay, oct.'11

11. Nature is not concave • Experimentalbindingenergies show a valley of stability, henceat least an approximateamount of concavity. But second differences, EA-1-2EA+EA+1 , canbenegative, because of, amongother causes, pairing and/or shelleffects. See, for instance, the sequence of tin isotopes: B. G. Giraud, IPhT Saclay, oct.'11

12. Nature is not concave Scatter plot of second differenceswith respect to N B. G. Giraud, IPhT Saclay, oct.'11

13. Nature is not concave Scatter plot of second differenceswith respect to Z B. G. Giraud, IPhT Saclay, oct.'11

14. Forcing Nature to be concave If the set of trial states carries good quantum numbers N and Z, one obtains a functional FN,Z[\rho]. But thereis no reason to expectthat, afterchanging N and Z, therewillbeconcavitywith respect to either N or Z. For instance, given the threeg.s. densities \rhoN,Z-1 , \rhoN,Z , \rhoN,Z+1 , itcanhappenthat FN,Z[\rhoN,Z] > (FN,Z-1[\rhoN,Z-1]+FN,Z+1[\rhoN,Z+1])/2. Namely, interpolations can overestimate the binding! One would prefer a universal functional F(\rho), valid for all values of N and Z as defined by integrating \rho. This is specially useful for Hartree-Bogoliubov cases, where N and Z are treated as constrained average values of operators N and Z, within trial states that do not have such good quantum numbers. As already stated, constrained minimizations make necessarily concave function(al)s. Let -c be the worst negative second difference observed experimentally. Then a term, c(N2+Z2)/2, added to H, increases every second differences by c. Physics is not changed, since the added term commutes with H. But, this way, “Nature has been made concave” and, now, it can be accounted for by a “universal” DF. B. G. Giraud, IPhT Saclay, oct.'11

15. Constructive DFT, via polynomials, from configuration mixing the densityat the origin and the densityat the mid-surface or in the tail. Then the free energy and the Lagrange multipliers are related by polynomial equations, B. G. Giraud, IPhT Saclay, oct.'11

16. Constructive DFT, via polynomials, from configuration mixing B. G. Giraud, IPhT Saclay, oct.'11

17. Constructive DFT, toy model, 1 constraint, subspace dimension 4 B. G. Giraud, IPhT Saclay, oct.'11

18. Constructive DFT, toy model, 1 constraint, subspace dimension 4 This final polynomial has order d x (d-1), with d the number of mixed states. It canbeunwieldy. But itensuresanalyticity, and the lowestrootmakes a concave branch. B. G. Giraud, IPhT Saclay, oct.'11

19. Summary of results • CM problemsolved; modestcost 1- and 2-body operatortrap. • Rotation problemsolved; RDFT brings 1-d simplification. • Existence of DFT for deformations in intrinsic frame, but complicated projection and J-dependence. • Compatibility betweenHamiltonian and concavity; modestcost N2+Z2term. • Constructive theory; costbigpolynomials. B. G. Giraud, IPhT Saclay, oct.'11

20. References and Thanks • Translation: Phys. Rev. C 77, 014311 (2008) • Rotation: Phys. Rev. C 78, 014307 (2008) • Deformedintrinsic: B.G. G., B.K. Jennings, B.R. Barrett, Phys. Rev. A 78, 032507 (2008) • Concavity: B.R. Barrett, B.G. G., B.K. Jennings, N. Toberg, Nucl. Phys. A 828, 267 (2009) • Constructive DFT: B.G.G., S. Karataglidis, Phys. Lett. B 703, 88 (2011) Andthanks to TRIUMF Laboratory (Canada), Rhodes University (South Africa) and University of Johannesburgh (South Africa) for partial support of thiswork. B. G. Giraud, IPhT Saclay, oct.'11