Exactly solvable Richardson-Gaudin models in nuclear structure

Transcription

1 Exactly solvable Rchardson-Gaudn models n nuclear structure Jorge Dukelsky In collaboraton wth people n audence: S. Pttel, P. Schuck, P. Van Isacker. And many others

2 Rchardson s Exact Soluton

3 Exact Soluton of the BCS Model P k k k Egenvalue equaton: H P H n g c c c c k, k ' k E k k ' k ' Ansatz for the egenstates (generalzed Cooper ansatz) M 0, 1 k 1 2 k E c c k k

4 Rchardson equatons M M g 2g 0, E E 2 E E E Propertes: k 0 k 1 1 Ths s a set of M nonlnear coupled equatons wth M unknowns (E ). The par energes are ether real or complex conjugated pars. There are as many ndependent solutons as states n the Hlbert space. The solutons can be classfed n the weak couplng lmt (g0). Exact solvablty reduces an exponental complex problem to an algebrac problem.

5 L k1 1 2 k E c c k k E= E= E= Evoluton of the real and magnary part of the par energes wth g. L=16, M=8. R. W. Rchardson, Phys. Rev. 141 (1966) 949. Solved numercal systems up to L=32, dm=10 8

6 The SU(2) Algebra Rank 1 and 1 quantum degree of freedom S, S S, S, S S, S, S 2S z z z The par realzatons s: 1 z j 1 S a a, S a a j jm jm j jm jm m m Other realzatons lke, two level atoms, spn, fnte center of mass pars, Holsten-Prmakoff or Schwnger, gve rse to dfferent physcal Hamltonans

7 Rchardson-Gaudn Models: Constructon of the Integrals of Moton J. D., C. Esebbag and P. Schuck, Phys. Rev. Lett. 87, (2001). The most general combnaton of lnear and quadratc generators, wth the restrcton of beng hermtan and number conservng, s X j 2 R S g S S S S Y S S j 2 z z z j j j j The ntegrablty condton R, Rj 0 leads to Yj X jk X jkyk X kx j These are the same condtons encountered by Gaudn (J. de Phys. 37 (1976) 1087) n a spn model known as the Gaudn magnet. 0

8 Gaudn (1976) found three solutons XXX (Ratonal) X j Y j 1 j XXZ (Hyperbolc Trgonometrc) 1 j j X j 2, Zj Cothx x j Snh x x Exact soluton j R j j Egenstates of the Ratonal and Hyperbolc Models Rchardson ansatz r 1 XXX S 0, XXZ S 0 E E

9 Any functon of the R operators defnes a vald ntegrable Hamltonan. The Hamltonan s dagonal n the bass of common egenstates of the R operators. Wthn the par representaton two body Hamltonans can be obtan by a lnear combnaton of R operators: H R, g The parameters g, s and s are arbtrary. There are 2 L+1 free parameters to defne an ntegrable Hamltonan n each of the famles. (L number of sngle partcle levels) The constant PM or reduced BCS Hamltonan solved by Rchardson can be obtaned by from the XXX famly by choosng =. For the same lnear combnaton n the Hyperbolc famly: l l l z H 2 S g S S BCS ï j j z H 2 S g S S Hyper ï ï j j j

10 Applcaton to Samarum sotopes G.G. Dussel, S. Pttel, J. Dukelsky and P. Sarrguren, PRC 76, (2007) Z = 62, 80 N 96 Selfconsstent Skyrme (SLy4) Hartree-Fock plus BCS n 11 harmonc oscllator shells. 40 to 48 pars n 286 double degenerate levels. Dm. of the parng Hamltonan matrx ~ to The strength of the parng force s chosen to reproduce the expermental parng gaps n 154 Sm ( n =0.98 MeV, p = 0.94 MeV) g n =0.106 MeV and g p =0.117 MeV. A dependence g=g n /A s assumed for the sotope chan.

11 Real Part Real Part 154 Sm C 2 C 5 C 4 C C C C C 1 G= ,0-0,5 0,0 0,5 1,0 G= Imagnary Part G= G= Imagnary Part

12 Correlatons Energes Mass Ec(Exact) Ec(PBCS Ec(BCS+H) Ec(BCS)

13 The Hyperbolc Model n Nuclear Structure J. Dukelsky, S. Lerma H., L. M. Robledo, R. Rodrguez-Guzman, S. Rombouts, Phys. Rev. C 84, (R) (2011) The separable ntegrable Hyperbolc Hamltonan Redefnng the 0 of energy the chemcal potental μ H S G S S z j j, j, absorbng the constant n H c c G c c c c j j j, j α s a new parameter that serves as an energy cutoff. In BCS approxmaton: The BCS Hamltonan has unphyscal Exactly solvable H wth nonconstant matrx elements G u v ' ' ' '

14 Mappng of the Gogny force n the Canoncal Bass We ft the parng strength G and the nteracton cutoff to the parng tensor u v and the parng gaps of the Gogny HFB egenstate n the Hartree-Fock bass. G u v uv ' ' ' ' Protons o Gogny _ Hyperbolco

15 M L D G E BCS corr E Exa corr 154 Sm x x U x x

16 Models derved from r = 1 RG [SU(2) and SU(1,1)] BCS or constant parng Hamltonan Generalzed Parng Hamltonans (Fermon and Bosons) Central Spn Model (Quantum dot) Gaudn magnets (Quantum magnetsm) Lpkn Model Two-level boson models (IBM, molecular, etc..) Atom-molecule Hamltonans (Feshbach resonances n cold atoms) Generalzed Jaynes-Cummngs models. Breached superconductvty. LOFF and breached LOFF states. p-wave parng n 2D lattces. Rchardson-Gaudn-Ktaev model of topologcal supeconductvty. Revews: J.Dukelsky, S. Pttel and G. Serra, Rev. Mod. Phys. 76, 643 (2004); G. Ortz, R. Somma, J. Dukelsky y S. Rombouts. Nucl. Phys. B 7070 (2005) 401

17 Exactly Solvable RG models for smple Le algebras Cartan classfcaton of Le algebras rank A n su(n+1) B n so(2n+1) C n sp(2n) D n so(2n) 1 su(2), su(1,1) parng so(3)~su(2) sp(2) ~su(2) so(2) ~u(1) 2 su(3) Three level Lpkns so(5), so(3,2) pn-parng sp(4) ~so(5) so(4) ~su(2)xsu(2) 3 su(4) Wgner so(7) FDSM sp(6) FDSM so(6)~su(4) color superconductvty 4 su(5) so(9) sp(8) so(8) parng T=0,1. Gnnoccho. S=3/2 fermons

18 Exactly Solvable Parng Hamltonans 1) SU(2), Rank 1 algebra 2) SO(5), Rank 2 algebra H n g P P j j H n g P P j j J. Dukelsky, V. G. Gueorguev, P. Van Isacker, S. Dmtrova, B. Errea y S. Lerma H. PRL 96 (2006) ) SO(8), Rank 4 algebra S. Lerma H., B. Errea, J. Dukelsky and W. Satula. PRL 99, (2007). 3) SO(6), Rank 3 algebra H n g P P j j B. Errea, J. Dukelsky and G. Ortz, PRA 79, (R) (2009). H n g S S g D D ST j j j j 1 1 S a a D a a 2 2,

19 Exact soluton of the SO(8) model M 1 E e u L l e e e 2 e g M M L ' ' ' M e 2 M M M M ' ' ' ' ' ' ' ' ' M M L ' ' ' ' M M L ' ' ' '

20 80 Nucleons n L=50 equdstant levels Quartet: 1e, 1, 1, 1 n-n Cooper par: 1e p-p Cooper par: 1e, 2, 1, 1

21 50 Even T G= Odd T 40 T=0 T=1 T=0,1 30 T=0 T=1 T=0,1 30 E T T T Analyss of the nuclear symmetry energy vs T n terms of the Isocrankng model (W. Satula and R. Wyss, PRL 86, 4488 (2001) and 87, (2001). e 1 o 1 ET T T, ET T T E 2J 2J T J T : so-moi, : Lnear enhancement factor (Wgner energy), E: 2qp exctaton (=2) T

22 Lnear enhancement factor λ Wgner lmt G=0.16 Inverse of the Iso-MoI G=.22 T=0 crcles, T=1damonds, T=0,1 trangles. Sold (open) -> even (odd) T

23 Pcket-Fence model and the thermodynamc lmt of p-n BCS G. F. Bertsch, J. Dukelsky, B. Errea, C. Esebbag, Ann. Phys. 325 (2019) 1340 Equdstant sngle partcle levels, 1,, 2 SU(4) symmetrc parng Hamltonan H n g S S D D ST j j j j Quarter fllng N, wth g 0.15 f 0.54 Thermodynamc lmt N,, N BCS equatons: 1/2 1/2 4 1 d 1, d g

24 Unlke the SU(2) RG model, we cannot derve analytcally the contnuous lmt. Proceed numercally by expandng the GS and quaspartcle energes as EGS b c d 4 a 2 3 1/ N N N N N E 4n E 4n 1 E 4n q GS GS 1 oe GS GS GS 2 g n 1 n 4n 1 2E 4n 1 E 4n E 4n 2 c 8 1, 160 N 1000, 40 n 250

25

26 levels, 200 partcles =0.5, g=-0.2 E correlacon exact 90 BCS T=(N-Z)/2

27 T=0,1 Parng Odd-Even Par effect as a sgnal of quartet correlatons 200 levels, g= E A+2 -E A -E A Exact p-n BCS Z=N

28 Summary For fnte systems, PBCS mproves sgnfcantly over BCS but t s stll far from the exact soluton. Typcally, PBCS msses ~ 1 MeV n bndng energy. The Isovector SO(5) and the SO(8) parng models are excellent benchmark models to study dfferent approxmatons dealng wth quartet correlatons, clusterzaton and condensaton. The SO(8) model can also descrbe spn 3/2 cold atoms where nuclear physcs could be explored n the lab. SO(5) has been used to test the QCM approxmaton n: N. Sandulescu, D. Negrea, J. Dukelsky, and C. W. Johnson Phys. Rev. C 85, (R) (2012) The exact GS energy of the T=0,1 parng Hamltonan goes to p-n BCS energy n the thermodynamc lmt. However, quartet correlatons are mportant for fnte systems. Alpha phases n nuclear matter requre more realstc nteractons: contact, schematc or realstc nuclear forces. Could they be explore wth cold atoms n optcal lattces?