Nonlinear QRPA. Adam Smetana. Institute of Experimental and Applied Physics Czech Technical University in Prague

Transcription

1 Nonlinear QRPA Adam Smetana Institute of Experimental and Applied Physics Czech Technical University in Prague Mikhail Krivoruchenko Fedor Šimkovic Jun Terasaki 06/21/2017 Workshop INT-17-2a, Seattle Adam Smetana Nonlinear QRPA 1/28

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6 Closer look to QRPA Derivation of QRPA starts with... the Schrödinger equation H n = E n n The hamiltonian is H = H(A, A,...) A = 0 where the state is the corresponding unperturbed vacuum. Adam Smetana Nonlinear QRPA 6/28

7 Phonon operator in QRPA...derivation of QRPA continues with... introducing the nth phonon operator Q n = Q n(x (n), Y (n) ) Q n 0 = n Q n 0 = 0 which creates nth excited state n from the ground state 0 This QRPA equation is fully equivalent to the Schrödinger equation 0 [δq, H, Q n] 0 = (E n E 0 ) 0 [δq, Q n] 0 where δq is some operator. Adam Smetana Nonlinear QRPA 7/28

8 QRPApproximation What makes the QRPA to be an approximation? 1. Truncation of the phonon operator Q 1 P 1(A, A,... ) X 1 A Y 1 A standard QRPA Q n P n (A, A,... ) Q n 1 multiphonon approach 2. Approximation of the true ground state 0 P 0 (A, A,... ) rpa = rpa = N e da A 3. Quasi-boson approximation violation of Pauli Exclusion Principle (PEP) 4. Not-satisfying the ground state condition: Q n rpa 0 Adam Smetana Nonlinear QRPA 8/28

9 QRPA equation 0 [δq, H, Q n] 0 = (E n E 0 ) 0 [δq, Q n] 0 Example phonon operator Q = XA YA δq {A, A} RPA ground state rpa = QRPA equation ( A B ) ( X ) B A Y ( 1 0 = (E 1 E 0 ) 0 1 ) ( X Y ) A = rpa [A, H, A ] rpa B = rpa [A, H, A] rpa Adam Smetana Nonlinear QRPA 9/28

10 Can (Q)RPA give us the exact solution? Yes, of course! but only for simple models 2 examples in the following 1 example by Jun Terasaki Adam Smetana Nonlinear QRPA 10/28

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12 pn-lipkin model H F = εc + λ 1 A A + λ 2 (A A + AA) decoupled systems of Ω odd and Ω even excited states finite number of excited states due to the Pauli exclusion principle A (2ω+1) = 0 j = 3/2, Ω = 3, 2 = 4, N = 4, χ = 0, Adam Smetana Nonlinear QRPA 12/28

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20 A phonon operator with just two amplitudes X and Y private to the nth excited state Adam Smetana Nonlinear QRPA 20/28

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22 Ω = 3 pn-lipkin model H F = εc + λ 1 A A + λ 2 (A A + AA) Adam Smetana Nonlinear QRPA 22/28

23 Exact QRPA for Ω = 3 pn-lipkin model we use full basis of operators for both Q and rpa : Q 135 = X 1A Y 1 A + X 3 A 3 Y 3 A 3 + X 5 A 5 Y 5 A 5 rpa = ( α 0 + α 2 A 2 + α 4 A 4 + α 6 A 6) we satisfy the ground state condition: Q 135 rpa =! 0 = α 2,4,6 = f 2,4,6 (X 1, X 3, X 5, Y 1, Y 3, Y 5 )α 0 we does not make bosonization we satisfy the Pauli exclusion principle We reproduce the exact solution! Adam Smetana Nonlinear QRPA 23/28

24 Towards realistic calculation truncation of the phonon operator: Q 135 = X 1 A Y 1 A + X 3 A 3 Y 3 A 3 + X 5 A 5 Y 5 A 5 Q 13 = X 1 A Y 1 A + X 3 A 3 Y 3 A 3 Q 1 = X 1 A Y 1 A PEP QRPA with Q rpa = 0 Adam Smetana Nonlinear QRPA 24/28

25 Energies j = 3/2, Ω = 3, 2 = 4, N = 4, χ = 0, No colapse in the range of κ 1.0 Adam Smetana Nonlinear QRPA 25/28

26 Beta transitions j = 3/2, Ω = 3, 2 = 4, N = 4, χ = 0, β (n) = rpa [Q(n), β ] rpa, β = 2Ω[u pv na + u nv pa] Adam Smetana Nonlinear QRPA 26/28

27 Conclusions Quality of QRPA approach depends on choice of Q and rpa For simple models with small configuration space the exact solution from QRPA is accessible For realistic calculation the configuration space must be truncated or approximated for both Q and rpa We want to go beyond the truncation of the level of linear phonon operator by including non-linear terms We have demonstrated the improvement from the phonon operator already with the next-order nonlinearities Our goal is to formulate and solve the realistic QRPA system with nonlinear phonon operator relevant for 0νββ We are now trying to understand the difficulties in numerical calculations Adam Smetana Nonlinear QRPA 27/28

28 Towards realistic calculation hamiltonian H = i ε i a i a i + 1 V ij,kl a i 4 a j a la k ijkl phonon operator Q k = c k + ph ground state + ph X k pha pa h + rpa = 1 N exp [ + Y k pha h a p + (p 1h 1)>(p 2h 2) (p 1h 1)>(p 2h 2)>(p 3h 3) (p 1h 1)>(p 2h 2) (p 1h 1)>(p 2h 2) C p1h 1p 2h 2 a p 1 a h1 a p 2 a h2 X k p 1h 1p 2h 2 a p 1 a h1 a p 2 a h2 Y k p 1h 1p 2h 2 a h 2 a p2 a h 1 a p1 C p1h 1p 2h 2p 3h 3 a p 1 a h1 a p 2 a h2 a p 2 a h2 ] Adam Smetana Nonlinear QRPA 28/28