Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. Guillaume SCAMPS Center for computational sciences, Tsukuba University November 6th 2017 Collaboration : Y. Hashimoto, T. Nakatsukasa Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 1 / 25
Recent evolution of the mean-field dynamics TDHF+BCS TDHFB TDHF Simplified pairing Full pairing No pairing correlations correlations Computational time Computational time × 1 . 5 × 1000 TDHF+BCS : S. Ebata, T. Nakatsukusa, et al., Phys. Rev. C 82, 034306 (2010). G. Scamps, D. Lacroix, Phys. Rev C 87, 014605 (2013). Time-dependent Hartree-Fock-Bogoliubov (TDHFB) : I. Stetcu, A. Bulgac, P. Magierski, and K. J. Roche, Phys. Rev. C 84, 051309(R) (2011). Y. Hashimoto, Phys. Rev. C 88, 034307 (2013). Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 2 / 25
Collision between two superfluid nuclei described with TDHFB with a Gogny force HFB breaks the particle-number symmetry → qp-vacuum states Results : have define gauge angles ) 0 1 / fm 1 45 20 O+ 20 O ( 90 P z / h 135 / 0 Relative momentum 20 O -1 20 O -2 4 6 8 10 12 14 16 18 20 22 at t=0, rotation of the gauge Relative distance R (fm) E cm = 11.41 MeV angle : Y. Hashimoto, G. Scamps, Phys. U → e i ϕ U θ ( z ) Rev. C 94, 014610 (2016) V → e − i ϕ V θ ( z ) Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 3 / 25
Evolution of two TDHFB calculation at the vicinity of the barrier κ ( r , ↑ , r , ↓ ) = | κ ( r , ↑ , r , ↓ ) | e 2 i ϕ ( r ) Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 4 / 25
Evolution of two TDHFB calculation at the vicinity of the barrier κ ( r , ↑ , r , ↓ ) = | κ ( r , ↑ , r , ↓ ) | e 2 i ϕ ( r ) Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 4 / 25
Josephson transfer 20 O+ 20 O 20 O 20 O J s ∝ sin (2 ϕ ) . Transfer Josephson effect The transfer of nucleons depends on the relative gauge angle. Y. Hashimoto, G. Scamps, Phys. Rev. C 94, 014610 (2016) Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 5 / 25
Problem Question The relative gauge angle is not a parameter of the reaction. Does those results of TDHFB are spurious ? Projection method We should restore the symmetry of the relative gauge angle | Ψ( t = 0) � = ˆ P N L − N R ( N L − N R ) | φ � , � 2 π P N L − N R ( N L − N R ) = 1 e i ϕ [ (ˆ N L − ˆ N R ) − ( N L − N R ) ] d ϕ ˆ 2 π 0 We have to consider an evolution of a mixture of HFB states N N 2 1 Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 6 / 25
Approximation Starting point M Evolution � | Ψ( t = 0) � = c n | φ n ( t = 0) � , n =1 | φ n ( t = 0) � = e i 2 n π ˆ N L | φ ( t = 0) � . M Assumption during the evolution Observables M O = � Ψ( t ) | ˆ O | Ψ( t ) � � | Ψ( t ) � = c n | φ n ( t ) � , � Ψ( t ) | Ψ( t ) � n =1 | φ n ( t ) � evolves with the TDHFB equation of motion We assume an evolution of a set of TDHFB trajectories with fixed coefficients. Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 7 / 25
Multi-nucleon transfer in the sub-barrier regime Motivations : Transfer under the barrier Montanari et al., PRL 113 (2014) Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 8 / 25
Calculation of the transfer probabilities Projection method with pairing Projection method for TDHF P L ( N ) = � Ψ( t ) | ˆ P L ( N )ˆ P ( N tot ) | Ψ( t ) � P L ( N ) = � Ψ( t ) | ˆ P L ( N ) | Ψ( t ) � � Ψ( t ) | ˆ P ( N tot ) | Ψ( t ) � � 2 π P L ( N ) = 1 e i ϕ (ˆ ˆ N L − N ) d ϕ 2 π 0 G. Scamps, D. Lacroix, PRC 87, (2013). *only when one of the fragment is C. Simenel, PRL 105 (2010). superfluid Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 9 / 25
Triple projection method Triple projection method P L ( N , t ) = � Ψ( t ) | ˆ P L ( N ) | Ψ( t ) � � Ψ( t ) | Ψ( t ) � � 2 π � 2 π 1 N − N tot ) ˆ d ϕ 2 e i ϕ 1 (ˆ U TDHFB ( t 0 , t ) e i ϕ 2 (ˆ N L − N L ) | φ ( t = 0) � | Ψ( t ) � = d ϕ 1 (2 π 2 ) 0 0 TDPHFB TDHFB t=t f t=0 t=0 t=t f Û Û Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 10 / 25
Triple projection : Pfaffian method Evolution Overlap G. F. Bertsch and L. M. Robledo, PRL 108, 042505 (2012) Optimized Pfaffian calculation : M. Wimmer, ACM Trans. Math Softw. 38, 30 (2012). Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 11 / 25
Test on toy model Simple model Exact solution Time-dependent V ( t ) = V 0 exp ( − α t 2 ) Multi-configuration method K. Dietrich, Phys. Let. B 32, 6 (1970). Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 12 / 25
Toy model Result 0.0045 Exact 0.004 0.0035 0.003 0.0025 P 2 0.002 0.0015 0.001 0.0005 0.0 -15 -10 -5 0 5 10 15 t [10 − 22 s] Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 13 / 25
Toy model Result 0.12 TDHFB 0.1 Exact x10 0.08 0.06 P 2 0.04 0.02 0.0 -0.02 -15 -10 -5 0 5 10 15 t [10 − 22 s] Conclusion Spurious result with the TDHFB evolution Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 13 / 25
Stationary condition Assumption during the evolution Stationary condition M � | Ψ( t ) � = c n | φ n ( t ) � , Without interaction : n =1 � φ n ( t + dt ) | φ n ( t ) � ≃ 0 | φ n ( t ) � evolves with the TDHFB equation of motion Important point We need to impose stationary condition in the TDHFB equation of motion Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 14 / 25
Modification of the equation of motion � U ( t ) � U ( t ) i � ∂ � � = H , V ( t ) V ( t ) ∂ t � h ∆ � H = (1) − ∆ ∗ − h ∗ � h − δλ L , R ( t ) − ǫ k ( t ) ∆ � H = (2) − h ∗ + δλ L , R ( t ) − ǫ k ( t ) − ∆ ∗ δλ L , R ( t ) = λ L , R ( t ) − λ L , R ( t = 0) 0.014 TDHFB eq. (1) 0.012 TDHFB eq. (2) Exact 0.01 0.008 P 2 0.006 0.004 0.002 0.0 -15 -10 -5 0 5 10 15 t [10 − 22 s] Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 15 / 25
Comparison Results 10 − 1 10 − 2 P 2 10 − 3 10 − 4 TDHFB exact 10 − 5 10 − 3 10 − 2 10 − 1 1 2 5 2 5 2 5 − V 0 [MeV] Conclusion The method works well in the toy model. We can expect a predictive power in realistic calculations. Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 16 / 25
TDHFB with Gogny interaction x z y N base = 2760 x and y direction : Harmonic oscillator basis n x + n y ≤ 4 In comparison, a full cartesian mesh is about z direction : Lagrange mesh nz= 46 100 000 degrees of freedom Cost of the calculation : one collision done in one day with 20 CPUs Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 17 / 25
Dispersion of the trajectories Evolution of the set of TDHFB trajectories Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 18 / 25
Dispersion of the trajectories Evolution of the set of TDHFB trajectories Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 18 / 25
Effect of the dispersion of the trajectories Non stable results Overlap 0.25 0.14 TDHFB 0.2 0.12 Intrinsic frame |� φ 1 ( t ) | φ 2 ( t ) �| 2 0.1 0.15 0.08 P 2 0.1 0.06 0.04 0.05 0.02 0.0 0.0 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 t [fm/c] t [fm/c] Guillaume SCAMPS Restoration of symmetry in time-dependent calculations. Josephson effect study with the Gogny TDHFB calculation. November 6th 2017 19 / 25
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