Basic Issues in Theories of Large Amplitude Collective Motion

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1 Basic Issues in Theories of Large Amplitude Collective Motion Takashi Nakatsukasa RIKEN Nishina Center) Adiabatic self-consistent collective coordinate ASCC) method GCM/GOA vs ASCC INT workshop: Quantitative Large Amplitude Shape Dynamics: fission and heavy ion fusion INT, Seattle, USA, October 3 th, 13

2 Brief introduction Theories of large amplitude collective motion, associated with TDHFB)

3 Adiabatic theories of LACM Baranger-Veneroni, Expansion with respect to χ Villars, Eq. for the collective subspace zero-th and first-order w.r.t. momenta) Non-uniqueness problem Validity condition Goeke- Reinhard, 1978-) ) ) ) t i t i e e t χ χ ρ ρ ) ) )], [ ) ) ) ) 1 Φ + Φ Φ Φ q q q im q Q H q q q Q q V H q δ δ Goeke, Reinhard, Rowe, NPA ) 48

4 Approaches to Non-uniqueness Problem 1) Yamamura-Kuriyama-Iida, 1984 Requirement of analyticity ex) Moya de Guerra-Villars, 1978) Therefore, in principle, we can determine a unique collective path in the ATDHF. The higher-order in p can be systematically treated. In practice, it is only applicable to simple models. ) Rowe, Mukhejee-Pal, 1981 Requirement of Point transf. and equations up to Op ) There is no systematic way to go beyond the second order in p. In practice, the method is applicable to realistic models as well.

5 Non-adiabatic theories of LACM Rowe-Bassermann, Marumori, Holzwarth-Yukawa, Local Harmonic Approach LHA) Curvature problem Correspondence between, Q,P Infinitesimal generator, is not guaranteed. δ Φ H V q Q Φ δ Φ [ H, Q ] + im 1 P Φ δ Φ [ H, P ] ic Q Φ Marumori et al, 198- Self-consistent collective coordinate SCC) method The problems of LHA are solved. The SCC equation is solved by the expansion with respect to q,p). δ Φq, p) Ĥ H q ˆQ H p H Φq, p) Ĥ Φq, p) ˆP Φ Adiabatic approx. LACM Matsuo, TN, Matsuyanagi, )

6 Adiabatic Self-consistent Collective Coordinate ASCC) method ASCC: Theory to define a collective submanifold with canonical coordinates. Equations expanded up to nd order in collective momenta Collective potential & collective masses leads to the following equations Constrained CMF) mean-field eq. LHE) Local harmonic eq. δ δ δ φ φ φ Hˆ V Qˆ φ [ ] Hˆ, iqˆ B Pˆ [ ˆ ˆ ˆ ] H V Q, P i 1 B φ C Qˆ [[ ] ] Hˆ, V Qˆ, Qˆ φ

7 Constrained mean-field calculation CMF equation δ β Ĥ λ ˆQ ) β, β ˆQ β β Potential energy surface V β) β Ĥ β Deformed non-equilibrium) MF states Slater determinants { β } Next step: Dynamics Staszczak et al.

8 GCM & GOA Construction of the Hilbert space δβ Ĥ λ ˆQ ) β HW eq. to a collective Hamiltonian! Slater determinants Generalized eigenvalue problem H β, β ') f β ')dβ ' ) f β ')dβ ' E I β, β ' Gaussian overlap approx. H β, β ') H GOA β, β ') I β, β ') I GOA β, β ') H coll β, β)ψβ) EΨβ) { β } H coll is constructed from Ĥ, with GOA mass Zero-point energy V β) β Ĥ β + ΔV β ) ) β Ĥ β ', I β, β ' ) β β ' H β, β '

9 ASCC method ASCC based on the TDVP ) β V q δ β Ĥ λ ˆQ! ) H q, p ) Ψ H Ψ ˆQ is determined by the local harmonic eq. LHE) LHE RPA at the potential minimum LHE RPA / TV) Mass obtained with No zero-point energy ΔV Some non-trivial?) remarks ˆQ is not merely the constraint operator Ĥ λ ˆQ LHE Hamiltonian is the one in a moving frame: ˆQ Q + Q α a + a + α ) α +Q α aa + Q 11 µ a + a ) α µ ) µ Ĥ λ ˆQ

10 Trivial example translation) TDHF eq. i t ρ t ) h ρ ) "# [ ], ρ t $ % Transformed TDHF i t ρ t ) h[ ρ ] v p, ρ t) $% & ' ρ t ) e i vt p ρ t Quasi-stationary solution in the moving frame #$ h[ ρ ] v p, ρ % & Due to Galilean symmetry ρ e Tr [ p ρ ] Amv im v r ρ e im v r Intrinsic motion is described by )e i vt p

11 Equation of collective motion TDHF TDVP) eq. δ Ψt) i t H Ψt) Collective variables Basic eq. of the SCC method Translational case q, p) δ Ψ qt), pt) ) i H Ψ qt), pt) ) t Quasi-stationary states in a moving frame δ Ψ q, p) Ĥ i q q i p p Ψ q, p ) δ Ψ q, p) Ĥ H p ˆP H q ˆQ Ψ q, p) δ Ψ q, p) Ĥ v P Ψ q, p) Marumori et al., PTP 64, ) R v, Ψ q, p q H p, " # ) ˆQ, ˆP P H p q $ % Ψ q, p ) i Determined by LHE

12 TDHFB) Classical Hamilton s form The TDHFB) equation can be described by the classical form. For instance, using the Thouless form Blaizot, Ripka, Quantum Theory of Finite Systems 1986) Yamamura, Kuriyama, Prog. Theor. Phys. Suppl )! z exp# 1 z c ph p " + c h $ & Φ % The TDHFB) equation becomes in a form i z 1+ zz + ) H z + 1+ z+ z) i z + 1+ z + z) H z 1+ zz+ ) The Holstein-Primakoff-type mapping ξ ph H π ph π ph H ξ ph Hz, z + ) z H z z z H ξ,π ) ξ + iπ ) ph! " z1+ z + z) 1/ # $ ph

13 Harmonic approximation in TDHF Small fluctuation around the HF state ξ,π H ξ,π Φ H Φ + 1 " c + p c h,c + h c p ) $ # In terms of classical variables ξ α, π ) ) ξ,π H ξ,π E + 1 ρ ph, ρ hp α! # " A B B * A * E + 1 Bαβ π α π β + 1 C αβξ α ξ β $! # &# % " A B % " c + % $ h c p ' B * A * ' $ & c + p c ' # h & ph index α ph) $ ρ hp & ρ &, ph % B αβ H π α π β π,ξξ ρ α 1 ξ α + iπ α ) E + 1 n p ) µ +ω n q µ ) * + µ Linear point transformation ξ α, π ) q, p ) α µ C αβ H ξ α ξ β π,ξξ Hξ α,π α ) Hq µ, p µ ) q δ µ µν µ µ α µ µ α µ µ X + Y ) α ξ, pµ ωµ X Y ) 1 ω q ξ α A B) αβ q ξ ν β, ω δ µ µν ξ q α µ α A + B) αβ α ξ q π β ν α

14 Harmonic approximation at non-equilibriums Second derivatives H c H q q c + H q c ξ α ξ β Σ q c ) ξ α ξ β q c ξ α ξ β Σ Σ Curvature C αβ Σ H H q c ξ α ξ β q c ξ α ξ β Σ Σ Mass B αβ H H q c Σ π α π β Σ q c π α π β Σ Second derivative are NOT tensors. n q q c ) Σ : q c, q n ; p c p n H Σ q n Σ Metric tensor K αβ q µ q µ ξ α ξ β C αβ αβ H H ξ α ξ Γ γ H β αβ ξ γ γ 1 γδ ) γ Γ K K + K K g f µ µ αβ δα, β δβ, α αβ, δ, µ, αβ Affine connection With this metric, the collective space is assumed to be flat.

15 Separation of Nambu-Goldstone modes Cyclic variable sym. op.) 1) Symmetry operator S momentum ) Symmetry operator S coordinate! " Ŝ, Ĥ # * p s, H $ * ) * * + { } PB p s { q s, H} qs PB ξ α B αγ C γβ q s TN, Walet, DoDang, PRC ) 143 TN, PTEP 1 1) 1A7 Ψt) S Ψt) q s ξt),πt) ) or p s ξt),πt) The Nambu-Goldstone modes become zero modes and are separated from the other modes. π α H π α π β ξ α Bαγ C γβ ξ β H ξ C ξ α α αβ q s q s π α π β q s ) H ξ α Bαβ q s ξ α

16 Local harmonic eq. in ASCC Hinohara et al, PTP 117 7) 451 TN, Walet, DoDang, PRC ) 143 Local Harmonic equation determines the normal modes: B αγ ξ β C γβ q ω ξ α B αγ q c C c q c γβ ξ ω q c α ξ β 1) Invariant property under the transformation: q c q c + cq s, p s p s + cp c V q V s q c V s q c Requires a gauge fixing of c. ) Strong canonicity condition prescription) [ ˆQ, Ŝ] to determine q ξ α ξ β and Problem q π α π β These quantities correspond to a + a-part pp-, hh-part) of ˆQ and Ŝ

17 Adiabatic SCC Symplectic LHE) Applications to simple models More realistic applications have been done with the P+Q Hamiltonian. Hinohara et al., PTP 119, 59 8); PRC 8, ); PRC 8, ); PRC 85, 433 1) Sato et al., NPA 849, ), PRC 86, 86, )

18 Applications to O4) models Model Hamiltonian Source of T-odd mean fields H P σ jm h 1 j m> # 1 "! 1 c Parameters Monopole+ Quadrupole pairing + Quadrupole int. G j m m P P + P P ) G P P + P P ) m ε, ε d 1 1 Ω 1 c jm, < Ω > Ω., d 14, Ω j j P / / 1., ε 3 1., d 1, Ω j m> σ 1. jm c j m c jm, Q 1 χq d j j ε 3 ε ε 1 m σ jm c + jm c jm Ω j+ 1 j

19 Larger G Hinohara, TN, Matsuo, Matsuyanagi, PTP115 6) 567 Potential Mass: M1/B Cranking Mass Effects of time-odd MF D φ Q φ

20 Exact Adiabatic SCC CHB with M cranking Time-odd effects are neglected in the cranking mass!

21 Curvature effects Neglected calc Full calc Qˆ µν Q + + aµ aν + h.c. ) + A B + µν Qµν aµ aν µν q ξ α ξ, q β π α π β In this model, requiring the gauge invariance, we can determine them. The curvature effects are weak.

22 Model of protons and neutrons T.N. & Walet, PRC ) 3397 G p.3 G p 1 Neutrons Protons Upper orbital has a larger quadrupole moment CHB+M cr LHE Exact

23 Adiabatic vs Diabatic Dynamics Review: Nazarewicz, NPA ) 489c The problem has been discussed since the paper by Hill and Wheeler 1953) The pairing interaction plays a key role for configuration changes at level crossings. ε ε ε β ε β β Specialization energy β Spontaneous fission life time is much larger for odd nuclei.

24 Summary Adiabatic Self-consistent Collective Coordinate ASCC) method to derive a collective submanifold and to determine the collective mass & potential. Difference between ASCC & GCM Zero-point energy and ) Collective mass RPA and GOA ) Hamiltonian H-λQ and H ) Applications to multi-o4) model ASCC vs Exact ASCC vs GCM vs GOA vs Exact Consistent calculation is desired for comparison between these GCM & ASCC ATDHFB).

25 Questions Can we justify the use of GOA? in GCM/ Perhaps, the small-amplitude limit is easy. Small-amplitude HF+GCM/GOA with complex generator coordinates of all ph-degrees of freedom reduces to RPA Jancovici & Schiff, 1964)) Then, how about the HFB+GCM? Does it reduce to QRPA? H λ ˆQ Where does the effect of come from? λ ˆN

Microscopic description of shape coexistence and shape transition

Microscopic description of shape coexistence and shape transition Nobuo Hinohara (RIKEN Nishina Center) Koichi Sato (RIKEN), Kenichi Yoshida (Niigata Univ.) Takashi Nakatsukasa (RIKEN), Masayuki Matsuo