Quantum Theory of Many-Particle Systems, Phys Dynamical self-energy in finite system

Transcription

1 Quantum Theory of Many-Particle Systems, Phys. 540 Self-energy in infinite systems Diagram rules Electron gas various forms of GW approximation Ladder diagrams and SRC in nuclear matter Nuclear saturation problem Other questions about last class and assignments? Comments? Dynamical self-energy in finite system Second-order already quite good for atoms (see later) For nuclei much more is required SRC as in nuclear matter Also long-range correlations (LRC) as will be discussed shortly Green s function Monte Carlo calculations available for nuclei up to A=12 energies of low-lying states and some transitions includes adjusted three-body force Other methods available for light nuclei No-core shell model Coupled-cluster method (also approximately for heavier nuclei) Hyperspherical expansion Doesn t clarify the physics and there is a lot...

2 Nuclei Self-energy should include ladder diagrams fully self-consistent implementation not yet available hybrid approaches available tame NN interaction -> G-matrix use as effective interaction in limited configuration space thereby incorporating import LRC study of SRC possible recent development of Faddeev-RPA technique (Carlo Barbieri) also applied to atoms Dynamic self-energy for nuclei Sum ladder diagrams schematically according to αβ G(E) γδ = αβ V γδ + 1 θ(σ M)θ(τ M) αβ V στ στ G(E) γδ 2 E ε στ σ ε τ with M identifying limit of model space Energy dependence relevant for calculating depletion Fermi sea Neglect for present purpose: study of LRC Employ nuclear-matter G-matrix -> operator -> finite nuclei second-order as in Ch. 11 including collective features from phtda phrpa pphhtda pphhrpa

3 G-matrix in second order Self-energy Σ (2) (γ, δ; E) = 1 γµ G ɛν ɛν G δµ 2 ɛµν { } θ(f ɛ)θ(f ν)θ(µ F ) θ(ɛ F )θ(ν F )θ(f µ) + E (ε ɛ + ε ν ε µ ) iη E (ε ɛ + ε ν ε µ )+iη practical calculation employed HO wave functions in a limited model space for proton removal from 48 Ca model space (all discrete) up to several major shell above the Fermi energy (both protons and neutrons) sp energies in self-energy experimental -> chosen such that major fragments have the experimental position generates hundreds of small fragments (and is not self-consistent) still: some qualitative features of the experiment are obtained Comparison with (e,e p) Height of peaks normalized to (2j+1) First peak Most of the rest Strength experiment nd order 0.73 More fragmentation and depletion needed n( d 3 ) = 0.89 with 0.80 in 2 experimental domain more fragmentation d 5 2 d 3 2 d 3 2 d 5 2 Results for 48 Ca

4 LRC effects 2 nd order overestimates total strength by 10-15% (SRC) Need for more fragmentation Several possibilities illustrated Allow any number of interactions inside model space separately for pp and hh ladder diagrams TDA (just diagonalize pp or hh) pp propagator G T pp DA (α, β; γ, δ; E) = n hh propagator G T hh DA (α, β; γ, δ; E) = m T A+2,n αβ T A+2,n γδ E (En A+2 E0 A)+iη α, β, γ, δ >F T A 2,m βα T A 2,m δγ E (E A 0 EA 2 m ) iη α, β, γ, δ <F TDA Numerator: components of eigenvectors of diagonalization in pp or hh space Denominator: corresponding eigenvalues Replace noninteracting pp or hh propagators by TDA version with Σ T pphh DA (α, β; E) = 1 2 Γ A+2,n ακ = κ<f,n + µ>f,ν>f Γ A 2,m ακ = µ<f,ν<f κ>f,m Results labeled pphhtda Γ A+2,n ακ Γ A+2,n βκ E ((En A+2 ω ((E A 0 E A 0 ) ε κ)+iη Γ A 2,m ακ Γ A 2,m βκ EA 2 m ακ G µν T A+2,n µν ακ G µν T A 2,m νµ ) ε κ ) iη

5 ph LRC Alternative: do the same in ph space Yields for three species of ph excitations Even N and Z: low-lying collective 2 + and 3 - isoscalar excitations Self-energy with Results labeled with phtda T A,n αβ Π T DA (α, β; γ, δ; E) = E (E A m 0 0 T A,n γδ Π T DA (α, β; γ, δ; E) = E (E A n 0 n E0 A)+iη α, γ >F; β, δ <F T A,m βα T A,m δγ Γ A+,n αµ = Σ T DA ph (α, β; E) = 1 2 ν>f,κ<f ακ G µν T A,n νκ µ>f,n 0 + EA m) iη Γ A+,n αµ Γ A+,n βκ E (ε µ +(En A E0 A )) + iη E (ε µ +(E0 A EA m)) iη Γ A,m αµ = ακ G µν Tκν A,m µ<f,m 0 Γ A,m αµ Γ A,m βκ ν<f,κ>f α, γ <F; β, δ >F Notes Replacing noninteracting polarization propagator with the TDA or RPA version is ambiguous Factor! reflects equivalent fermions No longer valid in higher order So either 3 rd and higher order underestimated with! or 2 nd order overestimated (without!) Faddeev-RPA solves this problem Adding pphh and ph is not possible either Requires subtracting double-counted 2 nd order term Near the corresponding poles the self-energy has the wrong slope and may yields so-called ghost solutions Require propagator with Lehmann representation in self-energy!

6 Spectral strength Results TDA more fragmentation needed RPA instabilities Protons in 48 Ca Various approximations LRC (low energy) affect deeply bound shells little but those near the Fermi energy substantially softening the Fermi surface SRC lead to a further ~10% depletion Illustrated for pphhtda results by using 0.9 factor for SRC Occupation numbers Shell Σ (2) Σ T DA ph Σ RP A ph Σ T DA pphh Σ RP A pphh 0s p p d d s f f p p g 1d 2s h 1f 2p Total

7 Treatment of SRC in finite nuclei Technically harder in finite nuclei than in nuclear matter where momentum conservation can be utilized In nuclei transformation between lab and center-of-mass system necessary and quite involved Sketch indirect approach that starts with a nuclear-matter G- matrix according to kl G SJS KLT k l = kl V SJS KLT k l + 1 dk (k ) 2 kl V SJS KLT k l 2 l Q(K, k ) E NM 2 K 2 4m 2 k 2 2m k l G SJS KLT k l Note quantum numbers of relative and center-of-mass motion and angle-averaged Pauli functions (product of step functions) Only depends on magnitude K so not on L but L is kept for future recoupling; density and energy choice not too important Development BHF with this G-matrix corrected according to diagram b) Double-counting in 2 nd order corrected in nuclear matter (real) Basis states: harmonic oscillators for holes and (orthogonalized) plane waves for particles so- Requires mixed representation involving lab states obtained from center-of-mass states with called vector brackets n 1 l 1 j 1 k 2 l 2 j 2 JT = 0 dk 1 k 2 1R n1,l 1 (k 1 ) k 1 l 1 j 1 k 2 l 2 j 2 JT k 1 l 1 j 1 k 2 l 2 j 2 JT klklλ k 1 l 1 k 2 l 2 λ Tp states in mixed representation (R -> oscillator wave function)

8 More BHF (no true self-consistency) self-energy [see a)] Σ BHF l 1 j 1 (k 1,k 1)= 1 2(2j 1 + 1) n 2 l 2 j 2 JT summation over occupied sp states in 16 O (2J + 1)(2T + 1) k 1 l 1 j 1 n 2 l 2 j 2 G JT k 1l 1 j 1 n 2 l 2 j 2 Imaginary part from b) and c) 2p1h self-energy given by Im Σ 2p1h l 1 j 1 (k 1,k 1; E) = 1 2(2j 1 + 1) n 2 l 2 j 2 using empirical hole energies (similar strategy for c) diagram) ll JST k 2 dk K 2 dk (2J + 1)(2T + 1) k 1 l 1 j 1 n 2 l 2 j 2 G JT klskl klskl G JT k 1l 1 j 1 n 2 l 2 j 2 πδ (E + ε n2l2j2 2 K 2 4m 2 k 2 ) m More Kinetic energy for particle states not realistic but sufficient for the study of SRC (not LRC) Real part corresponding to 2p1h term from Re Σ 2p1h l 1 j 1 (k 1,k 1; E) = P π Including correction for double counting we find DE solved in wave vector space Im Σ 2p1h l 1 j 1 (k 1,k 1; E ) E de E Σ = Σ BHF + Σ = Σ BHF + ( Re Σ 2p1h Σ c + Re Σ 1p2h) + i ( Im Σ 2p1h + Im Σ 1p2h) Quasiparticle states from usual EV equation (discretized -> w) N k i 2 ki 2 2m δ in + Σ BHF lj + Σ lj (E = ε lj m ) k n w(k n )zk m n lj = ε lj m zk m i lj n=1

9 More Sizable imaginary part starts below quasihole (qh) energies Discrete solutions separated from 2h1p continuum Spectroscopic factor z m α qh lj 2 = dk k 2 z m klj 2 Corresponding spectral function S qh (klj; E) = z m klj 2 δ(e ε lj m ) involves square of qh wave function Plot for p! in 16 O Normalized to S Can be used to describe (e,e p) cross section Cross section described by S = for p3/2 Calculated S = (only incorparates SRC) Shape is fine For p! is found Similar to numbers from experimental analysis with wave functions from adjusted Woods- Saxon potentials (to separation energy) Results Other calculations confirm that SRC deplete by about 10% LRC important!

10 Continuum results Use diagonal BHF propagators as starting point G (0) lj (α; E) = 1 E ε BHF αlj ± iη Iterate remaining Σ component of self-energy to get reducible self-energy according to (two-step procedure) α Σ red lj (E) β = α Σ lj (E) β + γ Propagator then given by G lj (α, β; E) =δ α,β G (0) lj Spectral function in wave vector basis is then given by S c (klj; E) = 1 ( π Im ) k α lj G lj (α, β; E) β k lj Total spectral function for removal α Σ lj (E) γ G (0) lj (α; E)+G(0) lj (α; E) α Σred α,β S lj (k; E) =S c (klj; E)+S qh (klj; E) (γ; E) γ Σred lj (E) β lj (E) β G (0) (β; E) lj Results p! spectral strength at different energies More high momenta at higher excitation energy in A-1 and no such components near the Fermi energy!

11 Confirmed by the momentum distribution after energy integration up to the Fermi energy of spectral functions Results Several curves exhibit momentum distribution when limit of integration is -100, -150, and the full limit, in addition to the quasihole contribution Latter near the Fermi energy and exhibits no high-momentum strength Other calculations with different interactions similar Where do high momenta occur? Consider time-ordered self-energy diagram Nuclear matter argument with momentum conservation Holds more or less similarly for nuclei External line k (large)! Intermediate holes < k F, say total momentum ~ 0.! Momentum conservation: intermediate particle -k! Energy intermediate state ~ " 2h - "(k)! The higher k the more negative the location of its strength! No high-momentum components near " F

12 High-momenta near " F? 208 Pb(e,e p) I. Bobeldijk et al., Phys. Rev. Lett. 73, 2684 (1994) NO! Green s function methods Relevance of continuum part of spectral strength Partial occupation numbers from divided in continuum and quasihole contributions qh -> s, p continuum -> d, f -> ˆn lj = 2(2j + 1) εf de 0 dk k 2 S(klj; E) lj ˆn qh ˆn c (E< 100) ˆn c ˆn ˆn/(2(2j + 1)) s p p d d f f

13 Energy contributions Partial contributions to energy sum rule for ground state BHF BHF+2p1h Total lj ɛ t E ɛ t E ɛ t E s 1 2 qh s 1 2 c p 3 2 qh p 3 2 c p 1 2 qh p 1 2 c l > 1c E/A(MeV) r (fm) % from quasiholes (last column) 63% from continuum representing only 11% of the nucleons!!! Still far from experiment MeV/A What are the rest of the protons doing? Jlab E Phys. Rev. Lett. 93, (2004) D. Rohe et al. 12C Location of high-momentum components Integrated strength agrees with theoretical prediction Phys. Rev. C49, R17 (1994)! 0.6 protons for 12 C Green s function methods

14 M. van Batenburg (thesis, 2001) & L. Lapikás from 208 Pb (e,e p) 207 Tl Occupation of deeply-bound proton levels from EXPERIMENT SRC LRC Up to 100 MeV missing energy and 270 MeV/c missing momentum Covers the whole mean-field domain for the FIRST time!! Confirmation of theory Depletion of the nuclear Fermi sea Location of single-particle strength in nuclei SRC & tensor Consistent with recent SRC JLab and Brookhaven expts. PRL96,082591(2006) PRL97,162504(2006) PRL99,072501(2007) SRC & tensor Green s function methods

15 We now essentially know what all the protons are doing in the ground state of a closed-shell nucleus!!! Unique for a correlated many-body system Information available for electrons in atoms (Hartree-Fock OK) Not for electrons in solids Not for atoms in quantum liquids Not for quarks in nucleons! Demonstrates how interesting the nucleus is as a quantum many-body system! How to do better? Faddeev summation developed in St. Louis (Thesis Barbieri)

16 Containing TDA RPA Describing excited states Essential for the electron gas! Plasmon But GW yields good total energy but not good self-energy SCGF

17 Application to Neon atom * Main peaks * Consistent with results from Heidelberg group Phys. Rev. A76, (2007) (also results with optimized small basis sets) Table: discretized continuum (complete space) Some results Distribution of s fragments in neon atom

18 Analysis Interference! Conclusion Self-energy of small atoms accurate with F-RPA Contains the relevant ingredients for the electron gas including a (possibly) correct self-energy! study background functionals for QP-DFT

19 Famous plot HF F-RPA # (2) Presentations Assignment