Excited states and tp propagator for fermions
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1 Excited states and tp propagator for fermions So far sp propagator gave access to E 0 Ground-state energy and all expectation values of 1-body operators E +1 Energies in +1 relative to ground state n E0 with corresponding addition amplitudes E 0 E 1 Energies in -1 relative to ground state m with corresponding removal amplitudes (& spectroscopic factors) Time to consider energies of excited states amplitudes identifying collective behavior and transition Additional information that may lead to improved description of self-energy and corresponding sp propagator Tp propagator contains information about excited states E k Instead of 4 times, only two-time version required
2 Tp propagator for excited states Relevant limit of four-time tp propagator G ph (, 1 ;, 1 ; t t ) lim t t + lim G II ( t, t, t, t ) t t + = i 0 T[a H (t)a H (t)a H (t )a H (t )] 0 involves time-reversed states and hole operators required for proper coupling to good total angular momentum Time-reversal operator generates time-reversed state T = i 0 T[b H (t)a H (t)a H (t )b H (t )] Form depends on chosen basis: particle with spin and momentum T p,m s p,m s =( 1) 1 2 +m s p, m s Contains product of unitary operator and complex conjugation 0 This unitary operator R y ( ): parity x rotation plus phase choice
3 Time-reversal Time-reversed states have bar over sp quantum numbers For fermions T = = Introduce operators that add or remove holes Making a hole b = a In sp basis of example b p,m s a p,ms =( 1) 1 2 +m s a p, ms Hole with momentum requires removal of particle with p,m s p, m s Consider coordinate space basis or angular momentum We know What about T p op T 1 T r op T 1 T r op p op T 1 T s op T 1 T j op T 1
4 Particle-hole propagator Two times require only one energy variable for FT Consider G ph (, 1 ;, 1 ; t t )= i 0 a a 0 0 a a 0 where ground-state contribution has been isolated since it is already contained in sp propagator Introduce polarization propagator to focus on excited states i (t t )e i (E 0 E n )(t t ) 0 a a n n a a n =0 + n =0 (t t)e i (E 0 E n )(t t) 0 a a n n a a 0 (, 1 ;, 1 ; t t )=G ph (, 1 ;, 1 ; t t )+ i 0 a a 0 0 a a 0 0
5 Familiar step Boson-like propagator FT polarization propagator Denominator: excitation energies for particles umerator: one-body transition amplitudes For example n Ô Ô = 0 2 = (, 1 ;, 1 ; E) = = O a a n =0 n =0 O O d(t t ) e i E(t t ) (, 1 ;, 1 ; t t ) 0 a a n n a a 0 E (E n E 0 )+i 0 a a n n a a 0 E +(E n E 0 ) i generates transition probability n a a 0 n a a 0 Most relevant for studying excited states ote information in second term
6 oninteracting polarization propagator Evaluate noninteracting limit replacing (0) (, 1 ;, 1 ; t t )=G (0) ph (, 1 ;, 1 ; t Employ sp basis in which H 0 is diagonal Excited states also eigenstates of Ĥ 0 (Ch. 3) Then oting Kramer s degeneracy for fermions Collect Ĥ 0 a a Ĥ by Ĥ0 and 0 by 0 t )+ i 0 a a 0 0 a a 0 0 = ( F ) (F ) + E 0 a a (0) (, 1 ;, 1 ; t t )= i (t t ) ( F ) (F ),, e i( )(t t )/ + (t t) (F ) ( F ),, e i( )(t t)/ Interpretation: first term represents independent propagation of a particle with from t to t and a hole from t to t so (ph) Second term reverses t and t and sp quantum numbers so (hp) = 0
7 Part a) forward propagation Part b) backward propagation Graphics Represented by noninteracting sp propagators Appropriate on account of being able to write (check) FT Or: (0) (, 1 ;, 1 ; t t )= i G (0) (, ; t t )G (0) (, ; t t) (0) (, 1 ;, 1 ; E) =,, ( F ) (F ) E ( )+i (0) (, 1 ;, 1 ; E) = check using contour integration Poles: excited states of noninteracting system Symmetric around 0 Feynman diagram for both terms (F ) ( F ) E +( ) i de 2 i G(0) (, ; E + E )G (0) (, ; E ),, (0) (, 1 ; E)
8 Random phase approximation (RPA) Higher-order terms can be evaluated using Wick s theorem as discussed in Ch. 9 (more general for 4 times) Either terms that dress the noninteracting sp propagators or terms that represent interaction between initial and final ph state Illustrated in first order in time formulation (1) (, 1 ;, 1 ; t t i 2 1 )= dt 1 V µ 4 µ 0 T a (t 1 )a (t 1)a (t 1 )a µ (t 1 )a (t)a (t)a (t )a (t ) 0 In HF basis dressing corrections vanish Keep interaction term (1) (, 1 ;, 1 ; t t ) (i ) 2 dt 1 µ V µ G (0) (, ; t t 1 )G (0) (µ, ; t 1 t)g (0) (, ; t 1 t )G (0) (, ; t t 1 )
9 Illustrated by RPA Representing direct and exchange contribution Several time-orderings still possible: f-f, f-b, b-f, b-b Introduce notation 1 V ph 1 V Using this definition and FT first-order correction one finds (1) (, 1 ;, 1 ; E) = (0) (, 1 ; E) 1 V ph 1 (0) (, 1 ; E) Feynman diagrams in energy formulation ot consistent with Lehmann representation Remember DE: generate all-order summation
10 Replace RPA (0) (, 1 ; E) 1 V ph 1 (0) (, 1 ; E) = (0) (, 1 ; E) 1 V ph 1 (0) (, 1 ;, 1 ; E) Iterates ph interaction to all orders RPA for excited states: add noninteracting term (0) (, 1 ; E) 1 V ph 1 RP A (, 1 ;, 1 ; E) RP A (, 1 ;, 1 ; E) = (0) (, 1 ;, 1 ; E) + (0) (, 1 ; E) 1 V ph 1 RP A (, 1 ;, 1 ; E) or equivalently RP A (, 1 ;, 1 ; E) = (0) (, 1 ;, 1 ; E) + (0) (, 1 ;, 1 ; E) 1 V ph 1 RP A (, 1 ;, 1 ; E)
11 Graphic RPA RPA in diagrams Diagrams generated have many names: ring, bubble, or sausage Iterate direct ph interaction Bubble represents both FW and BW propagation Only FW: 1 ph pair at any time Selection of many-p-many-h RPA...
12 RPA in finite system (schematic model) Lehmann representation (, 1 ;, 1 ; E) = also for RPA otation ow considered at the RPA level Lehmann RPA n=0 0 a a n n a a 0 X n n a a n E n E 0 E (E n E 0 )+i n=0 RP A (, 1 ;, 1 ; E) = 0 a a n n a a 0 E +(E n E 0 ) i 0 Y n n a a 0 = X n Focus on discrete, (bound) low-lying excited states RP Standard procedure lim (E n) A = (0) + (0) RP A V ph E n Poles of noninteracting propagator different from RPA one so n =0 X n (X n ) E n + i n =0 (Y n ) Y n E + n i X n = (0) (, 1 ; n) 1 V ph 1 X n
13 RPA Eigenvalue equation X n = (0) (, 1 ; n) Summation over ph and hp Also ph and hp for external quantum numbers For For ote minus sign and nonhermiticity (allows complex eigenvalues) ormalization from usual procedure including noninteracting propagator (for n > 0 ) X n 2 X n 2 =1 >F > <F < 1 V ph 1 X n >F > { n ( )}X n = 1 V ph 1 X n <F < { n +( )}X n = 1 V ph 1 X n
14 Simplified model Assume separability of interaction 1 V ph 1 = Q Q with a coupling constant and Q = Q Substitute { n ( )}X n = Q Q X n >F > { n +( )}X n = Q Immediately X n Q = n ( ) with constant Insert again X n = = 1 = >F > Q n ( ) Q X n Q 2 n ( ) Q X n <F < <F < >F > <F < Q 2 n ( )
15 EV equation Analysis only unknown quantities: excitation energies (eigenvalues) Truncated ph space: dimension D so 2D eigenvalues Plot right side 1 = >F > Solutions intersection with So sign is important ote reflection symmetry ote asymptotes at ± ph ote D-1 trapped solutions 1 is not irrespective of sign complex values: instability Q 2 n ( ) 1 <F < Q 2 n ( )
16 Analysis o instability when BW part is neglected: Tamm-Dancoff approximation (TDA) But: excitation energy can become negative (unphysical) RPA eigenvector for collective state explicit for degenerate case: all ph energies the same Define C = EV problem Remaining solution positive root still D-1 trapped Moves up or down depending on sign of coupling constant Amplitudes >F > Q 2 = <F < ph Q = C ph + ph X c ph = c Q ph ph c = 2 C ph + 2 ph ; X c hp = Q hp c + ph 1/2
17 more analysis Constant from normalization given by (check) Elaborate on collectivity Consider transition probability for Amplitude to excited state n ˆQ 0 = vanishes for noncollective states because (check) For collective state All strength combines into one state Can become very large for very attractive interaction (collective state approaching zero excitation energy) ote: energy-weighted strength conserved c ˆQ ˆQ = Q (X n ) 0 2 = ph = Electric quadrupole transition > 2 + in even-even nuclei c C Q a a >F > C ph c Q X n =0
18 Electric quadrupole transitions in nuclei Ratio to sp estimate Figure from BM Vol II
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