QMPT 540

Excited states and tp propagator for fermions• So far sp propagator gave access to

– Ground-state energy and all expectation values of 1-body operators

– Energies in N+1 relative to ground state with corresponding addition amplitudes

– Energies in N-1 relative to ground state with corresponding removal amplitudes (& spectroscopic factors)

• Time to consider energies of excited states and transition amplitudes identifying collective behavior

• Additional information that may lead to improved description of self-energy and corresponding sp propagator

• Tp propagator contains information about excited states

• Instead of 4 times, only two-time version required

EN0

EN+1n � EN

0

EN0 � EN�1

m

ENk

QMPT 540

Tp propagator for excited states• Relevant limit of four-time tp propagator

• involves time-reversed states and “hole” operators required for proper coupling to good total angular momentum

• Time-reversal operator generates “time-reversed” state

• Form depends on chosen basis: particle with spin and momentum

• Contains product of unitary operator and complex conjugation

• This unitary operator : parity x rotation plus phase choice

Gph(�,⇥�1; ⇤, ⌅�1; t� t⇤) ⇥ limt�⇥t+

limt⇥⇥t�+

GII(�t, ⌅t⇤,⇥t⇥ , ⇤t⇤)

= � i

� ⌅�N0 | T [a†

⇥H(t)a�H (t)a†⇤H

(t⇤)a⌅H(t⇤)] |�N

0 ⇧

⇥ � i

� ⌅�N0 | T [b⇥H (t)a�H (t)a†⇤H

(t⇤)b†⌅H(t⇤)] |�N

0 ⇧

T |�⇥ = |�⇥

T |p, ms⌅ ⇥ |p, ms⌅ = (�1)12+ms |�p,�ms⌅

�Ry(�)

QMPT 540

Time-reversal• Time-reversed states have bar over sp quantum numbers • For fermions

• Introduce operators that add or remove “holes”

• Making a hole • In sp basis of example

• Hole with momentum requires removal of particle with

• Consider coordinate space basis or angular momentum

• We know

• What about

T |�⇤ = |�⇤ = � |�⇤

b†� = a�

b†p,ms⇥ ap,ms = (�1)

12+msa�p,�ms

p, ms �p,�ms

T popT �1

T ropT �1 T rop � popT �1

T sopT �1 T jopT �1

QMPT 540

Particle-hole propagator• Two times require only one energy variable for FT • Consider

• where ground-state contribution has been isolated since it is already contained in sp propagator

• Introduce polarization propagator

• to focus on excited states

Gph(�,⇥�1; ⇤, ⌅�1; t � t⇥) = � i

� ⇥�N0 | a†

⇥a� |�N

0 ⇤ ⇥�N0 | a†⇤a⌅ |�N

0 ⇤

� i

�

�⇧

⇤⌥

n ⇤=0

⇧(t � t⇥)ei� (EN

0 �ENn )(t�t�) ⇥�N

0 | a†⇥a� |�N

n ⇤ ⇥�Nn | a†⇤a⌅ |�N

0 ⇤

+⌥

n ⇤=0

⇧(t⇥ � t)ei� (EN

0 �ENn )(t��t) ⇥�N

0 | a†⇤a⌅ |�Nn ⇤ ⇥�N

n | a†⇥a� |�N

0 ⇤

⇥⌃

⌅

�(�,⇥�1; ⇤, ⌅�1; t � t⇥) = Gph(�,⇥�1; ⇤, ⌅�1; t � t⇥) +i

� ⇥⇥N0 | a†

⇥a� |⇥N

0 ⇤ ⇥⇥N0 | a†⇤a⌅ |⇥N

0 ⇤

QMPT 540

FT polarization propagator• Familiar step

• Boson-like propagator

• Denominator: excitation energies for N particles

• Numerator: one-body transition amplitudes • For example generates transition probability

• Most relevant for studying excited states • Note information in second term

�(�,⇥�1; ⇤, ⌅�1;E) =⇥ ⇤

�⇤d(t � t⇥) e

i� E(t�t�)�(�,⇥�1; ⇤, ⌅�1; t � t⇥)

=�

n ⌅=0

⇥⇥N0 | a†

⇥a� |⇥N

n ⇤ ⇥⇥Nn | a†⇤a⌅ |⇥N

0 ⇤E � (EN

n � EN0 ) + i⇧

��

n ⌅=0

⇥⇥N0 | a†⇤a⌅ |⇥N

n ⇤ ⇥⇥Nn | a†

⇥a� |⇥N

0 ⇤E + (EN

n � EN0 ) � i⇧

O =�

�⇥

��| O |⇥⇥ a†�a⇥

�����Nn | O |�N

0 ⇥���2

=⇥

�⇥

⇥

⇤⌅

�⇤| O |⌅⇥ ��| O |⇥⇥� ��Nn | a†⇤a⌅ |�N

0 ⇥ ��Nn | a†�a⇥ |�N

0 ⇥�

QMPT 540

Noninteracting polarization propagator• Evaluate noninteracting limit replacing

• Employ sp basis in which is diagonal

• Excited states also eigenstates of (Ch. 3) • Then

• Noting Kramer’s degeneracy for fermions

• Collect

• 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)

H by H0 and |⇥N0 � by |�N

0 �

�(0)(�,⇥�1; ⇤, ⌅�1; t � t⇥) = G(0)ph (�,⇥�1; ⇤, ⌅�1; t � t⇥) +

i

� ⇥⇥N0 | a†

⇥a� |⇥N

0 ⇤ ⇥⇥N0 | a†⇤a⌅ |⇥N

0 ⇤

H0

H0

H0 a†�a⇥ |�N0 ⇥ = ⇤(� � F )⇤(F � ⇥)

�⌅� � ⌅⇥ + E�N

0

⇥a†�a⇥ |�N

0 ⇥

�� = ��

�(0)(�,⇥�1; ⇤, ⌅�1; t� t⇥) =

� i

�

�⇧(t� t⇥)⇧(�� F )⇧(F � ⇥)⌅�,⇤⌅⇥,⌅e

�i(⇧��⇧⇥)(t�t�)/�

+ ⇧(t⇥ � t)⇧(F � �)⇧(⇥ � F )⌅�,⇤⌅⇥,⌅e�i(⇧⇥�⇧�)(t��t)/�

⇥

� �

• Part a) “forward propagation” • Part b) “backward propagation”

• Represented by noninteracting sp propagators

• Appropriate on account of being able to write (check)

• FT

• Or:

• check using contour integration

• Poles: excited states of noninteracting system

• Symmetric around 0

• Feynman diagram for both terms

QMPT 540

Graphics

�(0)(�,⇥�1; ⇤, ⌅�1; t� t⇥) = �i�G(0)(�, ⇤; t� t⇥)G(0)(⌅, ⇥; t⇥ � t)

�(0)(�,⇥�1; ⇤, ⌅�1;E) =�

dE⇥

2⇧iG(0)(�, ⇤;E + E⇥)G(0)(⌅, ⇥;E⇥) � ⌅�,⇤⌅⇥,⌅�(0)(�,⇥�1;E)

⇧(0)(↵,��1; �, ��1;E) = �↵,���,�

⇢✓(↵� F )✓(F � �)

E � ("↵ � "�) + i⌘� ✓(F � ↵)✓(� � F )

E + ("� � "↵)� i⌘

�

QMPT 540

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

• In HF basis dressing corrections vanish

• Keep interaction term

�(1)(�,⇥�1; ⇤, ⌅�1; t� t⇥) =��i

�

⇥2⌅ ⇤

�⇤dt1

14

⇤

⇧⌃µ�

⌅⇧⌃|V |µ�⇧

⇥ ⌅⇥N0 | T

⇧a†⇧(t1)a†⌃(t1)a�(t1)aµ(t1)a†⇥(t)a�(t)a†⇤(t⇥)a⌅(t

⇥)⌃|⇥N

0 ⇧

�(1)(�,⇥�1; ⇤, ⌅�1; t� t⇥) ⇤ (i�)2⇥ ⇤

�⇤dt1

�

�⇥µ⌅

⌅⇧⌃| V |µ�⇧

⇥ G(0)(�,⇧; t� t1)G(0)(µ,⇥; t1 � t)G(0)(�, ⇤; t1 � t⇥)G(0)(⌅,⌃; t⇥ � t1)

QMPT 540

RPA• Illustrated by

• Representing direct and exchange contribution • Several time-orderings still possible: f-f, f-b, b-f, b-b

• Introduce notation

• Using this definition and FT first-order correction one finds

• Feynman diagrams in energy formulation

• Not consistent with Lehmann representation • Remember DE: generate all-order summation

⇥�⇥�1| Vph |⇤⌅�1⇤ � ⇥�⌅| V |⇥⇤⇤

�(1)(�,⇥�1; ⇤, ⌅�1;E) = �(0)(�,⇥�1;E) ��⇥�1| Vph |⇤⌅�1⇥�(0)(⇤, ⌅�1;E)

�(1)(�,⇥�1; ⇤, ⌅�1; t� t⇥) ⇤ (i�)2⇥ ⇤

�⇤dt1

�

�⇥µ⌅

⌅⇧⌃| V |µ�⇧

⇥ G(0)(�,⇧; t� t1)G(0)(µ,⇥; t1 � t)G(0)(�, ⇤; t1 � t⇥)G(0)(⌅,⌃; t⇥ � t1)

QMPT 540

RPA• Replace

• Iterates ph interaction to all orders

• RPA for excited states: add noninteracting term

• or equivalently

�(0)(�,⇥�1;E) ⇥�⇥�1| Vph |⇤⌅�1⇤�(0)(⇤, ⌅�1;E) =

�(0)(�,⇥�1;E)�

�⇥

⇥�⇥�1| Vph |⇧⌃�1⇤�(0)(⇧, ⌃�1; ⇤, ⌅�1;E)

� �(0)(�,⇥�1;E)�

�⇥

⇥�⇥�1| Vph |⇧⌃�1⇤�RPA(⇧, ⌃�1; ⇤, ⌅�1;E)

�RPA(�,⇥�1; ⇤, ⌅�1;E) = �(0)(�,⇥�1; ⇤, ⌅�1;E)

+ �(0)(�,⇥�1;E)�

�⇥

��⇥�1| Vph |⇧⌃�1⇥�RPA(⇧, ⌃�1; ⇤, ⌅�1;E)

�RPA(�,⇥�1; ⇤, ⌅�1;E) = �(0)(�,⇥�1; ⇤, ⌅�1;E)

+�

�⇤⇥⌅

�(0)(�,⇥�1; ⌃, ��1;E) �⌃��1| Vph |⇧⌥�1⇥�RPA(⇧, ⌥�1; ⇤, ⌅�1;E)

QMPT 540

RPA in diagrams• Graphic RPA

• 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...

QMPT 540

RPA in finite system (schematic model)• Lehmann representation

• also for RPA

• Notation

• Now considered at the RPA level

• Lehmann RPA

• Focus on discrete, (bound) low-lying excited states

• Standard procedure

• Poles of noninteracting propagator different from RPA one so

�(�,⇥�1; ⇤, ⌅�1;E) =�

n ⇥=0

⇥⇥N0 | a†

⇥a� |⇥N

n ⇤ ⇥⇥Nn | a†⇤a⌅ |⇥N

0 ⇤E � (EN

n � EN0 ) + i⇧

��

n ⇥=0

⇥⇥N0 | a†⇤a⌅ |⇥N

n ⇤ ⇥⇥Nn | a†

⇥a� |⇥N

0 ⇤E + (EN

n � EN0 ) � i⇧

Xn�⇥ � ⇤�N

n | a†�a⇥ |�N0 ⌅� Yn

�⇥ ⇥ ⇧�Nn | a†

⇥a� |�N

0 ⌃�

= �Xn⇥�

��n ⇥ EN

n � EN0

�RPA(�,⇥�1; ⇤, ⌅�1;E) =�

n ⇤=0

Xn�⇥(Xn

⇤⌅)⇥

E � ⌃⇧n + i⇧

��

n ⇤=0

(Yn�⇥)⇥Yn

⇤⌅

E + ⌃⇧n � i⇧

limE�⇥�

n

(E � ��n)

��RPA = �(0) + �(0) Vph �RPA

⇥

Xn�⇥ = �(0)(�,⇥�1; ⇧⇧

n)�

⇤⌅

⇥�⇥�1|Vph |⇤⌅�1⇤Xn⇤⌅

RPA

QMPT 540

• Eigenvalue equation • Summation over ph and hp

• Also ph and hp for external quantum numbers

• For

• For

• Note minus sign and nonhermiticity (allows complex eigenvalues)

• Normalization from usual procedure including noninteracting propagator (for )

Xn�⇥ = �(0)(�,⇥�1; ⇧⇧

n)�

⇤⌅

⇥�⇥�1|Vph |⇤⌅�1⇤Xn⇤⌅

� > F > ⇥

{⇧⇧n � (⇧� � ⇧⇥)}Xn

�⇥ =�

⇤⌅

⇧�⇥�1|Vph |⇤⌅�1⌃Xn⇤⌅

� < F < ⇥

��n > 0 �

�>F>⇥

|Xn�⇥ |2 �

�

�<F<⇥

|Xn�⇥ |2 = 1

{⇧⇧n + (⇧⇥ � ⇧�)}Xn

�⇥ = ��

⇤⌅

⇧�⇥�1|Vph |⇤⌅�1⌃Xn⇤⌅

QMPT 540

Simplified model• Assume separability of interaction • with a coupling constant and

• Substitute

• Immediately

• with constant

• Insert again

��⇥�1| Vph |⇤⌅�1⇥ = ⇧Q�⇥Q⇥⇤⌅

|Q�⇥ | = |Q⇥�|

{⇥⇧n � (⇥� � ⇥⇥)}Xn

�⇥ = �Q�⇥

�

⇤⌅

Q�⇤⌅Xn

⇤⌅ � > F > ⇥

{⇥⇧n + (⇥⇥ � ⇥�)}Xn

�⇥ = ��Q�⇥

�

⇤⌅

Q�⇤⌅Xn

⇤⌅ � < F < ⇥

Xn�⇥ = N Q�⇥

�⇤n � (�� � �⇥)

� > F > ⇥

Xn�⇥ = �N Q�⇥

�⇤n � (�� � �⇥)

� < F < ⇥

N = ��

�⇥

Q��⇥Xn

�⇥

1�

=�

�>F>⇥

|Q�⇥|2

⇥⇤n � (⇥� � ⇥⇥)

��

�<F<⇥

|Q�⇥|2

⇥⇤n � (⇥� � ⇥⇥)

QMPT 540

Analysis• EV equation

• only unknown quantities: excitation energies (eigenvalues)

• Truncated ph space: dimension D so 2D eigenvalues • Plot right side

• Solutions intersection with

• So sign is important • Note reflection symmetry

• Note asymptotes at

• Note D-1 trapped solutions

• 1 is not irrespective of sign • complex values: instability

1�

=�

�>F>⇥

|Q�⇥|2

⇥⇤n � (⇥� � ⇥⇥)

��

�<F<⇥

|Q�⇥|2

⇥⇤n � (⇥� � ⇥⇥)

1�

±�ph

QMPT 540

Analysis• No 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

• EV problem still D-1 trapped

• Remaining solution positive root

• Moves up or down depending on sign of coupling constant • Amplitudes

C =�

�>F>⇥

|Q�⇥|2 =�

�<F<⇥

|Q�⇥|2�ph

1�

= C�

1⇥� � ⇥ph

� 1⇥� + ⇥ph

⇥

⇥�c =

�2�C⇥ph + ⇥2

ph

⇥1/2

X cph = N Qph

��c � �ph

; X chp = �N Qhp

��c + �ph

QMPT 540

more analysis• Constant from normalization given by (check) • Elaborate on collectivity

• Consider transition probability for

• Amplitude to excited state

• 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)

• Note: energy-weighted strength “conserved”

• Electric quadrupole transition 0+ --> 2+ in even-even nuclei

|N | = �

�C⇥ph

⇥�c

Q =�

�⇥

Q�⇥a†�a⇥

⇥�Nn | Q |�N

0 ⇤ =�

�⇥

Q�⇥(Xn�⇥)�

�

�>F>⇥

Q��⇥Xn

�⇥ = 0���⇥�N

c | Q |�N0 ⇤

���2

=�ph

��c

C

QMPT 540

Electric quadrupole transitions in nuclei• Ratio to sp estimate • Figure from BM Vol II

QMPT 540

Excited states in atoms• Excited states in atoms do not exhibit strong collective features

and can be described as rather pure ph excitations • One important “collective” feature does occur in that the mean

field is changed for particles above the Fermi energy

• Note that HF sp energies for neutral atoms are used in RPA

• As discussed in the HF chapter, this potential is too short-ranged and does not support bound particle states

• The HF ph spectrum is therefore continuous and there is no Rydberg spectrum of bound excited states

• Cured in RPA (or TDA)!

�p,p�⇥p = �p| T |p�⇥ +�

h��

�ph��| V |p�h��⇥

QMPT 540

RPA for atoms• Consider contribution of time-forward diagrams in A-matrix • In particular, the diagonal sub-block with the same hole orbital

• This matrix can be diagonalized with a unitary transformation among unoccupied orbitals

• and therefore

• generating new sp energies for the unoccupied orbitals

• Eigenstates of modified mean field by omitting hole state under consideration

Aph,p�h = �p,p�(⇥p � ⇥h) � ⇥ph| V |p�h⇤

= ��p,p�⇥h + ⇥p| T |p�⇤ +�

h�� ⇥=h

⇥ph��| V |p�h��⇤

|p(h)� =�

p�

U (h)p,p� |p��

�p(h)|T |p�(h)⇥ +�

h�� ⇥=h

�p(h)h��|V |p�(h)h��⇥ = �p,p�⇥(h)p

QMPT 540

RPA for atoms• For nuclei this effect is small, especially for heavier nuclei • For atoms critical since the “ ” potential is ion-like and thus

decays like 1/r supporting a Rydberg series of unoccupied states

• In this new basis

• omitting the interaction term in the A-matrix with

• Example: excited states in neon involving the promotion of the 2p orbit to an unoccupied state

V N�1

Aph,p�h� = �p,p��h,h�(⇥(h)p � ⇥h) + (1 � �h,h�)⇥p(h)h�1|Vph|p⇥(h

�)h⇥�1⇤

Bph,p�h� = ⇥p(h)p⇥(h�)|V |h h⇥⇤

h = h�

ELS = �(h)p � �h + ⇥p(h)h�1LS|Vph|p(h)h�1LS⇤

��

L�S�

(2L⇥ + 1)(2S⇥ + 1)4(2⌃h + 1)(2⌃p + 1)

⇥p(h)h�1L⇥S⇥|Vph|p(h)h�1L⇥S⇥⇤QMPT 540

RPA for neon• Left column: experiment • Next: modified mf ph

spectrum with maintaining spherical symmetry: 2*3 -> 2*3*⅚

already bound 3s, 3p, 4s, and 4p levels

• Next: shift spectrum by

to get exp. hole energy

• Next: lift degeneracy by calculating interaction energy in each LS configuration

�h = �0.850

� = .057

QMPT 540

RPA in atoms• One illustration with transitions to the continuum • Photo absorption spectrum of xenon

• HF for the width of “giant resonance” peak too wide

• RPA (solid) • compared with data