excited states and tp propagator for fermionswimd/q540-17-13.pdfqmpt 540 excited states and tp...
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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