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 E N 0 E N +1 n - E N 0 E N 0 - E N -1 m E N k
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Excited states and tp propagator for fermionswimd/Q540-17-13.pdfQMPT 540 Excited states and tp propagator for fermions • So far sp propagator gave access to – Ground-state energy
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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