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Chalmers University of Technology
Many Body Solid State Physics
2007
Mattuck chapter 5 - 7
Chalmers University of Technology
Contents
• Quantum Vacuum: how to solve the
equations of nothing
• Birds eye of Diagrams: start of the
elementary part of the book (page 118->)
• Learning how to count: occupation number
formalism
• Any questions ?!?
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Quantum vacuum
• Meaning of the vacuum of amplitude
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Vacuum amplitude
E0  W0  Φ0|H 1|Φ0   ...
Fermi vacuum   0 | 11..111000...
R(t) = probability (amplitude) that if
the system at t=0 is in the
Fermi vacuum, then at t = t the
system is in the Fermi vacuum
= “no particle propagator”
R (t )
| 0   | 0 
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Vacuum amplitude
R(t )  0 | U (t ) |  0  eiW0t
U(t) = time development operator
d
E0  W0  lim i ln R(t )
t  (1i ) dt
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Pinball vacuum amplitude
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Pinball vacuum amplitude
G
+
P=
O
+
O
O
+
O
+…
G
L
G
+ L
O
+…
O
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Quantum one-particle vacuum
amplitude
2
p
H 0
 U (r )
2m
“Vacuum polarisation” or “vacuum fluctuation”
t
Zeroth
First
Second
Third
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Quantum one-particle vacuum
amplitude
=-
“Nevertheless it is important to retain such diagrams which violates
conservation of particle number to prove the linked cluster theorem.”
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Quantum one-particle vacuum
amplitude
Topological equivalence
t3
t3
t3
  
t3
t2
t2
t2
t1
t1
t1
t2
t1
t
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Quantum one-particle vacuum
amplitude
R  1 +
+
+
+
+…
+
+
+…
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Quantum one-particle vacuum
amplitude
Linked cluster theorem
ln R(t )   All linked diagrams
Which can be shown via entities like
+
+
=
x
These gives us the possibility to
get the ground state energy even
when the perturbation in strong.
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The many body case
R=1+
+
+ … = all diagrams starting and
beginning in the ground state
Again E0 is only sum over linked diagrams
We can get E0 in some approximation, eg. Hartree-Fock:
E0 = W0 +
+
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Bird’s eye view of MBP
Field theoretic ingredient
Significance in MB theory
Occupation number formalism
Express arbitrary state of MB
system
Primitive operators from which all
MB operators can be built
Quasi particle energies, momentum
distribution and more
Ground state energy
Collective excitations, non
equilibrium properties
Equilibrium thermodynamic
properties
Temperature dependent properties
Creation and destruction operators
Single particle propagator
Vacuum amplitude
Two-particle propagator
Finite temperature vacuum amplitude
Finite temperature propagator
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Second quantization (again)
• A way to write the wave function in a
compact way (no Slater determinant crap)
• A way to treat the particle type
automatically (fermions and bosons)
• Can refer to any basis (momentum, real…)
• A way to vary particle number
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Second quantization (again)
Extended
Hilbert
| 000...
space =
| 001...

| 010...
| 100...
No particle One particle
| 011...

| 101...
 ...
| 110...
Two particles …
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Chalmers University of Technology