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Projects for PHYS 5520
Ming Gong
Email: skylark.gong@gmail.com
TA: Deng Naijing
njdeng@phy.cuhk.edu.hk
Chap. 3 Discuss the finite T Green’s function
Results at zero temperature
S-matrix

Gellmann-Low theorem
Feynman diagrams and Wick contraction.
What can I do if I do not like eqs?
 Most of the students for experiments do not like Eqs.
 It doesn’t mean that you can not understand theoretical physics.
Punchline
1.
2.
3.
4.
5.
At Zero T, the low-order self-energies can be well reproduced by
perturbation theory (Lippmann-Schwinger Eq.)
The high-order terms can not be accessed by perturbation theory, but can
be calculated using numerical methods.
The perturbation theory is just an approximation, not a systematical/selfconsistent theory. The many-body physics is a self-consistent theory.
The perturbation results maybe divergent, thus need renormalization
(not discussed here).
The perturbation theory can not take the temperature effect into account
( finite temperature Green’s function)
How to study the finite T Green’s function?
Please always remember the basic methods used in zero T
Green’s functions
 The same methods should be used at finite Temperature (1-1
correspondence) though small modifications are required.

I plan to finish this chapter in one week (at most two weeks),
which is very fast.
 We need time to discuss the ideas in many-body physics.
Many-body physics is not a theory about complex math, but
instead, it is a theory about ideas.

Important results in T=0 Green’s function
 Interaction picture
 Ground state wave function (at T = 0) and S-matrix, Gell-
Mann-Low theorem.
 Wick’s contraction  The operator arrays can be decoupled
by 2-body contractions. The final results equal ALL the
possible pairings/contractions between the operators.
 Linked cluster theorem and Feynman diagrams. The Green’s
function is determined by ONLY the connected diagrams.
 Dyson Equation enables to include all the possible diagrams to
infinity orders.
Finite Temperature T

We consider the partition function Z, instead of the ground
state.

Not only the ground state, but also the excited states have
contributions to the final results (any physical observations).
 Good news because the trace is basis-independent.
We do not have the Gellmann-Low theorem at finite T.
 Most of the results at zero temperature will fail at finite T.

Physical observations
A is any operator.
 The denominator is the partition function Z.
 Work in grand canonical ensemble, thus H should be replaced by

Chemical potential and # of particle

Omega: grand canonical potential/thermodynamic potential. The
system is said to be open in the sense that the system can exchange
energy and particles (N is not conserved) with a reservoir, so that
various possible states of the system can differ in both their total
energy and total number of particles (from wiki).
Homework (Finish in 2 weeks)

Please read the textbook about statistical physics carefully
and answer the following questions.
Why we have different statistics for fermion and boson?
 Why all physical observations can be derived from the
partition function Z?

Failure of Gell-Mann-Low theorem

Two possible conditions that this theorem can fail:
degeneracy of the wave function and energy level
crossing.
Hydrogen atoms
ground state |1s> is unique.
Excited states maybe degenerate.
We may have this conclusion. In most
of the systems, the ground state can be unique
While the excited states can be degenerate.
Good news: The trace is basis independent.
Trace is basis independent.

A standard result in quantum mechanics.
We do not need to care about the
Basis OR We can choose any basis.
Important results in T=0 Green’s function
 Interaction picture
 Ground state wave function (at T = 0) and S-matrix, Gell-
Mann-Low theorem.
 Wick’s contraction  The operator arrays can be decoupled
by 2-body contractions. The final results equal ALL the
possible pairings/contractions between the operators.
 Linked cluster theorem and Feynman diagrams. The Green’s
function is determined by ONLY the connected diagrams.
 Dyson Equation enables to include all the possible diagrams to
infinity orders.
Definition of Green’s function
 At zero temperature
 Green’s function at finite temperature (following the
basic idea in Mahan’s book).
 Next, basic properties of these Green’s functions: Interaction pic.
About time-ordering operator

I did not mention the following point at zero temperature for
Fermion and boson couplings.

At finite temperature, the value of tau is chosen in the
following regime.
Time-ordering can be well-defined in this case.
Wick rotation Wick’s second contribution to physics
Direct connection between evolution operator and partition
function.
 Wick’s rotation is the most important result for finite T Green’s
function.

Rotation of Pi/2

Without the Trace operator (Tr), we have
This result is correct for any
t (real/complex numbers).
• Related to Feynman diagrams
• The Trace can be made based on basis of H0
Because the eigenvectors of H are totally unknown.
Real vs imaginary time Green Function
Interaction picture
Key Eqs in finite T Green’s function
'
Then
0, with , ' [0, ]
' [ ,0] and
'
[0, ]
Boson particle: A symmetric function
Fermion particle: An anti-symmetric function
We only care about the physics for tau in [0, \beta].
Fourier transformations of G
2mpi and (2m+1)pi are called Matsubara frequencies (1955). A
New Approach to Quantum-Statistical Mechanics, T. Matsubara,
Prog. Theor. Phys. 14 (1955) 351
Two different Fourier transformations

Type I, in the whole space
Type II, f(t) is a periodic function f(t)=f(t+T)
These waves widely happen
In engineering.

Periodic signals
No picture for Matsubara is available
1955
Two methods to calculate the last result

Standard method used in Mahan’s book.
Two methods to calculate the last result

Equation of motion method
Can be directly determined
From Wick’s rotation.
The same method can be
used in T=0 case, which
We didn’t discuss in details.
Homework (Finish in 2 weeks)

Please show that the above results can also be obtained
directly using Wick’s rotation.
 Now calculate the result in energy
space.
Connection between zero and finite T

A direct connection between Green’s function at zero
temperature and at finite temperature.
An important trick used in future because
is not directly related to physical observations.

Homework (Finish in 2 weeks). Please explain this general
relation from the Wick’s rotation viewpoint.
Electron-phonon
From standard definition
Chemical potential
of phonon = 0
 We have emphasized three different tricks to get this result.
Conclusion for the Green’s function for
H0 at finite T
In both zero and finite temperature, the Green’s functions
for non-interacting particles have a simple form.
2. The Green’s function in real time and imaginary time has
the following simple connection  Wick’s rotation.
1.
Point 1 is useful for Feynman diagrams in future and point 2 is
useful for physical observations.
Important results in T=0 Green’s function
 Interaction picture
 Ground state wave function (at T = 0) and S-matrix, Gell-
Mann-Low theorem.
 Wick’s contraction  The operator arrays can be decoupled
by 2-body contractions. The final results equal ALL the
possible pairings/contractions between the operators.
 Linked cluster theorem and Feynman diagrams. The Green’s
function is determined by ONLY the connected diagrams.
 Dyson Equation enables to include all the possible diagrams to
infinity orders.
Dyson Equation
 Definition of Green’s function
 We hope to have the following similar result
Some mathematcal background
 Definition of Green’s function
S-matrix
True for tau < 0
Homework (Finish in 2 weeks)

Please verify the following result by yourself.

I have only shown the result for tau < 0 and I have emphasized
that this result is also correct when tau > 0.

Please show this result using two different ways. (1) Wick’s
rotation; and (2) Direct exact calculation.
Where

This result is identical to that at zero temperature

The temperature effect is replaced by
NO adiabatic evolution, thus do not have the arbitrary phase
Adiabatic evolution at zero T

Ground state is unique. This phase need to be cancelled
exactly.
Q: how to evaluate the following expression?
At zero temperature, the operators are calculated over the ground
state wave functions.  Wick’s contraction.
 At finite temperature,

Not only the ground state but also the excited states have contributions
 Major difference between zero and finite T.
Wick’s theorem at T = 0

Because of the normal ordering
?
= all possible Green’s functions
YES, this theorem is correct even at finite temperature.
Discussion in Mahan’s book
Not sufficient, we need to explain the validity of Wick’s theorem at finite T
more carefully We need to understand this result from different angles.
An exact proof is also lacked in Abrikosov’s book.
Exact proof can be found in Matsubara’s original paper (1955).
We try to understand this result from different angles
1. From intuitive physical pictures.
A carefully analysis show that
is not necessary to be the
ground state. For excited states we still have this conclusion.
This argument is used in most of the textbooks.
We try to understand this result from different angles
2. Mathematical proof
http://users.physik.fu-berlin.de/~romito/qft2011/set5.pdf
Results at T = 0
Definition in
this note.
 The most crucial observation from the above result.
 The above result can be obtained from the following identity.
 Final result for Wick’s contraction at finite T.
= all possible pairs.
For Boson, we do not
have the minus sign.
= All possible pairs.
Each pair = Green’s function
= all connected diagrams because the denominator
can be exactly cancelled due to linked cluster theorem.
Dyson Equation of course can be applied.
Important results in T=0 Green’s function





Normal orderings are modified
Interaction picture
We can have Wick contraction
Ground state wave function (at
T = 0)explicitly
and S-matrix,
Gell-the
without
involving
Mann-Low theorem.
Normal-ordering
Wick’s contraction  The operator arrays can be decoupled
by 2-body contractions. The final results equal ALL the
possible pairings/contractions between the operators.
Linked cluster theorem and Feynman diagrams. The Green’s
function is determined by ONLY the connected diagrams.
Dyson Equation enables to include all the possible diagrams to
infinity orders.
Next, two major things
Linked cluster theorem and Dyson Equation (Homework, finished
in 2 weeks)
 Fourier transformation from imaginary time space to energy
space.


The following is the trick in Arbriksov’s book, page 70-71.

(1) This trick is independent of time t; and (2) Wick’s rotation.
This homework should be explained from these two angles.
Next, two major things
Linked cluster theorem and Dyson Equation.
 Fourier transformation from imaginary time space to energy
space.

Wick’s rotation

Only consider the diagram above
Convolution theorem
can still be used.
Conservation of energy/momentum
Q: How to sum over omega?
A summary for zero and finite T (please remember this table)
Zero temperature
Finite Temperature
Interaction picture in real time Interaction picture in imaginary time
(a direct wick rotation, tau = it)
Gellmann-Low theorem find the
ground state of H from H0.
Trace is independent of basis, not
necessary to find the ground state
Wick contraction is FULLY based on Wick contraction, but do not need to
normal ordering
based on normal ordering.
Feynman diagrams
The Feynman diagrams at finite temperature
is identical to that at zero temperature.
Linked cluster theorem,
denominator = exp(iL), which is
exactly cancelled.
G(p, E), convolution theorem
Conservation of E and P.
Linked cluster theorem, denominator
is exactly cancelled from Wick’s
rotation.
G(p, iE), convolution theorem
Conservation of E and P.
A summary for zero and finite T (please remember this table)
1. The ground state and related normal ordering are not necessary.
2. Wick contraction need to be understood using a new way
3. All the other things can be understood based on wick rotation.
Gian-Carlo Wick from wiki
GIAN-CARLO WICK, 1909–1992
 A Biographical Memoir by
MAURICE JACOB

Interested students can read the following material to get a better
understanding for the contributions of Gian-Carlo Wick in physics.
http://www.nasonline.org/publications/biographicalmemoirs/memoir-pdfs/wick-gian-carlo.pdf
Frequency summations One special thing at
finite temperature
The most special Eq.

Most of the above results can be directly verified using mathematic or
other tools. DON’T WORRY.
A lot of open codes can be found in internet.
We need one method to
obtain the above result.
.
Contour integration

Eq. 3.219, see previous slide.
Over some loop
along the Im z axis.
We find that the singularity of distribution functions are identical to their
Matsubara frequencies. From the same reason: statistics of F/B.
One simple example
Construct a closed loop
Homework (finish in 2 weeks). Please show
Fermion case
Ref: Introduction to Many-body quantum theory in condensed matter
physics, BY Henrik Bruus and Karsten Flensberg
After summation,
Take tau  0.

Conditional converged series

Since the summation does not converge, the result may differ by a
constant upon different choice of the Matsubara weighting function
About the definition of
Homework(Finish in 2 weeks)

Please verify the following result
This result is from the note by Henrik Bruus and Karsten
Flensberg, page 174.
Note that the following result for fermion is also correct for
fermion.
Take tau = 0+ gives the distribution function of boson.
Homework (Finish in 2 weeks)

Please verify the following using two basic methods.
Method 1
Basic Eq.
See the note by Henrik Bruus and Karsten Flensberg, page 174

Method 2: Please verify the above
result using Mathematica.
 Explain why this diagram is the
lowest-order self-energy for phonon.

Some personal comments about the
Matsubara frequency summations
Historically, the Matsubara frequencies and related summations are
important in many-body physics.
 A lot of tables can be found in wiki and other open materials.
http://en.wikipedia.org/wiki/User:EverettYou/Matsubara_Frequ
ency
For unknown reasons, these tables can not be found in textbooks.
 Now, the summation is not so important in modern physics due to
the progresses in symbolic calculations using mathematic, matlab as
well as maple, or other softwares.

Some personal comments: Wick vs Matsubara
Wick’s rotation seems to be essential in finite temperature Green’s
function.
 But this theory is totally developed by Matsubara.
 Wick’s rotation maybe developed in 1954 and published in APS,
while Matsubara’s theory was published in a Japan journal in 1955
(one year later), which is not well-known at that time.
 Matsubara is stimulated by the Wick contraction and normalordering but not stimulated by Wick rotation (his paper was not
directly cited in Matsubara’s original paper).
 Matsubara found/developed the Wick’s rotation by himself.
* The above history maybe incorrect, but is interesting and deserve
to be examined more carefully in future. This short history should
be remembered.

End of chapter 3 for finite T GFs: Almost 1-1 correspondences
between zero and finite T GFs; Remember this table.
Zero temperature
Finite Temperature
Interaction picture in real time Interaction picture in imaginary time
(a direct wick rotation, tau = it)
Gellmann-Low theorem find the
Trace is independent of basis, not
ground state of H from H0.
necessary to find the ground state
Wick contraction is FULLY based on Wick contraction, but do not need to
normal ordering
based on normal ordering.
Feynman diagrams
The Feynman diagrams at finite temperature
is identical to that at zero temperature.
Linked cluster theorem,
denominator = exp(iL), which is
exactly cancelled.
G(p, E), convolution theorem
Conservation of E and P.
Linked cluster theorem, denominator
is exactly cancelled from Wick’s
rotation.
G(p, iE), convolution theorem
Conservation of E and P.
The following things will not be discussed
Introduction to Many-body quantum theory in condensed matter
physics, BY Henrik Bruus and Karsten Flensberg
1. Green’s function and slater determinant.

The last result can be expressed using slater determinant.
2. Time evolution of Green’s function. This method is extremely
useful in approximations, instead of establish an exact theory.
Definiton of G(t)
Not be discussed
in my course.
Equation of motion
For G(t)
Exact expression of
G(t) using
interaction picture
Approximations
Fourier transformations
Maybe the same results
A complete &
self-consistent
theory