§2.1 Single-Particle Green's Functions at Zero Temperature
Hamiltonian with interaction can not always be solved exactly, the first few orders of perturbation theory cannot provide an adequate description of an interacting many-particle system. So we must find other methods to obtain all orders in perturbation theory. Luckily, we can get a formal solution through Green’s functions.
1. Definition, symmetry and physical significance of the single-particle Green’s functions
1.1. Definition
The single-particle Green’s function is defined as:
where
And
The
where the upper (lower) sign refers to bosons (fermions).
More generally, the
product of several operators orders them from right to left in ascending time order and adds a factor , where is the number of interchanges of fermion operators from the original given order.
1.2. Symmetry
If the Hamiltonian does not depend on time which means that
, then the Green’s function depends only on the time difference,If the system is translational invariance (i.e.
), then the Green’s function depends only on the position difference,Proof of 1. and 2.
Here we just prove the case of fermions. Expand the definition of the Green’s function,
It is obviously to see that the Green’s function only depend on the time difference
. Since the total momentum operator commute with , they have common eigenstatesFor ground state, we have
And we find
This relation is true for both bosons and fermions. Its solution is
Then we can rewrite the Green’s function as
where we have used
. So it also only depend on the position difference . Thus . The same is true for the bosons.∎
If the Hamiltonian is independent of particle’s spin, then
And if
, then . Assume the spin quantum number is , then and , soIf all three conditions above are met, then
1.3. Physical significance
Without loss of generality, we can assume
Starting from the ground state
Similarly for the state
So Green’s function is actually proportional to this probability amplitude
2. Relation to observables
2.1. Ground state expectation value of single-particle operators
The expectation value of any single-particle operator in the ground state of the system can be expressed via Green’s function.
where the upper (lower) sign refers to bosons (fermions).
Proof
Recall the previous section, we have obtained the expression of single-particle operator in second quantization:
Then the ground state of a single operator
Note that
where the upper (lower) sign refers to bosons (fermions). Then
∎
If
For total kinetic energy operator:
For total momentum operator:
For local density operator:
And total particle number operator:
For spin density operator:
And total spin operator:
For electrical current density operator:
2.2. Ground state energy
We have obtain the total kinetic energy of ground state in
We assume that
where
Then we can express the total ground state energy solely in terms of the single-particle Green’s function,
Proof
Consider the Heisenberg equation of Heisenberg field operator
where
For definiteness, consider the fermion case which is more complicated, so
So the Heisenberg equation becomes
Multiply both sides left by
In the limit
For boson case, the negative sign becomes positive sign. Add the total kinetic energy, and we finally get the total ground state energy,
∎
We can write the ground state energy in another form
where the Hamiltonian is written with a variable coupling constant
Proof
The time-independent Schrödinger equation for an arbitrary value
Take the derivative of
Multiply both sides left by
Thus we have proof the Hellman-Feynman theorem,
Substitute
Integrate
∎
If the Hamiltonian does not depend on time, and the system is homogeneous, we can write the ground state energy in a simpler form. First, take the Fourier transform of the single-particle Green’s function
And the corresponding inverse Fourier transform is
Then the ground state energy is
And another form of the ground state energy is
3. Free fermions
As an example of the above formalism, we will consider the single-particle Green’s function for a noninteracting homogeneous system of fermions. The Hamiltonian of the system is
In plane wave representation, we have
where
Then
where
With box normalization which means that
Then the summation of
and
These are well-known results in previous study (solid state physics). Now we will use single-particle Green’s function to re-obtain above results to verify that our theory is correct. The single-particle Green’s function for fermions is defined as
where we use
where we have set
Exercise: Calculate
Use Baker-Campbell-Hausdorff formula
Solve Heisenberg equation with condition
,
The result is
where
where
Then we find
Thus we finally obtain the single-particle Green’s function for free fermions
The Fourier transform of Green’s function is
Proof
We can use an integral representation for the step function
If
Contour for the step function.
TikZ code
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\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,decorations.markings,calc,bending,shadows}
\begin{document}
\begin{tikzpicture}[>={Kite[inset=0pt,length=0.32cm,bend]},
decoration={markings,
mark= at position .1 with {\arrow{>}},
mark= at position .3 with {\arrow{>}},
mark= at position .45 with {\arrow{>}},
mark= at position 0.8 with {\arrow{>}},
}]
\def\radius{4}
\filldraw[yshift=-4pt,postaction ={decorate}, thick,fill=gray!40]
(-\radius,0) -- (\radius,0) arc (0:-180:\radius) --cycle;
\filldraw[yshift=4pt,thick,postaction={decorate}, fill=gray!40]
(-\radius,0)--(\radius,0) arc (0:180:\radius)--cycle;
\fill (0,0.8) node[above right]{$i\eta$} circle (2pt) ;
\fill (0,-0.8) node[below right]{$-i\eta$} circle (2pt) ;
\node at (140:\radius+1) {$t<t'$ contour};
\node at (-40:\radius+1) {$t>t'$ contour};
\draw[-Latex] (-\radius*1.5,0) -- (\radius*1.5,0) node[below]{$\mathrm{Re}(\omega)$};
\draw[-Latex] (0,-\radius*1.2) -- (0,\radius*1.2)node[right]{$\mathrm{Im}(\omega)$} ;
\end{tikzpicture}
\end{document}
So the Green’s function in
which immediately yields
∎
And the ground state energy can be calculated via
which is the same as
where we choose the upper-half
Contour for the third line integral.
TikZ code
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\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,decorations.markings,calc,bending,shadows}
\begin{document}
\begin{tikzpicture}[>={Kite[inset=0pt,length=0.32cm,bend]},
decoration={markings,
mark= at position .1 with {\arrow{>}},
mark= at position .3 with {\arrow{>}},
mark= at position .45 with {\arrow{>}},
mark= at position 0.8 with {\arrow{>}},
}]
\def\radius{4}
\filldraw[thick,postaction={decorate}, fill=gray!40]
(-\radius,0)--(\radius,0) arc (0:180:\radius)--cycle;
\fill (1.5,0.8) node[above right]{$\epsilon_{\mathbf{k}} +i\eta$} circle (2pt) ;
\fill (1.5,-0.8) node[below right]{$\epsilon_{\mathbf{k}} +i\eta$} circle (2pt) ;
\draw[-Latex] (-\radius*1.5,0) -- (\radius*1.5,0) node[below]{$\mathrm{Re}(z)$};
\draw[-Latex] (0,-\radius*0.5) -- (0,\radius*1.2)node[right]{$\mathrm{Im}(z)$} ;
\end{tikzpicture}
\end{document}
This result is also the same as
4. References
- Fetter, A. L., Walecka, J. D. Quantum Theory of Many-Particle Systems. Courier Corporation (2002).
- Mahan, G. D., Many-Particle Physics. Springer Science+Business Media, LLC. (2000).
- Altland, A., Simons, B. D., Condensed Matter Field Theory. Cambridge University Press. (2010)
- 金彪. 量子多体理论. 2024年春