Feynman Rules for Imaginary-Time Green's Functions

Hello! Welcome to the third lesson in our module on Green's functions and diagrammatic perturbation theory.

In the last two lessons, we established a powerful conceptual framework. We started by defining the single-particle Green's function and its connection to the spectral function, $A(k,\omega )$. Then, we saw how the experimental technique of ARPES provides a direct measurement of this spectral function, revealing the energy and lifetime of quasiparticles, which are governed by the real and imaginary parts of the self-energy, $\Sigma (k,\omega )$.

This leaves us with a critical question: how do we actually calculate the Green's function and the self-energy for a system where electrons interact? Today, we will answer that question by developing the central computational tool of many-body physics. Our learning outcome is to formulate the Feynman rules for diagrammatic perturbation theory of the Green's function in the imaginary-time formalism. This will provide a systematic "recipe" for calculating corrections to the electronic properties arising from interactions.

Given your background in QFT, you'll find the underlying logic familiar. Our focus will be on adapting these ideas to the finite-temperature, imaginary-time (or Matsubara) formalism, which is the standard language for condensed matter problems.

1. The Perturbation Expansion: From Interacting to Non-Interacting

Our strategy is to start with a Hamiltonian we can solve, $H_0$ (the non-interacting part), and treat the interaction part, $V$, as a perturbation. We want to express the full, interacting Green's function, $G$, in terms of the known non-interacting Green's function, $G_0$, and a power series in $V$.

The key is to work in the imaginary-time interaction picture. In this picture, operators evolve in imaginary time $\tau$ according to the non-interacting Hamiltonian $H_0$. This allows us to separate the known dynamics of $H_0$ from the unknown effects of $V$. This procedure leads to a celebrated result known as the finite-temperature Gell-Mann-Low theorem, which expresses the full Matsubara-Green function as:

$$G_{\alpha \beta }(\tau -{\tau }^{\prime })=-\frac{⟨T_{\tau }[{\tilde{c}}_{\alpha }(\tau ){\tilde{c}}_{\beta }^{†}({\tau }^{\prime })e^{-{\int }_0^{\beta }d{\tau }_1\tilde{V}({\tau }_1)}]{⟩}_0}{⟨T_{\tau }[e^{-{\int }_0^{\beta }d{\tau }_1\tilde{V}({\tau }_1)}]{⟩}_0}$$

Let's unpack this formidable expression:

• The operators $\tilde{c}$ and $\tilde{V}$ are in the interaction picture (evolving with $H_0$).
• $T_{\tau }$ is the imaginary-time ordering operator.
• The averages $⟨\dots {⟩}_0$ are taken with respect to the non-interacting system.
• $\beta =1/k_{B}T$ is the inverse temperature.

This formula is exact but not directly computable. The next step is to expand the exponential in a power series of the interaction $\tilde{V}$. This generates an infinite series of terms, each involving an increasingly complex average of creation and annihilation operators.

To build intuition for how this expansion works and how it naturally leads to diagrams, please watch the following video. While it's set in the context of T=0 scalar field theory using path integrals, the fundamental logic of expanding the interaction term and evaluating correlation functions is identical to our problem.

2. Wick's Theorem: Taming the Expansion

As the video demonstrates, the perturbation expansion leaves us with a series of complicated terms, each requiring the calculation of a time-ordered product of many field operators, like $⟨T_{\tau }[{\tilde{c}}_{\alpha }(\tau ){\tilde{c}}_{\beta }^{†}({\tau }^{\prime })\tilde{V}({\tau }_1)\tilde{V}({\tau }_2)\dots ]{⟩}_0$.

The tool that makes this calculation tractable is Wick's Theorem. It provides a remarkable simplification:

Wick's Theorem: The thermal average of a time-ordered product of an even number of creation and annihilation operators is equal to the sum of all possible fully-contracted pairings of these operators, with an appropriate sign for fermions.

A contraction of two operators is simply their time-ordered thermal average in the non-interacting system:

$$\text{Contraction of }\tilde{A}({\tau }_1)\text{ and }\tilde{B}({\tau }_2)\equiv ⟨T_{\tau }[\tilde{A}({\tau }_1)\tilde{B}({\tau }_2)]{⟩}_0$$

For our problem with fermions, the only non-zero contraction is between a creation and an annihilation operator. This contraction is, by definition, the non-interacting Matsubara-Green function:

$$⟨T_{\tau }[{\tilde{c}}_k(\tau ){\tilde{c}}_{k^{\prime }}^{†}({\tau }^{\prime })]{⟩}_0=-G_0(k,\tau -{\tau }^{\prime }){\delta }_{k{k}^{\prime }}$$

This is the central insight: Wick's theorem systematically reduces the problem of calculating the interacting Green's function $G$ to evaluating sums of products of the known non-interacting Green's function $G_0$.

3. Feynman Rules for the Imaginary-Time Formalism

Each term in the Wick expansion corresponds to a specific way of pairing up operators. A Feynman diagram is simply a graphical representation of one of these terms. The set of Feynman rules provides the dictionary to translate from a diagram to its corresponding mathematical expression.

For practical calculations, it is most convenient to work in momentum and Matsubara frequency space. The following rules allow you to compute the $n$-th order correction to the self-energy, ${\Sigma }^{(n)}$, for a system of interacting electrons. A detailed derivation can be found in the provided notes by Chang (Resource LINK) and Georges (Resource LINK).

Feynman Rules (Momentum-Matsubara Frequency Space):

1. Draw the Diagrams: For the $n$-th order self-energy, draw all topologically distinct, connected diagrams with two external lines (one incoming, one outgoing) and $n$ interaction vertices.

2. Assign Momenta and Frequencies: Assign a momentum $\mathbf{k}$ and a fermionic Matsubara frequency $i{\omega }_m$ to each solid line (propagator). Assign a momentum $\mathbf{q}$ and a bosonic Matsubara frequency $i{\nu }_n$ to each wavy interaction line.

3. Apply Conservation Laws: At each vertex, momentum and Matsubara frequency must be conserved. The sum of incoming momenta/frequencies must equal the sum of outgoing momenta/frequencies.

4. Write Down Factors for Each Element:

   • Fermion Propagator: For each solid line with momentum $\mathbf{k}$ and frequency $i{\omega }_m$, write a factor of the bare propagator:
     $G_0(\mathbf{k},i{\omega }_m)=\frac{1}{i{\omega }_m-({ϵ}_{\mathbf{k}}-\mu )}$
   • Interaction Line: For each interaction vertex representing a two-body potential $V(\mathbf{q})$, write a factor of $-V(\mathbf{q})$. (The minus sign comes from the expansion of $e^{-V}$).

5. Integrate/Sum Over Internal Variables: For any momentum/frequency that is not fixed by the external variables and conservation laws (i.e., for any internal loop), perform an integration over momentum and a summation over Matsubara frequency:
   $\int \frac{d^{d}k}{(2\pi {)}^d}\frac{1}{\beta }{\sum }_{i{\omega }_m}$

6. Include the Sign Factor: Multiply the entire expression by $(-1{)}^F$, where $F$ is the number of closed fermion loops in the diagram.

7. Symmetry Factors: If a diagram is symmetric under the permutation of identical lines, you may need to divide by a symmetry factor. For self-energy diagrams, this is often handled by a careful definition of the interaction vertex. We will not dwell on this finer point for now.

Let's see this in action for the first-order corrections to the self-energy.

First-order self-energy diagrams: (left) The "Hartree" term. The interaction is with the average density of all other electrons. (right) The "Fock" term, an exchange interaction. These diagrams are taken from the notes by Chang (Resource LINK).

Using our rules for the Fock diagram (right):

• There is one vertex ($n=1$).
• There is one closed fermion loop ($F=1$). The sign is $(-1{)}^1=-1$.
• The interaction vertex gives a factor $-V(\mathbf{k}-\mathbf{q})$.
• The internal fermion line has a propagator $G_0(\mathbf{q},i{\omega }_n)$.
• We sum over the internal variables $\mathbf{q}$ and $i{\omega }_n$.

Putting it all together, the expression is:

$${\Sigma }_F(\mathbf{k})=(-1)\int \frac{d^{3}q}{(2\pi {)}^3}\frac{1}{\beta }\sum _{i{\omega }_n}[-V(\mathbf{k}-\mathbf{q})]G_0(\mathbf{q},i{\omega }_n)e^{i{\omega }_n{0}^{+}}$$

The summation over the Matsubara frequency simply yields the particle number operator $n_{\mathbf{q}}$, recovering the familiar Hartree-Fock expression at finite temperature.

Test your understanding!

Consider the "bubble" diagram shown below, which is a second-order ($n=2$) contribution to the interaction potential itself (not the self-energy). Try to write down the mathematical expression for it, which is denoted ${\Pi }_0(q,i{\nu }_n)$. Don't worry about the overall sign, just focus on the propagators and the internal loop sum/integral.

The "bubble" diagram for the polarization operator ${\Pi }_0$. This diagram represents the creation and subsequent annihilation of a virtual electron-hole pair.

Show answer

The diagram has one internal loop. Let's label the momentum and frequency of the bottom line as $\mathbf{k}$ and $i{\omega }_m$. Due to conservation at the vertices, the top line must have momentum $\mathbf{k}+\mathbf{q}$ and frequency $i{\omega }_m+i{\nu }_n$.

Following the rules:

1. Propagators: We have two propagators: $G_0(\mathbf{k},i{\omega }_m)$ and $G_0(\mathbf{k}+\mathbf{q},i{\omega }_m+i{\nu }_n)$.
2. Internal Variables: We must sum over the internal momentum $\mathbf{k}$ and Matsubara frequency $i{\omega }_m$.
3. Sign: There is one closed fermion loop, so there's a factor of $(-1)$.

The expression is (ignoring other prefactors):
${\Pi }_0(\mathbf{q},i{\nu }_n)\propto (-1)\int \frac{d^{3}k}{(2\pi {)}^3}\frac{1}{\beta }{\sum }_{i{\omega }_m}{G}_0(\mathbf{k},i{\omega }_m)G_0(\mathbf{k}+\mathbf{q},i{\omega }_m+i{\nu }_n)$
This object is called the "polarization bubble" and is fundamental to understanding screening in metals.

Conclusion

Today, we have constructed a powerful and systematic engine for calculating the properties of interacting electron systems. By combining the interaction picture, Wick's theorem, and a graphical notation, we arrived at a concrete set of Feynman rules that allow us to translate any interaction process into a computable mathematical expression.

Key Takeaways:

• The interacting Green's function can be expressed as a perturbation series in the interaction potential $V$.
• Wick's theorem is the key that allows us to evaluate each term in the series by decomposing it into products of non-interacting propagators, $G_0$.
• Feynman diagrams provide an intuitive visualization of the terms in this expansion.
• The Feynman rules are a precise dictionary for translating any diagram into a momentum-frequency space integral and sum, accounting for propagators, vertices, conservation laws, and sign conventions.

Preview of the Next Lesson

We have seen how to calculate corrections to the Green's function order by order. However, this raises a new question: which diagrams do we need to calculate? There are infinitely many! In the next lesson, we will introduce a profound organizing principle. We will see that all "one-particle irreducible" diagrams can be summed up into a single object: the self-energy $\Sigma$. This will lead us to the Dyson equation, a self-consistent equation that relates the full Green's function $G$ to the bare Green's function $G_0$ and the self-energy $\Sigma$.

The Dyson equation, $G=G_0+G_0\Sigma G$, is one of the most important equations in many-body physics, as it resums an infinite class of diagrams and provides a non-perturbative way to understand the behavior of interacting systems.