Green's Functions and Quasiparticle Excitations

Hello! Welcome to the third lesson in our module on many-body formalism.

In our previous lesson, we derived the Hartree-Fock equations, which provide a static, mean-field picture of electrons in a solid. This approximation reduces the interacting many-body problem to a set of single-particle equations, giving us a set of orbitals and their corresponding energies. While powerful, this picture is incomplete. It doesn't tell us about the dynamics of the system: What happens when we add or remove an electron? How do these excitations propagate through the crystalline sea of other electrons?

This lesson introduces the central mathematical tool for answering these questions: the single-particle Green's function. Our goal is to define the single-particle Green's function and relate its poles to quasiparticle excitations. We will see how this function acts as a "propagator" for electrons and how its mathematical structure reveals the energies and lifetimes of the collective excitations, known as quasiparticles, which dominate the low-energy physics of interacting systems.

1. What is a Green's Function? An Intuitive Start

Before diving into the many-body world, let's build some intuition. In general, a Green's function is a tool for solving inhomogeneous linear differential equations. It represents the system's response to a point-like "kick" or source.

The following video introduces the concept, first using the familiar Poisson equation from electrodynamics and then extending it to the single-particle Schrödinger equation. It provides a clean physical interpretation of the Green's function as a propagator.

As you saw, for a single particle, the Green's function G(r,t; r',t') is directly related to the time-evolution operator:

$$G(r,t;r^{\prime },t^{\prime })\propto ⟨r|e^{-i\hat{H}(t-t^{\prime })}|r^{\prime }⟩\text{for }t>t^{\prime }$$

This describes the amplitude for a particle starting at position r' to be found at position r a time t-t' later. But how do we generalize this to a system with $10^{23}$ interacting electrons?

2. The Many-Body Green's Function

In a many-body system, we can't track a single particle. Instead, we ask: what is the amplitude for a particle added to the system at (r', t') to be removed at (r, t)? This process is described by the single-particle Green's function, defined using the creation and annihilation operators you're familiar with.

The next segment of the video you just watched provides a clear and concise definition.

Let's formalize the key definitions for a system of fermions. The retarded Green's function is:

$$G_{AB}^R(t,t^{\prime })=-i\theta (t-t^{\prime })⟨\{c_A(t),c_B^{†}(t^{\prime }){\}}_{+}⟩$$

Here, $c_B^{†}(t^{\prime })$ creates a particle in state $B$ at time $t^{\prime }$, and $c_A(t)$ annihilates a particle in state $A$ at time $t$. The angle brackets denote a thermal average over the many-body system, and $\theta (t-t^{\prime })$ is the Heaviside step function enforcing causality ($t>t^{\prime }$).

The anticommutator $\{c_A(t),c_B^{†}(t^{\prime }){\}}_{+}=c_A(t)c_B^{†}(t^{\prime })+c_B^{†}(t^{\prime })c_A(t)$ is crucial. It can be broken down into two physically distinct processes:

1. The "Greater" Green's function, $G^{>}$:

   $$G_{AB}^{>}(t,t^{\prime })=-i⟨c_A(t)c_B^{†}(t^{\prime })⟩$$

   This describes the propagation of an added particle. We create a particle at t' and annihilate it at t.

2. The "Lesser" Green's function, $G^{<}$:

   $$G_{AB}^{<}(t,t^{\prime })=+i⟨c_B^{†}(t^{\prime })c_A(t)⟩$$

   This describes the propagation of a hole. We annihilate a particle at t (creating a hole) and then fill it by creating a particle at t'.

The full retarded Green's function, $G^R=\theta (t-t^{\prime })(G^{>}-G^{<})$, therefore describes the propagation of either a particle-like or a hole-like excitation.

3. Poles of the Green's Function and the Spectral Representation

The time-domain Green's function contains all the dynamics, but it's often more transparent to work in the frequency domain via a Fourier transform. For a system where the Hamiltonian is time-independent, the Green's function only depends on the time difference, $G(t,t^{\prime })=G(t-t^{\prime })$, and we can define:

$$G(\omega )={\int }_{-\infty }^{\infty }d\tau G(\tau )e^{i\omega \tau },\text{where }\tau =t-t^{\prime }$$

A remarkable property of the Green's function is that its poles in the complex frequency plane correspond to the exact excitation energies of the interacting system. To see this, one can derive the Lehmann Representation of the Green's function.

The derivation involves inserting a complete set of the exact many-body eigenstates $|n⟩$ of the Hamiltonian $\hat{H}$ into the definition of the Green's function. While the full derivation is lengthy, the resulting expression is what matters.

The following video provides a detailed blackboard derivation of the Lehmann representation and introduces the spectral function. Since you are comfortable with formal derivations, this will provide the theoretical backbone for our discussion.

The Lehmann representation (at zero temperature for simplicity, with ground state $|N,0⟩$) schematically looks like this:

$$G(k,\omega )=\sum _m\frac{|⟨N+1,m|c_k^{†}|N,0⟩{|}^2}{\omega -(E_{N+1,m}-E_{N,0})+i\eta }+\sum _n\frac{|⟨N-1,n|c_k|N,0⟩{|}^2}{\omega -(E_{N,0}-E_{N-1,n})-i\eta }$$

Let's unpack this:

• The first term has poles at $\omega =E_{N+1,m}-E_{N,0}$. These are the energies required to add one electron with momentum $k$ and leave the system in an excited state $|N+1,m⟩$. These are the particle-like excitations.
• The second term has poles at $\omega =E_{N,0}-E_{N-1,n}$. These are the energies required to remove one electron with momentum $k$, leaving the system in state $|N-1,n⟩$. These are the hole-like excitations.

The poles of the single-particle Green's function are the single-particle excitation energies of the interacting many-body system.

The information contained in these poles is captured by the spectral function, defined as:

$$A(k,\omega )=-\frac{1}{\pi }\text{Im}{G}^R(k,\omega )$$

Using the identity $\frac{1}{x\pm i\eta }=\mathcal{P}\frac{1}{x}\mp i\pi \delta (x)$, the spectral function becomes a series of delta peaks located precisely at the excitation energies.

4. Quasiparticles and the Self-Energy

For a non-interacting system, adding an electron with momentum $k$ creates a new eigenstate with energy ${ϵ}_k$. The spectral function is just a single, sharp delta function: $A(k,\omega )=\delta (\omega -{ϵ}_k)$.

In an interacting system, a bare electron added to the system collides with other electrons, creating a complex cloud of excitations. It's no longer an eigenstate. However, in many cases (particularly in metals), this composite object—the electron plus its surrounding "polarization cloud"—behaves like a well-defined particle, but with modified properties. This is a quasiparticle.

This concept is formalized through the self-energy, $\Sigma (k,\omega )$. The self-energy encapsulates all the complex interaction effects. The full Green's function can be written in a compact form called the Dyson equation:

$$G(k,\omega )=G_0(k,\omega )+G_0(k,\omega )\Sigma (k,\omega )G(k,\omega )$$

Solving for $G(k,\omega )$, where $G_0(k,\omega )=\frac{1}{\omega -{ϵ}_k+i\eta }$ is the non-interacting Green's function, gives:

$$G(k,\omega )=\frac{1}{\omega -{ϵ}_k-\Sigma (k,\omega )}$$

The following text explains how the properties of the self-energy give rise to the quasiparticle concept.

As the text explains, the poles of the Green's function are now the solutions to $\omega -{ϵ}_k-\Sigma (k,\omega )=0$. The self-energy is a complex function, $\Sigma =\text{Re}\Sigma +i\text{Im}\Sigma$.

• Quasiparticle Energy $E_k$: The real part, $\text{Re}\Sigma$, shifts the energy of the bare particle. The quasiparticle energy $E_k$ is the solution to $E_k={ϵ}_k+\text{Re}\Sigma (k,E_k)$.
• Quasiparticle Lifetime ${\tau }_k$: The imaginary part, $\text{Im}\Sigma$, gives the pole a finite imaginary component. This broadens the delta-function peak in the spectral function into a Lorentzian. The width of this Lorentzian is ${\Gamma }_k\propto |\text{Im}\Sigma |$, and the quasiparticle lifetime is ${\tau }_k=\hslash /{\Gamma }_k$. A non-zero $\text{Im}\Sigma$ means the quasiparticle is not a true eigenstate and will eventually decay.

The spectral function for a stable quasiparticle is approximately:

$$A(k,\omega )\approx \frac{Z_k}{\pi }\frac{{\Gamma }_k/2}{(\omega -E_k{)}^2+({\Gamma }_k/2{)}^2}$$

where $Z_k$ is the quasiparticle weight, a measure of the overlap between the "dressed" quasiparticle and the original "bare" electron.

Test your understanding!

Imagine a system where, at the Fermi energy ($\omega=0$), the imaginary part of the self-energy $\text{Im}\Sigma(k_F, 0) = 0$. What does this imply about the lifetime of a quasiparticle exactly at the Fermi surface? Why is this physically reasonable?

Show answer

If $\text{Im}\Sigma(k_F, 0) = 0$, the quasiparticle lifetime $\tau = \hbar/\Gamma \propto 1/|\text{Im}\Sigma|$ becomes infinite. This means that a quasiparticle excitation right at the Fermi surface is stable and does not decay.

This is physically reasonable due to phase space constraints. An electron at the Fermi surface can only scatter to an unoccupied state. If it loses an infinitesimal amount of energy, the electron it scatters off must also be excited to an unoccupied state. At T=0, all states below the Fermi surface are full and all states above are empty. The available phase space for such scattering processes shrinks to zero as the electron's energy approaches the Fermi energy, leading to an infinite lifetime. This is a cornerstone of Landau's Fermi liquid theory.

5. Experimental Observation: Photoemission Spectroscopy

This entire formalism might seem abstract, but it has a direct and profound connection to experiment. As someone with experience in spectroscopy, you know that experiments measure the response of a system to a probe. The Green's function is precisely the theoretical object that describes this response.

Angle-Resolved Photoemission Spectroscopy (ARPES) is an experimental technique that measures the kinetic energy and emission angle of electrons ejected from a material by high-energy photons. By conserving energy and momentum, one can deduce the binding energy $\omega$ and crystal momentum $k$ the electron had inside the solid. The measured intensity, $I(k,\omega )$, is directly proportional to the single-particle spectral function, $A(k,\omega )$.

Therefore, an ARPES experiment is a direct visualization of the spectral function. The bright bands in an ARPES map are the quasiparticle dispersion relations $E_k$, and the width of these bands reveals their lifetime ${\tau }_k$.

This powerful connection is why the Green's function formalism is indispensable in modern condensed matter physics.

Conclusion

This lesson has been a deep dive into the heart of many-body theory. We have moved from a static, mean-field description to a dynamic picture of propagating excitations.

Key Takeaways:

• The single-particle Green's function describes the propagation of an added particle or hole in an interacting many-body system.
• Its poles in the complex frequency plane correspond to the system's exact single-particle excitation energies.
• The self-energy $\Sigma (k,\omega )$ accounts for all interaction effects, modifying the properties of a bare electron.
• In many systems, the excitations are long-lived quasiparticles, which behave like particles with a renormalized energy $E_k$ (from $\text{Re}\Sigma$) and a finite lifetime ${\tau }_k$ (from $\text{Im}\Sigma$).
• The spectral function $A(k,\omega )=-\frac{1}{\pi }\text{Im}{G}^R(k,\omega )$ gives the probability distribution of single-particle states and is directly measured in ARPES experiments.

Preview of the Next Lesson

We have defined the Green's function and the self-energy and seen their immense physical importance. But how do we actually calculate the self-energy? It contains all the complexity of the interactions. The next lessons will be dedicated to developing approximation schemes for $\Sigma (k,\omega )$. We will start by introducing the crucial concept of dielectric screening and the Random Phase Approximation (RPA), which is our first and most fundamental model for calculating the effective interaction between electrons in a solid.