ARPES and the Spectral Function

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

In our last session, we established a crucial theoretical foundation. We derived the Lehmann representation and defined the single-particle spectral function, $A(k,\omega )$, as the probability density of finding a single-particle excitation with momentum $k$ and energy $\omega$. We concluded that $A(k,\omega )$ is essentially a theoretical map of the allowed electronic states in an interacting system.

Today, we will bridge the gap between this abstract theoretical concept and tangible experimental results. Our goal is to relate the single-particle spectral function to angle-resolved photoemission spectroscopy (ARPES) intensity maps. You'll learn how ARPES works and see that the rich, colorful plots it produces are, in fact, direct visualizations of the spectral function. This lesson is central to connecting many-body theory with the electronic and optical properties of materials, a key aspect of your research interests.

1. The ARPES Experiment: A Momentum-Resolved Look at Electrons

At its core, ARPES is a sophisticated application of the photoelectric effect. High-energy photons strike a material, kicking out electrons. What makes ARPES so powerful is that it doesn't just count the ejected electrons; it precisely measures their kinetic energy and the angle at which they emerge.

To get a quick and intuitive overview of the experimental setup and the underlying physics, please watch this short video.

As the video explains, the experiment measures two key quantities for each photoemitted electron:

1. Kinetic Energy ($E_{kin}$): From energy conservation, we can find the electron's initial binding energy $E_B$ relative to the Fermi level ($E_F$):
   $E_{kin}=h\nu -\varphi -E_B$
   where $h\nu$ is the incident photon energy and $\varphi$ is the material's work function. We are interested in the occupied states, so we typically plot the intensity versus binding energy, where $E_B>0$ for states below $E_F$. It's common practice to define the energy variable $\omega$ as the electron's energy relative to the Fermi level, so $\omega =-E_B$.

2. Emission Angle ($\theta ,\varphi$): The momentum of the electron is conserved parallel to the crystal surface as it escapes into the vacuum. This is the key to momentum resolution. By measuring the angle and $E_{kin}$, we can find the electron's momentum components inside the solid:
   $k_{x,\text{crystal}}=k_{x,\text{vacuum}}=\frac{1}{\hslash }\sqrt{2m_e{E}_{kin}}\sin \theta \cos \varphi$
   $k_{y,\text{crystal}}=k_{y,\text{vacuum}}=\frac{1}{\hslash }\sqrt{2m_e{E}_{kin}}\sin \theta \sin \varphi$
   The momentum component perpendicular to the surface, $k_z$, is not conserved, which is a complication we'll set aside for now. By focusing on 2D materials or clever experimental geometries, we can often map out the full $E(k_x,k_y)$ relationship, known as the band structure or dispersion.

2. Theoretical Basis: Linking Photoemission Intensity to the Spectral Function

Now for the central question: what determines the intensity $I(k,\omega )$ that ARPES measures? The answer lies in applying Fermi's golden rule to the photoemission process, and as we'll see, the result beautifully connects to the Green's function formalism.

The process is often conceptualized using the three-step model:

1. An electron in the bulk absorbs a photon and is excited to a higher energy state.
2. The excited electron travels to the surface.
3. The electron escapes from the surface into the vacuum.

The crucial information about the initial electronic structure is contained in the first step. The derivation involves some formalism that should be familiar from your QFT studies, particularly the "sudden approximation." The following resource provides a clear walkthrough of this derivation.

Let's summarize the key physical argument. The photoemission intensity $I(k,\omega )$ is proportional to the total transition rate from an initial N-electron state ${\Psi }_i^N$ to all possible final states ${\Psi }_f^N$, governed by Fermi's Golden Rule:

Here's where the sudden approximation comes in: the photon removes an electron so quickly that the remaining $N-1$ electrons don't have time to react.

• The initial state is the $N$-electron ground state, $|{\Psi }_i^N⟩$.
• The final state is approximated as a product of the free photoelectron state $|{\varphi }_{k_f}⟩$ and the state of the remaining $N-1$ electrons, $c_k|{\Psi }_i^N⟩$. This remaining state is generally not an eigenstate of the $(N-1)$-particle Hamiltonian.

To get the full transition rate, we must sum over all the actual eigenstates $|{\Psi }_m^{N-1}⟩$ of the $(N-1)$-particle system. The expression for the intensity then involves terms like ${\sum }_m|⟨{\Psi }_m^{N-1}|c_k|{\Psi }_i^N⟩{|}^2$. This is precisely the structure that defines the single-particle spectral function!

After some algebra, we arrive at the master equation for ARPES:
$I(k,\omega )\propto |M_{fi}(k,\nu ,\mathbf{A}){|}^2\cdot f(\omega )\cdot A(k,\omega )$
Let's break this down:

1. $A(k,\omega )$: The Spectral Function. This is the quantity we defined in the last lesson. It contains all the information about the electronic excitations: their energies and lifetimes.
2. $f(\omega )$: The Fermi-Dirac Distribution. This function ensures that we only see signals from initially occupied states ($\omega <0$, or $E_B>0$).
3. $|M_{fi}{|}^2$: The Matrix Element. This factor describes the probability of the photon-electron interaction itself. It depends on the initial and final state wavefunctions, the photon energy $h\nu$, and the light polarization $\mathbf{A}$.

In many introductory discussions, the matrix element is treated as a constant, leading to the simplified statement that ARPES measures the spectral function of the occupied states. While this is a powerful starting point, it's important to remember that matrix element effects can strongly modulate the observed intensity, sometimes making certain bands invisible for a given experimental geometry. This is discussed in the introduction of the chinook paper (LINK).

This image provides a perfect summary of our discussion, showing real ARPES data alongside the theoretical expression for the spectral function.

3. Interpreting ARPES Intensity Maps

With the relation $I\propto A(k,\omega )$ in hand, we can now understand what an ARPES map tells us about the electrons in a material.

As we saw in the last lesson, for non-interacting electrons, the self-energy $\Sigma =0$, and the Green's function is simply $G_0(k,\omega )=(\omega -{ϵ}_k^0+i\eta {)}^{-1}$. This gives a spectral function composed of delta functions:
$A_0(k,\omega )=\delta (\omega -{ϵ}_k^0)$
In this case, the ARPES map would show infinitely sharp lines tracing the bare band dispersion ${ϵ}_k^0$.

For interacting electrons, the self-energy $\Sigma (k,\omega )={\Sigma }^{\prime }(k,\omega )+i{\Sigma }^{\prime \prime }(k,\omega )$ is non-zero. The spectral function takes the form of a Lorentzian (or more complex peak shape):
$A(k,\omega )=\frac{1}{\pi }\frac{-{\Sigma }^{\prime \prime }(k,\omega )}{(\omega -{ϵ}_k^0-{\Sigma }^{\prime }(k,\omega ){)}^2+({\Sigma }^{\prime \prime }(k,\omega ){)}^2}$

This expression is incredibly rich:

• Peak Position (Energy Renormalization): The peaks in $A(k,\omega )$ are no longer at the bare energy ${ϵ}_k^0$. They are shifted by the real part of the self-energy, ${\Sigma }^{\prime }(k,\omega )$. The observed quasiparticle dispersion is given by the solution to ${\omega }_k={ϵ}_k^0+{\Sigma }^{\prime }(k,{\omega }_k)$.
• Peak Width (Lifetime): The peaks now have a finite width, determined by the imaginary part of the self-energy, ${\Sigma }^{\prime \prime }(k,\omega )$. This broadening is not just an experimental artifact; it is an intrinsic consequence of interactions. The quasiparticle is no longer a true eigenstate and can decay into other excitations. Its lifetime ${\tau }_k$ is directly related to the peak width:
  ${\tau }_k=\frac{\hslash }{2|{\Sigma }^{\prime \prime }(k,{\omega }_k)|}$

Therefore, an ARPES map of an interacting system shows "fuzzy" or broadened bands. The centerline of a band traces the renormalized quasiparticle energy, and its "fuzziness" (width) tells you how quickly that quasiparticle decays. This provides a direct experimental window into the many-body interactions within the solid.

Conclusion

Today, we have forged a critical link between the abstract world of many-body theory and the concrete reality of experiment. This connection is one of the triumphs of modern condensed matter physics.

Key Takeaways:

• ARPES measures the kinetic energy and emission angle of photoemitted electrons to determine their binding energy $\omega$ and momentum $k$ inside the solid.
• The measured ARPES intensity is directly proportional to the single-particle spectral function: $I(k,\omega )\propto |M_{fi}{|}^{2}f(\omega )A(k,\omega )$.
• An ARPES intensity map is a direct visualization of the $k$-resolved electronic structure.
• The peak positions in the map reveal the renormalized quasiparticle energies, shifted from the bare band energies by ${\Sigma }^{\prime }(k,\omega )$.
• The peak widths are determined by ${\Sigma }^{\prime \prime }(k,\omega )$ and are inversely proportional to the quasiparticle lifetime, providing a direct measure of interaction strength.

Preview of the Next Lesson

We now have a complete conceptual picture: the Green's function contains the self-energy, the self-energy shapes the spectral function, and the spectral function is what ARPES measures. The missing piece is a practical method to actually calculate the self-energy for a given interacting Hamiltonian. This is the task of perturbation theory. In the next lesson, we will begin to assemble this computational toolkit by formulating the Feynman rules for diagrammatic perturbation theory, which provide a systematic way to calculate the Green's function and self-energy order by order.