Types and Physical Interpretation of Single-Particle Green's Functions

Hello! Welcome to the first lesson of our new module on Diagrammatic Many-Body Perturbation Theory.

In the previous module, we explored wavefunction-based methods like Configuration Interaction (CI). We saw that while powerful, they quickly become computationally intractable. The core challenge is the need to approximate the entire N-body wavefunction, an object of immense complexity.

Today, we pivot to an entirely different and powerful formalism, one that is the language of modern condensed matter theory. Instead of trying to compute the full wavefunction, we will focus on calculating objects called correlation functions, which describe the relationships and dynamic behavior of particles within the many-body system. The most fundamental of these is the single-particle Green's function. Our goal is to define the various types of single-particle Green's functions and, most importantly, to build a solid intuition for their physical meaning.

1. The Green's Function: From Mathematical Tool to Physical Propagator

You've likely encountered Green's functions in your studies of differential equations, perhaps in electrodynamics when solving the Poisson equation. They are a general mathematical tool for solving inhomogeneous linear differential equations. The core idea is to find the response of the system to a point-like source (a delta function). Once you know this fundamental response, you can construct the solution for any arbitrary source by treating it as a sum of many point sources.

In quantum mechanics, the differential equation we care about is the Schrödinger equation. By applying the same logic, we can define a Green's function that solves the Schrödinger equation with a delta-function source. This leads to a profound physical interpretation.

As the video explains, for a single particle, the Green's function $G(r,t;r^{\prime },t^{\prime })$ gives the probability amplitude for a particle that is at position $r^{\prime }$ at time $t^{\prime }$ to be found at position $r$ at time $t$. It is the quantum mechanical propagator.

This idea of a propagator is the key. We are now going to generalize this concept from a single particle in a vacuum to a single particle moving through the complex, interacting environment of a many-body system.

2. Single-Particle Green's Functions in Many-Body Systems

In a many-body system, we are no longer interested in the propagation of a single, isolated particle. Instead, we want to describe the propagation of a particle that is added to or removed from the N-particle system. We therefore redefine the Green's function in the language of second quantization, using creation ($c^{†}$) and annihilation ($c$) operators. The definitions involve a quantum and thermal average, denoted by $⟨\dots ⟩$, over the many-body states.

We will define several related Green's functions. For simplicity, we'll use a general index $\nu$ which could represent momentum and spin, $(k,\sigma )$, or position and spin, $(r,\sigma )$. The operators are in the Heisenberg picture, so their time dependence is given by $c_{\nu }(t)=e^{iHt}{c}_{\nu }{e}^{-iHt}$.

Let's break down these definitions. (We'll use the fermionic case, with the anticommutator, as is most common in condensed matter).

Retarded and Advanced Green's Functions

The retarded Green's function is defined as:

$$G^R(\nu ,t;{\nu }^{\prime },t^{\prime })=-i\theta (t-t^{\prime })⟨\{c_{\nu }(t),c_{{\nu }^{\prime }}^{†}(t^{\prime })\}⟩$$

• The Heaviside step function $\theta (t-t^{\prime })$ ensures this is only non-zero for $t>t^{\prime }$. It enforces causality: the particle is created at $t^{\prime }$ before it is annihilated at $t$. It describes propagation forward in time.
• The anticommutator $\{A,B\}=AB+BA$ is used for fermions.

The advanced Green's function is defined as:

$$G^A(\nu ,t;{\nu }^{\prime },t^{\prime })=+i\theta (t^{\prime }-t)⟨\{c_{\nu }(t),c_{{\nu }^{\prime }}^{†}(t^{\prime })\}⟩$$

• The step function $\theta (t^{\prime }-t)$ ensures this is only non-zero for $t^{\prime }>t$. It describes a process that can be interpreted as propagating backward in time.

"Greater" and "Lesser" Green's Functions

The anticommutator in the definitions above can be expanded. This leads to two more fundamental quantities.

The "greater" Green's function:

$$G^{>}(\nu ,t;{\nu }^{\prime },t^{\prime })=-i⟨c_{\nu }(t)c_{{\nu }^{\prime }}^{†}(t^{\prime })⟩$$

This corresponds to the process of first creating a particle in state ${\nu }^{\prime }$ at time $t^{\prime }$, and later annihilating a particle in state $\nu$ at time $t$. It describes the propagation of a particle added to the system.

The "lesser" Green's function:

$$G^{<}(\nu ,t;{\nu }^{\prime },t^{\prime })=+i⟨c_{{\nu }^{\prime }}^{†}(t^{\prime })c_{\nu }(t)⟩$$

This corresponds to first annihilating a particle from the system in state $\nu$ at time $t$, and later creating one in state ${\nu }^{\prime }$ at time $t^{\prime }$. This describes the propagation of a hole (the absence of a particle) left behind in the system.

Notice that the retarded and advanced functions are just combinations of these:

• $G^R(t,t^{\prime })=\theta (t-t^{\prime })[G^{>}(t,t^{\prime })-G^{<}(t,t^{\prime })]$
• $G^A(t,t^{\prime })=\theta (t^{\prime }-t)[G^{<}(t,t^{\prime })-G^{>}(t,t^{\prime })]$

The Time-Ordered (Feynman) Green's Function

Finally, we define the Green's function that is most common in quantum field theory, the time-ordered or causal Green's function.

$$G^c(\nu ,t;{\nu }^{\prime },t^{\prime })=-i⟨T[c_{\nu }(t)c_{{\nu }^{\prime }}^{†}(t^{\prime })]⟩$$

The time-ordering operator $T$ arranges the operators so that the one with the latest time argument is on the left, adding a minus sign for every interchange of fermionic operators.

$$T[A(t)B(t^{\prime })]=\{\begin{array}{ll}A(t)B(t^{\prime })& \text{if }t>t^{\prime }\\ -B(t^{\prime })A(t)& \text{if }{t}^{\prime }>t\end{array}$$

This single, compact definition elegantly combines the particle and hole propagation pictures:

• If $t>t^{\prime }$, $G^c(t,t^{\prime })=-i⟨c_{\nu }(t)c_{{\nu }^{\prime }}^{†}(t^{\prime })⟩=G^{>}(t,t^{\prime })$. It describes a particle propagating forward in time.
• If $t^{\prime }>t$, $G^c(t,t^{\prime })=+i⟨c_{{\nu }^{\prime }}^{†}(t^{\prime })c_{\nu }(t)⟩=G^{<}(t,t^{\prime })$. It describes a hole propagating forward in time (which is equivalent to a particle propagating backward in time).

This is why $G^c$ is often simply called the Feynman propagator. It contains the dynamics of both particles and holes.

Test your understanding!

Using the definitions above, express the time-ordered Green's function, $G^c$, in terms of the greater ($G^>$) and lesser ($G^<$) Green's functions and the Heaviside step function.

Show answer

Based on the definition of the time-ordering operator $T$, we can write $G^c$ as: $$ G^c(t, t') = \theta(t-t') G^>(t, t') + \theta(t'-t) G^<(t, t') $$ This expression clearly shows how the time-ordered propagator contains both the particle-propagation part ($G^>$ for $t>t'$) and the hole-propagation part ($G^<$ for $t'>t$).

3. Physical Interpretation: The Lehmann Representation and the Spectral Function

So far, these are formal definitions. Where is the physics? The deep physical content of the Green's function is revealed by expressing it in a basis of the exact many-body eigenstates of the Hamiltonian. This is known as the Lehmann representation.

The derivation is a bit algebraic, but the result is profoundly important. It shows that the Green's function, when transformed into the frequency domain, has poles at frequencies corresponding to the exact excitation energies of the system.

Let's summarize the key result from the Lehmann representation of the retarded Green's function in the frequency domain, $G^R(\nu ,\omega )$. Its imaginary part defines a new, crucial quantity called the spectral function:

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

The Lehmann representation shows that the spectral function can be written as:

$$A(\nu ,\omega )=\frac{1}{Z}\sum _{n,m}|⟨m|c_{\nu }^{†}|n⟩{|}^2(e^{-\beta E_n}+e^{-\beta E_m})\delta (\omega +E_n-E_m)$$

This looks complicated, but its meaning is beautiful:

• The delta function $\delta (\omega +E_n-E_m)$ ensures that $A(\nu ,\omega )$ is only non-zero when the frequency $\omega$ exactly matches an energy difference $E_m-E_n$. These are the energies required to excite the system from state $|n⟩$ to state $|m⟩$ by adding one particle.
• The matrix element $|⟨m|c_{\nu }^{†}|n⟩{|}^2$ gives the probability of this transition occurring.

Therefore, the spectral function $A(\nu ,\omega )$ is the probability density of finding a particle-like excitation with quantum numbers $\nu$ and energy $\omega$ in the interacting system.

For a non-interacting system, the particle-like excitations are just the particles themselves. The spectral function becomes a sharp delta peak: $A_0(k,\omega )=\delta (\omega -{ϵ}_k)$. In an interacting system, this peak can broaden, shift, and break up into multiple peaks, reflecting the complex nature of excitations (now called "quasiparticles"). This is precisely what is measured in experiments like Angle-Resolved Photoemission Spectroscopy (ARPES), which essentially maps out the spectral function of a material.

Conclusion

This lesson has been a deep dive into a new formalism. We have moved away from the N-body wavefunction and introduced the single-particle Green's function as our main object of interest.

Key Takeaways:

• Green's functions are propagators: They give the probability amplitude for a particle (or hole) to propagate from one space-time point to another within a many-body system.
• Different "flavors" for different purposes:
  • Retarded/Advanced ($G^R,G^A$): Used for calculating linear response and have strict causal ordering.
  • Greater/Lesser ($G^{>},G^{<}$): Describe the dynamics of adding particles vs. creating holes.
  • Time-Ordered ($G^c$): The Feynman propagator, which combines particle and hole propagation and is the natural object for perturbation theory.
• Physical information is in the poles: The Lehmann representation shows that the poles of the Green's function in the frequency domain correspond to the system's true single-particle excitation energies.
• The Spectral Function ($A(k,\omega )$) is the imaginary part of $G^R$ and represents the energy- and momentum-resolved probability of finding a particle-like excitation. It is a direct link between theory and experiment.

Preview of the Next Lesson:

We have defined what Green's functions are and what they mean. But how do we calculate them for an interacting system where we don't know the exact eigenstates? The answer lies in perturbation theory. In the next lesson, we will learn how to expand the Green's function in powers of the interaction potential. This will lead us to the famous Feynman diagrams, which provide an intuitive, pictorial way to organize and calculate the terms in this expansion.