Understanding the Bethe-Salpeter Equation

Hello! Let's continue our journey into the many-body physics of excitons.

Introduction

In our last lesson, we formulated the effective Hamiltonian for an interacting electron-hole pair using the language of second quantization. We identified the kinetic energy terms, the attractive direct Coulomb interaction, and the repulsive exchange interaction. This gave us the operator $H_{eff}$ that governs the exciton's existence.

However, having the Hamiltonian is only the first step. The real challenge is to solve for its eigenstates and eigenvalues, which correspond to the exciton's wavefunction and energy. While one could try to directly diagonalize $H_{eff}$ , a more powerful and systematic approach used in modern condensed matter physics is based on many-body Green's functions. This machinery is perfectly suited to describe quasiparticle excitations.

Today's learning outcome is to describe the structure and physical meaning of the Bethe-Salpeter equation (BSE) for the two-particle Green's function. The BSE is the central equation of motion for an electron-hole pair. By the end of this lesson, you will understand what the BSE is, what its components represent physically, and how it provides a rigorous framework for calculating exciton properties. This will connect the Hamiltonian we derived last time to the computational methods used in cutting-edge research.

Why Green's Functions? The Two-Particle Correlation Function

From your work in Advanced Quantum Mechanics and QFT, you'll recall that Green's functions (or propagators) are powerful tools. The single-particle Green's function, $G$ , describes the propagation of a single electron or hole through the interacting system. Its poles in the frequency domain give the quasiparticle energies—the energies to add or remove an electron, which are measured in photoemission and inverse photoemission experiments.

An exciton, however, is not a single-particle excitation. It is a correlated two-particle (electron-hole) excitation. Therefore, we need a more complex object: a two-particle Green's function or correlation function.

Introduction to many-body Green-function theory

The following text from a set of lecture notes by Julien Toulouse provides a formal definition of the key quantities we need: the two-particle Green's function and the related four-point linear-response function.

Please read Section 1.2, 'Two-particle Green function and linear-response function'. Pay close attention to the definition of the four-point linear-response function, denoted as χ in Eq. (8), and its physical meaning as a descriptor of particle-hole propagation. Note how its poles in the Lehmann representation (Eq. 11) correspond to neutral excitation energies.

Let's summarize the key ideas from the reading. The object we are interested in is the four-point linear-response function, which we'll denote as $L$ to match the notation of our main resource later on. (It is called χ in the resource you just read). It is defined as:

L (1, 2; 1^{'}, 2^{'}) = - G_{2} (1, 2; 1^{'}, 2^{'}) + G (1, 1^{'}) G (2, 2^{'})

where $G_{2}$ is the two-particle Green's function and $G$ is the single-particle Green's function. The variables $1, 2, \dots$ are shorthand for spacetime coordinates $(r_{1}, t_{1}), (r_{2}, t_{2}), \dots$ .

Physically, $L$ describes the correlated part of the motion of two particles. When we choose the time ordering appropriate for an electron-hole pair, $L$ becomes the electron-hole correlation function. It tells us how the probability of finding an electron at spacetime point $1$ and a hole at spacetime point $2$ evolves.

As shown in Eq. (11) of the reading, the poles of this function in frequency space occur at the neutral excitation energies ( $E_{N, n} - E_{N}$ ) of the many-body system. This is precisely what we want: the energies of excitons are the poles of $L$ . Our goal is therefore to find an equation that allows us to calculate $L$ . That equation is the Bethe-Salpeter Equation.

The Structure of the Bethe-Salpeter Equation

The BSE is a Dyson-like equation for the two-particle correlation function $L$ . It has a structure that should be familiar to you from single-particle field theory.

Feynman Diagram of the Bethe-Salpeter Equation — A common diagrammatic representation of the Bethe-Salpeter Equation. The full two-particle interaction, Γ (which corresponds to our L), is the sum of an irreducible interaction K (our L₀) and a term where the irreducible interaction is followed by the propagation of two particles (S S) and another full interaction (Γ).

In algebraic form, the BSE can be written as an integral equation:

L (1, 2, 1^{'}, 2^{'}) = L_{0} (1, 2, 1^{'}, 2^{'}) + \int d 3 d 4 d 5 d 6 L_{0} (1, 5, 1^{'}, 3) Ξ (3, 6, 5, 4) L (4, 2, 6, 2^{'})

Don't be intimidated by the variables and integrals. Symbolically, this is much simpler:

L = L_{0} + L_{0} Ξ L

Let's break down the three components:

$L$ : The Full Electron-Hole Propagator. This is the quantity we want to find. It describes the propagation of an electron and a hole that are interacting with each other and with the rest of the crystal.
$L_{0}$ : The Non-Interacting Propagator. This is our starting point. It describes the propagation of an electron and a hole that do not interact with each other. It is simply the product of the individual single-particle Green's functions for the electron and the hole: $L_{0} (1, 2, 1^{'}, 2^{'}) = i G (1, 1^{'}) i G (2^{'}, 2)$ .
$Ξ$ : The Interaction Kernel. This is the most important part, as it contains all the physics of the electron-hole interaction. It represents an irreducible "scattering event" between the electron and the hole. The term $L_{0} Ξ L$ represents a process where the non-interacting pair propagates ( $L_{0}$ ), scatters via the interaction kernel ( $Ξ$ ), and then continues to propagate as a fully interacting pair ( $L$ ).

The BSE is a self-consistent equation. It states that the total propagation of an interacting pair ( $L$ ) is equal to its non-interacting propagation ( $L_{0}$ ) plus the sum of all possible interaction processes.

The Physical Meaning of the Interaction Kernel ( $Ξ$ )

So, what is the kernel $Ξ$ physically? Formally, it is defined as the functional derivative of the electronic self-energy $Σ$ with respect to the single-particle Green's function $G$ :

Ξ = i \frac{δ Σ}{δ G}

While this definition is exact, its physical meaning becomes clearer when we use an approximation for $Σ$ , such as the $G W$ approximation which is standard for these calculations. Within this framework, the kernel $Ξ$ naturally splits into two physically distinct parts.

Linear Response and More: the Bethe-Salpeter Equation

The following chapter by Lucia Reining provides an excellent, physically motivated discussion of the BSE and the components of its kernel. It connects the formal many-body concepts to the physics of excitons.

Please read Section 4, 'Self-energies and generalized response: the Bethe-Salpeter equation', and Section 5, 'The Bethe-Salpeter equation from the GW approximation'. Focus on the structure of the BSE as presented in Eq. (26) and the decomposition of the kernel Ξ in Eq. (27) and Eq. (44). Pay special attention to Figure 4, which diagrammatically explains the two main contributions to the interaction kernel.

As the reading explains, the kernel $Ξ$ for an electron-hole pair consists of two main contributions:

The Screened Direct Coulomb Attraction (- $W$ ): This term arises from the correlation part of the self-energy. It represents the attractive force between the negatively charged electron and the positively charged hole. Crucially, this is not the bare Coulomb interaction $v_{c} = e^{2} / r$ . It is a dynamically screened interaction $W$ , which accounts for the fact that all other electrons in the crystal rearrange themselves to screen the charge of the electron and hole. This screening significantly weakens the interaction. This term is directly analogous to the $H_{direct}$ term from our previous lesson's Hamiltonian.
The Bare Exchange Interaction ( $v_{c}$ ): This term comes from the Hartree part of the self-energy. It is a purely quantum mechanical effect due to the Pauli exclusion principle and the indistinguishability of electrons. It is a short-range, repulsive interaction that corresponds to the virtual annihilation and recreation of the electron-hole pair. This term is responsible for the energy splitting between exciton states with different spin configurations (e.g., singlet and triplet excitons). This is the many-body version of the $H_{exchange}$ term from our last lesson.

A diagram of the two main terms in the BSE interaction kernel, adapted from Figure 4 of Reining's article. The wiggly line represents the attractive screened Coulomb interaction (-W) between the electron (e) and hole (h). The dashed line represents the repulsive bare exchange interaction (vc).

The interplay between the attractive direct term and the repulsive exchange term determines the final energy and character of the exciton.

Test your understanding!

Consider solving the Bethe-Salpeter Equation $L = L_{0} + L_{0} Ξ L$ . What would the solution for $L$ be if the interaction kernel $Ξ$ were set to zero? What would the excitation energies (the poles of $L$ ) correspond to in this case?

Show answer

If $Ξ = 0$ , the BSE simplifies to $L = L_{0}$ . This means the fully interacting propagator is just the non-interacting propagator.

The poles of $L_{0}$ are determined by the poles of the single-particle Green's functions it is made from. The excitation energies would be simple differences between uncorrelated electron and hole quasiparticle energies, e.g., $ϵ_{c} (k) - ϵ_{v} (k^{'})$ . Physically, this corresponds to creating an electron-hole pair that does not interact. There would be no bound states (excitons), and the optical absorption would just be a continuum starting at the band gap. The kernel $Ξ$ is therefore essential for describing the binding of the electron and hole to form an exciton.

From Integral Equation to an Effective Eigenvalue Problem

Solving a four-point integral equation like the BSE seems formidable. However, by transforming it into a suitable basis, we can turn it into a more familiar problem. The natural basis to use is that of non-interacting electron-hole pairs, which are the eigenstates of $L_{0}$ .

When we do this, the BSE transforms from an integral equation for $L$ into a matrix eigenvalue equation for an effective two-particle Hamiltonian, $H^{2p}$ .

Linear Response and More: the Bethe-Salpeter Equation

This final reading from Reining's article shows how the BSE is solved in practice by converting it into a two-body Schrödinger equation.

Please read Section 6, 'A two-body Schr̈odinger equation'. Focus on how the BSE is transformed into the matrix equation (56), H A = E A. Understand what the components of the Hamiltonian matrix (Eq. 53) and the final expression for the absorption spectrum (Eq. 59) represent.

The key result of this transformation is the eigenvalue problem:

v^{'} c^{'} k^{'} \sum H_{v c k, v^{'} c^{'} k^{'}}^{2p} A_{v^{'} c^{'} k^{'}}^{λ} = E_{λ} A_{v c k}^{λ}

Let's dissect this equation:

$H^{2p}$ : This is the matrix representation of the effective two-particle Hamiltonian.
- Its diagonal elements are the energies of the non-interacting electron-hole pairs, $ϵ_{c} (k) - ϵ_{v} (k)$ . These are the energies of the absorption continuum.
- Its off-diagonal elements are the matrix elements of the interaction kernel, $Ξ$ . These elements mix the different non-interacting pairs together, creating new, correlated states.
$E_{λ}$ : These are the eigenvalues of the Hamiltonian. They are the energies of the interacting electron-hole system. For energies below the band gap, these are the exciton binding energies.
$A^{λ}$ : These are the eigenvectors. Each eigenvector describes an exciton state $λ$ as a coherent superposition of non-interacting electron-hole pairs. The coefficients $A_{v c k}^{λ}$ give us the exciton wavefunction in momentum space.

This is a profound result. Solving the abstract Bethe-Salpeter Equation is equivalent to diagonalizing a Hamiltonian for the electron-hole pair. The BSE provides the rigorous, many-body prescription for constructing this Hamiltonian matrix.

Conclusion

Today, we have formally introduced the Bethe-Salpeter Equation, the central theoretical tool for describing excitons in solids.

Key Takeaways:

Excitons, as two-particle excitations, are described by a two-particle correlation function, $L$ . The poles of $L$ yield the exciton energies.
The Bethe-Salpeter Equation (BSE) is a Dyson-like integral equation for $L$ , with the symbolic form $L = L_{0} + L_{0} Ξ L$ .
The non-interacting propagator, $L_{0}$ , describes an electron and hole that do not feel each other's presence.
The interaction kernel, $Ξ$ , contains the crucial physics of the electron-hole interaction, primarily the attractive screened Coulomb interaction (-W) and the repulsive exchange interaction (v_c).
In practice, solving the BSE is equivalent to diagonalizing an effective two-particle Hamiltonian, whose eigenvalues are the exciton energies and whose eigenvectors are the exciton wavefunctions.

Preview of the Next Lesson

We have now seen the full, powerful, and complex many-body formalism of the BSE. But how does this connect back to the simple, intuitive hydrogen model of the Wannier-Mott exciton we discussed in Module 2? In the next lesson, we will show that by applying the effective mass approximation to the BSE, it elegantly reduces to the Wannier equation. This will not only unify the simple and advanced pictures but also give us deep confidence in the validity of this sophisticated formalism.

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