Bethe-Salpeter Equation, Tamm-Dancoff and Random Phase Approximation

1. The Bethe-Salpeter Equation

The Bethe-Salpeter equation (BSE) is the exact equation of motion for the two-particle Green's function $L$. It was introduced by Salpeter and Bethe in 1951 (Physical Review 84, 1232) in the context of relativistic bound states, and later became a cornerstone of many-body perturbation theory in condensed matter physics.

1.1 Two-Particle Green's Function

The central object is the four-point correlation function (two-particle Green's function):

\[L(1,2;1',2') = G_2(1,2;1',2') - G(1,1')\,G(2,2')\]

where $G_2$ is the full two-particle Green's function and $G$ is the one-particle Green's function. $L$ describes the correlated propagation of two particles (or a particle-hole pair) beyond what is already captured by independent propagation.

1.2 The Exact BSE

The BSE is a Dyson-like equation for $L$:

\[L = L_0 + L_0\, \Xi\, L\]

where:

• \[L_0 = G\,G\]

  is the non-interacting two-particle propagator (a product of two dressed one-particle Green's functions)

• \[\Xi\]

  is the irreducible particle-hole interaction kernel — the sum of all two-particle irreducible diagrams in the particle-hole channel

This equation is exact: no approximation has been made. The approximation enters through the choice of $G$ and $\Xi$.

1.3 What Makes BSE Exact (and Hard)

The exact BSE requires:

1. Fully dressed propagators: $G$ should be the exact interacting single-particle Green's function, including all self-energy corrections $\Sigma$ to infinite order.

2. Irreducible kernel $\Xi$: must include all diagrams that are two-particle irreducible in the particle-hole channel — not just the bare interaction, but also vertex corrections, screening, and higher-order processes.

3. Frequency dependence: both $G$ and $\Xi$ are frequency-dependent, making the BSE an integral equation in both momentum and frequency.

In practice, no one solves the exact BSE. Instead, different levels of approximation to $G$ and $\Xi$ define a hierarchy of methods.

2. Hierarchy of Approximations

2.1 Level 0: Hartree-Fock + Bare Interaction

Replace:

• \[G \to G_{\text{HF}}\]

  : the Hartree-Fock Green's function (single-particle energies and wavefunctions from SCF)

• \[\Xi \to V_{\text{bare}}\]

  : the bare (unscreened) Coulomb or Hubbard interaction

This is the level implemented in MeanFieldTheories.jl. It gives the standard RPA and TDA of condensed matter and nuclear physics.

2.2 Level 1: GW + Screened Interaction

Replace:

• \[G \to G_{\text{GW}}\]

  : quasiparticle Green's function with self-energy $\Sigma = iGW$ (the GW approximation)

• \[\Xi \to W\]

  : the screened Coulomb interaction (within RPA screening)

This is the standard GW-BSE approach widely used in computational materials science for excitonic spectra and optical absorption. The screening of the interaction and the self-energy correction to the band gap are both crucial for quantitative accuracy in real materials.

2.3 Level 2: Beyond GW

Include vertex corrections in both $\Sigma$ and $\Xi$:

• \[\Sigma\]

  includes vertex corrections beyond GW (e.g., the GWΓ approximation)

• \[\Xi\]

  includes diagrams beyond the screened interaction (e.g., second-order exchange, T-matrix contributions)

This level is largely a frontier of current research and is rarely attempted in practice.

2.4 Summary Table

Going down the table trades accuracy for computational simplicity. MeanFieldTheories.jl operates at Level 0, which is appropriate for model Hamiltonians (Hubbard, Heisenberg-like) where the interaction is already short-ranged and the primary interest is in qualitative collective mode structure (magnon dispersions, Goldstone modes, excitation gaps).

3. TDA and RPA: Two Approximations Within the BSE

Once the propagator and kernel are fixed (e.g., at Level 0), there is still a choice of which excitation channels to include. This gives two further levels of approximation.

3.1 Tamm-Dancoff Approximation (TDA)

The TDA restricts the excitation operator to forward (particle-hole) processes only:

\[\hat{O}^\dagger_{\mu\mathbf{q}} = \sum_{\mathbf{k}, n_0, n} \psi^{n_0 n}_\mathbf{k}\, f^\dagger_{\mathbf{k}+\mathbf{q}, n}\, f_{\mathbf{k}, n_0}\]

where $n_0 \in$ occ and $n \in$ unocc. The variational principle leads to a Hermitian eigenvalue problem:

\[\mathcal{A}(\mathbf{q})\, \boldsymbol{\psi}_\mu = \varepsilon_\mu\, \boldsymbol{\psi}_\mu\]

where the $\mathcal{A}$ matrix contains:

• Diagonal: mean-field particle-hole energy $E^n_{\mathbf{k}+\mathbf{q}} - E^{n_0}_\mathbf{k}$
• Off-diagonal: residual interaction (exchange minus direct kernels)

Properties:

• Hermitian $\Rightarrow$ real eigenvalues, orthogonal eigenvectors
• All eigenvalues are non-negative (for a stable HF ground state)
• Simple and numerically robust

Limitation: ignores ground-state correlations (quantum fluctuations). The HF ground state $|G\rangle$ is treated as the exact vacuum with no zero-point particle-hole pairs.

3.2 Random Phase Approximation (RPA)

The RPA extends the excitation operator to include both forward and backward processes:

\[\hat{O}^\dagger_{\mu\mathbf{q}} = \sum_{\mathbf{k}, n_0, n} \left[ X^{n_0 n}_\mathbf{k}\, f^\dagger_{\mathbf{k}+\mathbf{q}, n}\, f_{\mathbf{k}, n_0} - Y^{n_0 n}_\mathbf{k}\, f^\dagger_{\mathbf{k}, n_0}\, f_{\mathbf{k}-\mathbf{q}, n} \right]\]

The backward term $Y$ allows the excitation to annihilate virtual particle-hole pairs already present in the correlated ground state. This leads to the Bosonic BdG eigenvalue problem:

\[\begin{pmatrix} \mathcal{A}(\mathbf{q}) & \mathcal{B}(\mathbf{q}) \\ -\mathcal{B}(-\mathbf{q})^* & -\mathcal{A}(-\mathbf{q})^* \end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix} = \varepsilon \begin{pmatrix} X \\ Y \end{pmatrix}\]

where:

• \[\mathcal{A}(\mathbf{q})\]

  : same TDA matrix (forward-forward coupling)

• \[\mathcal{B}(\mathbf{q})\]

  : forward-backward coupling (ground-state correlations)
• The lower-left block $-\mathcal{B}(-\mathbf{q})^*$ and lower-right block $-\mathcal{A}(-\mathbf{q})^*$ follow from the symmetry relations $\mathcal{C}(\mathbf{q}) = -\mathcal{B}(-\mathbf{q})^*$ and $\mathcal{D}(\mathbf{q}) = -\mathcal{A}(-\mathbf{q})^*$, which hold for any Hermitian Hamiltonian

Properties:

• Non-Hermitian, but eigenvalues come in $\pm\varepsilon$ pairs
• Guarantees Goldstone modes: for spontaneously broken continuous symmetries, the RPA spectrum is exactly gapless at the ordering wavevector. TDA generically gives a small spurious gap.
• Symplectic normalization: $X^\dagger X - Y^\dagger Y = I$

When RPA matters:

• Antiferromagnets and other symmetry-broken states (Goldstone theorem)
• Systems where $\mathcal{B}$ is non-negligible (strong ground-state correlations)
• When exact sum rules or conservation laws must be satisfied

3.3 Comparison

3.4 Relation to the Single-Mode Approximation

As a historical note, Feynman's single-mode approximation (SMA) for superfluid $^4\text{He}$ (1954) can be viewed as a special case of TDA where the envelope function $\psi^n_\mathbf{k}$ is frozen to a predetermined form (the density operator $\rho_\mathbf{q}$), leaving no variational freedom. The SMA gives one energy per wavevector via the celebrated formula $\varepsilon(\mathbf{q}) = f(\mathbf{q}) / S(\mathbf{q})$. The TDA/BSE generalizes this by variationally optimizing $\psi$, yielding a complete multi-branch excitation spectrum.

4. Implementation in MeanFieldTheories.jl

The function solve_ph_excitations implements both TDA and RPA at the HF + bare interaction level (Level 0):

• solver = :TDA: diagonalizes the Hermitian $\mathcal{A}$ matrix
• solver = :RPA: constructs the full Bosonic BdG matrix and solves via Cholesky decomposition (with fallback to direct diagonalization)

The detailed derivations of the $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{D}$ matrices are given in:

• Tamm-Dancoff Approximation: derivation of the $\mathcal{A}$ matrix
• Random Phase Approximation: derivation of the $\mathcal{B}$ and $\mathcal{D}$ matrices, symplectic normalization, and the Bosonic BdG eigenvalue problem

References

[1] E. E. Salpeter and H. A. Bethe, A Relativistic Equation for Bound-State Problems, Phys. Rev. 84, 1232 (1951).

[2] D. J. Rowe, Methods for Calculating Ground-State Correlations of Vibrational Nuclei, Phys. Rev. 175, 1283 (1968).

[3] G. Onida, L. Reining, and A. Rubio, Electronic excitations: density-functional versus many-body Green's-function approaches, Rev. Mod. Phys. 74, 601 (2002).