Irreducible Vertices and Bethe–Salpeter Equations

A Bethe–Salpeter equation reorganizes a four-point correlation function into repeated propagation joined by a kernel irreducible in a chosen two-particle channel. It converts microscopic interactions into collective enhancement, bound-state poles, or instabilities. The channel, leg ordering, and approximation to the kernel must be declared because there is no single channel-independent “irreducible vertex”; the condensed-matter channel decomposition is reviewed in Bickers 2004, §6.

Required background. Current Vertices and Ward-Consistent Response defines the functional kernel $\delta\Sigma/\delta G$, and Finite-Density Diagrammatics and Medium Insertions fixes diagrammatic signs and medium lines.

Helpful background. Bethe–Salpeter and Faddeev Bound-State Equations develops the general functional-equation setting.

Channel irreducibility

Let $L(12;34)$ be a connected two-particle propagator and $L_0(12;34)=-G(1,3)G(4,2)$ the product appropriate to a particle–hole convention. A diagram is particle–hole irreducible if it cannot be separated by cutting the two propagators that carry that channel’s total transfer. The Bethe–Salpeter equation is

$$L=L_0+L_0KL,$$

where products imply integration and internal-index sums. Formally,

$$L=(1-L_0K)^{-1}L_0.$$

In a response derived by differentiating Dyson’s equation, $K=\delta\Sigma/\delta G$ with index ordering determined by the chosen definitions. In a particle–particle channel the two propagators and exchange signs are reordered, and the corresponding irreducibility notion changes.

A separable example

Suppose a scalar channel has a momentum-independent kernel $g$ and a projected bubble $\chi_0(q)$. Then

$$\chi(q)=\chi_0(q)+\chi_0(q)g\chi(q) =\frac{\chi_0(q)}{1-g\chi_0(q)}.$$

A zero of $1-g\chi_0$ is a pole of the projected response if the numerator and analytic continuation are regular there. In the particle–particle vacuum problem it can describe a bound state; in a medium it can signal a pairing instability. In a particle–hole channel it can describe a density or magnetic collective mode.

The same algebra appears in RPA, but the physical content depends on the kernel and channel. Calling every geometric response “RPA” obscures whether exchange, ladder, or screening processes are being repeated.

Eigenvalue formulation

After discretizing internal momentum, frequency, and indices, write

$$\sum_j M_{ij}(q)\,\varphi_j=\lambda(q)\varphi_i, \qquad M=L_0K.$$

The resolvent becomes large as an eigenvalue approaches one. This is a useful diagnostic, but $\lambda=1$ on a finite grid is not by itself proof of a thermodynamic phase transition. One must demonstrate resolution and cutoff convergence, impose the correct symmetry and analytic continuation, and distinguish a finite-system pole from a divergent susceptibility in the thermodynamic limit.

Left and right eigenvectors differ for a non-Hermitian real-frequency kernel. Their overlap fixes residues and sensitivity near defective points; inspecting only the largest absolute eigenvalue can miss the physically relevant mode.

Crossing and double counting

A kernel irreducible in one channel contains diagrams reducible in another. Summing several channel equations independently can therefore double count common diagrams. Parquet equations resolve this by decomposing the full vertex into a fully irreducible part plus contributions reducible in each inequivalent channel. Simpler ladder calculations remain useful, but their omitted and repeated diagram classes should be explicit.

Crossing symmetry is a separate constraint from conservation. A self-consistent channel approximation may conserve number yet violate crossing relations among four-point functions.

Numerical validity checks

For a computed Bethe–Salpeter solution, vary the internal frequency range, momentum cutoff, quadrature, and basis size. Check exchange symmetries, the noninteracting limit, relevant Ward identities, and sum rules. Near a pole, report both its position and residue; an apparent divergence caused by an ill-conditioned discretization is not a physical collective mode.

Common pitfalls

Leaving the channel unnamed. Particle–hole and particle–particle irreducibility are different definitions and resum different processes.

Reading an eigenvalue crossing without convergence. Kernel eigenvalues depend on grid, cutoff, and normalization. Show stability before interpreting unity.

Combining ladders by addition. Independently resummed channels share diagrams and can double count them.

Exercises

Solve a rank-one kernel

Let $K_{ij}=g\,u_i u_j$ and $L_{0,ij}=\ell_i\delta_{ij}$. Find the condition for a pole.

Solution

$L_0K$ has one nonzero eigenvalue $\lambda=g\sum_i\ell_i u_i^2$. The inverse $(1-L_0K)^{-1}$ is singular when $1-g\sum_i\ell_i u_i^2=0$. The residue projects onto the associated right and left rank-one vectors.

Distinguish two criteria

Why can a large particle–particle ladder response occur without an established finite-temperature ordered phase in two dimensions?

Solution

The ladder measures strong pair correlations in its selected channel. Long-range order additionally depends on phase fluctuations and the thermodynamic limit; in two dimensions continuous symmetry forbids ordinary finite-temperature long-range order, while a Berezinskii–Kosterlitz–Thouless transition requires a stiffness analysis not contained in the bare pairing criterion.

Continue

Hedin Equations, Screened Interactions, and GW embeds a vertex equation in a closed one- and two-particle hierarchy. Conserving Approximations and Φ-Derivable Functionals derives compatible kernels. Cooper Instability and Pairing applies the particle–particle channel.

References

• Bickers, N. E. “Self-Consistent Many-Body Theory for Condensed Matter Systems.” In Theoretical Methods for Strongly Correlated Electrons, edited by David Sénéchal, André-Marie Tremblay, and Claude Bourbonnais, 237–296. New York: Springer, 2004. DOI.

Further reading

• Kadanoff, Leo P., and Gordon Baym. Quantum Statistical Mechanics: Green’s Function Methods in Equilibrium and Nonequilibrium Problems. New York: W. A. Benjamin, 1962. Publisher record.
• Salpeter, Edwin E., and Hans A. Bethe. “A Relativistic Equation for Bound-State Problems.” Physical Review 84 (1951): 1232–1242. DOI.