Small-N Limit: Eikonal vs. Maximally Crossed Diagrams

Table of Links

• Prologue
• Diagrammatic(s) Rules
• Straight-forward Eikonal
• Legacy Bosonization
• Wonton Holography
• Holographic Propagators
• Strange Cuprates
• Stranger Things
• Epilogue

Straight-forward Eikonal

Another approach to the problem of fermions with the interactions (1) was proposed in the small-N limit where there is no fermion back-reaction as any fermionic loops are suppressed. This limit was first addressed in the context of summing over the so-called ’maximally crossed’ diagrams [13]. However, shortly after its first applications it was realized that this method represents a simplified form of the general eikonal approach [14].

The most straightforward (in the geometrical sense as well) eikonal calculation focuses exclusively on the Debye-Waller ’damping factor’ picked up by the fermion’s wave function while the latter is moving down its 3 straight-line semiclassical path. Computing the average < exp(i R drA) > over the bosonic fluctuations with the interaction function D(ω, k) =< AA > given by (1) one obtains

where the exponential ’damping factor’ reads

First, neglecting the FS curvature one readily finds the lowest-order eikonal action

whereas for η = 1 the frequency integration produces an additional factor ∼ ln |r − iτ|.

For example, in d = 2 and for η = 2/3 (11) gives rise to the asymptotics

which were obtained in [7, 15–17], barring minor differences (see next section). The expressions (12,13) yield the ’off-FS’ Fourier transform

demonstrates a cusp or even a true singularity, depending on the anomalous exponent η2 (at large N one finds η2 ∼ 1/N, though) [14, 16, 18].