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 Prologue Prologue Diagrammatic(s) Rules Diagrammatic(s) Rules Straight-forward Eikonal Straight-forward Eikonal Legacy Bosonization Legacy Bosonization Wonton Holography Wonton Holography Holographic Propagators Holographic Propagators Strange Cuprates Strange Cuprates Stranger Things Stranger Things Epilogue 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 over the bosonic fluctuations with the interaction function D(ω, k) = 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]. Author: (1) D. V. Khveshchenko, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599. Author: Author: (1) D. V. Khveshchenko, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv available on arxiv