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arXiv:cond-mat/9701073

Perturbation Theory around Non-Nested Fermi Surfaces II. Regularity of the Moving Fermi Surface: RPA contributions

Joel Feldman, Manfred Salmhofer B, Eugene Trubowitz B

Published Jan 10, 1997• 4 citations
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Abstract

Regularity of the deformation of the Fermi surface under short-range interactions is established for all contributions to the RPA self-energy (it is proven in an accompanying paper that the RPA graphs are the least regular contributions to the self-energy). Roughly speaking, the graphs contributing to the RPA self-energy are those constructed by contracting two external legs of a four-legged graph that consists of a string of bubbles. This regularity is a necessary ingredient in the proof that renormalization does not change the model. It turns out that the self--energy is more regular when derivatives are taken tangentially to the Fermi surface than when they are taken normal to the Fermi surface. The proofs require a very detailed analysis of the singularities that occur at those momenta p where the Fermi surface S is tangent to S+p. Models in which S is not symmetric under the reflection p to -p are included.

DOI
10.48550/arxiv.cond-mat/9701073

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Regularity of the moving Fermi surface: RPA contributions

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Regularity of the deformation of the Fermi surface under short-range interactions is established for all contributions to the RPA self-energy (it is proven in an accompanying paper that the RPA graphs are the least regular contributions to the self-energy). Roughly speaking, the graphs contributing to the RPA self-energy are those constructed by contracting two external legs of a four-legged graph that consists of a string of bubbles. This regularity is a necessary ingredient in the proof that renormalization does not change the model. It turns out that the self--energy is more regular when derivatives are taken tangentially to the Fermi surface than when they are taken normal to the Fermi surface. The proofs require a very detailed analysis of the singularities that occur at those momenta p where the Fermi surface S is tangent to S+p. Models in which S is not symmetric under the reflection p to -p are included.
Regularity of the deformation of the Fermi surface under short-range interactions is established for all contributions to the RPA self-energy (it is proven in an accompanying paper that the RPA graphs are the least regular contributions to the self-energy). Roughly speaking, the graphs contributing to the RPA self-energy are those constructed by contracting two external legs of a four-legged graph that consists of a string of bubbles. This regularity is a necessary ingredient in the proof that renormalization does not change the model. It turns out that the self-energy is more regular when derivatives are taken tangentially to the Fermi surface than when they are taken normal to the Fermi surface. The proofs require a very detailed analysis of the singularities that occur at those momenta p where the Fermi surface S is tangent to S + p. Models in which S is not symmetric under the reflection p → − p are included. © 1998 John Wiley & Sons, Inc.
Regularity of the deformation of the Fermi surface under short-range interactions is established for all contributions to the RPA self-energy (it is proven in an accompanying paper that the RPA graphs are the least regular contributions to the self-energy). Roughly speaking, the graphs contributing to the RPA self-energy are those constructed by contracting two external legs of a four-legged graph that consists of a string of bubbles. This regularity is a necessary ingredient in the proof that renormalization does not change the model. It turns out that the self-energy is more regular when derivatives are taken tangentially to the Fermi surface than when they are taken normal to the Fermi surface. The proofs require a very detailed analysis of the singularities that occur at those momenta p where the Fermi surface S is tangent to S + p. Models in which S is not symmetric under the reflection p → − p are included. © 1998 John Wiley & Sons, Inc.
Regularity of the deformation of the Fermi surface under short-range interactions is established to all orders in perturbation theory. The proofs are based on a new classification of all graphs that are not doubly overlapping. They turn out to be generalized RPA graphs. This fact provides a simple extension to all orders of the regularity theorem of the Fermi surface movement proven in [2]. Models in which S is not symmetric under the reflection p → −;p are included. © 1999 John Wiley & Sons, Inc.
The naive perturbation expansion for many-fermion systems is infrared divergent. One can remove these divergences by introducing counterterms. To do this without changing the model, one has to solve an inversion equation. We call this procedure Fermi surface renormalization (FSR). Whether or not FSR is possible depends on the regularity properties of the fermion self--energy. When the Fermi surface is nonspherical, this regularity problem is rather nontrivial. Using improved power counting at all orders in perturbation theory, we have shown sufficient differentiability to solve the FSR equation for a class of models with a non-nested, non-spherical Fermi surface. I will first motivate the problem and give a definition of FSR, and then describe the combination of geometric and graphical facts that lead to the improved power counting bounds. These bounds also apply to the four--point function. They imply that only ladder diagrams can give singular contributions to the four--point function.
We prove regularity properties of the self-energy, to all orders in perturbation theory, for systems with singular Fermi surfaces which contain Van Hove points where the gradient of the dispersion relation vanishes. In this paper, we show for spatial dimensions d ≥ 3 that despite the Van Hove singularity, the overlapping loop bounds we proved together with E. Trubowitz for regular non-nested Fermi surfaces [J. Stat. Phys.84 (1996) 1209–1336] still hold, provided that the Fermi surface satisfies a no-nesting condition. This implies that for a fixed interacting Fermi surface, the self-energy is a continuously differentiable function of frequency and momentum, so that the quasiparticle weight and the Fermi velocity remain close to their values in the noninteracting system to all orders in perturbation theory. In a companion paper, we treat the more singular two-dimensional case.